This document provides an introduction to biostatistics. It defines biostatistics as the application of statistical methods to biological and health sciences. Biostatistics is divided into descriptive and inferential biostatistics. Descriptive biostatistics summarizes and analyzes data, while inferential biostatistics draws conclusions about populations from samples. The document also defines key biostatistics terms and concepts, including populations, samples, parameters, statistics, and different types of data and variables.
Statistics is the science of collecting, organizing, analyzing, and drawing conclusions from data. Biostatistics applies statistical methods to biological topics like public health, clinical trials, genetics, and ecology. Descriptive statistics summarizes and presents data, while inferential statistics allows generalizing from samples to populations through hypothesis testing, determining relationships among variables, and making predictions. Key concepts include data, variables, populations, samples, measurement scales, and sampling methods. Common graphs for presenting data include histograms, bar charts, line graphs, and pie charts.
The document discusses hypothesis testing and statistical methods. It defines key terms like the null hypothesis, alternative hypothesis, dependent and independent variables. It then outlines the steps in hypothesis testing which include stating the hypotheses, selecting a test statistic, determining confidence levels, computing and plotting the test statistic to make conclusions. It discusses using p-values and provides an example comparing blood pressure before and after exercise. It also explains how to compare means for one and two independent groups as well as two dependent groups using t-tests.
This document provides an overview of Epi Info, statistical software developed by the CDC for epidemiology. It discusses the software's history, beginning as an MS-DOS program in 1985. Modern versions are available for Windows, Android, iOS, and as web/cloud apps. The software allows users to create electronic surveys, enter data, and perform analyses including t-tests, regression, and analyses of complex survey data. It has been open source since 2008.
This document provides an introduction to biostatistics. It defines key concepts such as statistics, data, variables, populations, and samples. It discusses different types of variables including quantitative and qualitative variables. It also describes different measurement scales including nominal, ordinal, interval and ratio scales. Sources of data and descriptive statistics are introduced. Descriptive statistics help summarize and organize data using tables, graphs, and numerical measures.
This document discusses the role of biostatistics in public health and research. It covers topics such as the definition of statistics, the collection and analysis of data, basic statistical concepts like variables and populations, different measurement scales, sampling methods, and statistical inference. Biostatistics is involved in formulating scientific questions, designing experiments, collecting and screening data, analyzing and interpreting results, and presenting findings. It also discusses the importance of probability and non-probability sampling techniques as well as using computer software programs for statistical analysis in health sciences research.
This document discusses determining the appropriate size of a sample from a population. It explains that a sample should be small enough to be practical but large enough to accurately represent the population within a desired level of precision and confidence level. The key factors in determining sample size are the homogeneity of the population, number of classes or groups in the population, nature of the study, type of sampling method, desired accuracy and confidence level, availability of resources, and other considerations like the nature and size of units, size of the population, and time constraints. The document concludes by mentioning an approach to determining sample size based on precision rate and confidence level.
This document provides an overview of key concepts in biostatistics. It begins with introductions to terminology, sources and presentation of data, and measures of central tendency and dispersion. It then discusses the normal curve, sampling techniques, and types of tests of significance including t-tests, ANOVA, and non-parametric tests. The document provides examples and explanations of commonly used statistical analyses for comparing means and assessing relationships in data.
Statistics is the science of collecting, organizing, analyzing, and drawing conclusions from data. Biostatistics applies statistical methods to biological topics like public health, clinical trials, genetics, and ecology. Descriptive statistics summarizes and presents data, while inferential statistics allows generalizing from samples to populations through hypothesis testing, determining relationships among variables, and making predictions. Key concepts include data, variables, populations, samples, measurement scales, and sampling methods. Common graphs for presenting data include histograms, bar charts, line graphs, and pie charts.
The document discusses hypothesis testing and statistical methods. It defines key terms like the null hypothesis, alternative hypothesis, dependent and independent variables. It then outlines the steps in hypothesis testing which include stating the hypotheses, selecting a test statistic, determining confidence levels, computing and plotting the test statistic to make conclusions. It discusses using p-values and provides an example comparing blood pressure before and after exercise. It also explains how to compare means for one and two independent groups as well as two dependent groups using t-tests.
This document provides an overview of Epi Info, statistical software developed by the CDC for epidemiology. It discusses the software's history, beginning as an MS-DOS program in 1985. Modern versions are available for Windows, Android, iOS, and as web/cloud apps. The software allows users to create electronic surveys, enter data, and perform analyses including t-tests, regression, and analyses of complex survey data. It has been open source since 2008.
This document provides an introduction to biostatistics. It defines key concepts such as statistics, data, variables, populations, and samples. It discusses different types of variables including quantitative and qualitative variables. It also describes different measurement scales including nominal, ordinal, interval and ratio scales. Sources of data and descriptive statistics are introduced. Descriptive statistics help summarize and organize data using tables, graphs, and numerical measures.
This document discusses the role of biostatistics in public health and research. It covers topics such as the definition of statistics, the collection and analysis of data, basic statistical concepts like variables and populations, different measurement scales, sampling methods, and statistical inference. Biostatistics is involved in formulating scientific questions, designing experiments, collecting and screening data, analyzing and interpreting results, and presenting findings. It also discusses the importance of probability and non-probability sampling techniques as well as using computer software programs for statistical analysis in health sciences research.
This document discusses determining the appropriate size of a sample from a population. It explains that a sample should be small enough to be practical but large enough to accurately represent the population within a desired level of precision and confidence level. The key factors in determining sample size are the homogeneity of the population, number of classes or groups in the population, nature of the study, type of sampling method, desired accuracy and confidence level, availability of resources, and other considerations like the nature and size of units, size of the population, and time constraints. The document concludes by mentioning an approach to determining sample size based on precision rate and confidence level.
This document provides an overview of key concepts in biostatistics. It begins with introductions to terminology, sources and presentation of data, and measures of central tendency and dispersion. It then discusses the normal curve, sampling techniques, and types of tests of significance including t-tests, ANOVA, and non-parametric tests. The document provides examples and explanations of commonly used statistical analyses for comparing means and assessing relationships in data.
Epidemiological statistics and study designs were discussed. The key points are:
1. Epidemiology deals with disease patterns in populations and epidemiological statistics uses sampling and statistical methods for research.
2. The stages of epidemiological investigations are diagnostic, descriptive, investigative, experimental, analytical, intervention, decision-making, and monitoring phases.
3. Major types of epidemiological studies include descriptive epidemiology, observational studies like cohort, case-control, and cross-sectional studies, and experimental randomized controlled trials. Each type has advantages and disadvantages for investigating different research questions.
The document discusses key concepts in public health methodologies and biostatistics. It defines data as facts that can be processed by computers. Statistics is described as the study of collecting, summarizing, analyzing and interpreting data. Biostatistics applies statistical techniques to health-related fields like medicine. Descriptive statistics refers to methods used to describe data, while inferential statistics are used to draw conclusions from numeric data. Variables, grouped vs. ungrouped data, and types of variables are also outlined.
This document provides an introduction to biostatistics, including definitions, types of data, scales of measurement, and methods of data collection. It defines biostatistics as statistics applied to biological and health problems. Descriptive statistics are used to collect, organize, summarize and present data, while inferential statistics generalize from samples to populations using probabilities. Data can be qualitative, involving categories without numerical values, or quantitative, having numerical values. Methods of collecting data include observation, interviews, questionnaires, and reviewing documentation. The appropriate method depends on the type of data needed and available resources.
Clinical Research Statistics for Non-StatisticiansBrook White, PMP
Through real-world examples, this presentation teaches strategies for choosing appropriate outcome measures, methods for analysis and randomization, and sample sizes as well as tips for collecting the right data to answer your scientific questions.
Topic: Types of Data
Student Name: Duwa
Class: B.Ed. 2.5
Project Name: “Young Teachers' Professional Development (TPD)"
"Project Founder: Prof. Dr. Amjad Ali Arain
Faculty of Education, University of Sindh, Pakistan
This document provides an introduction to statistics and biostatistics in healthcare. It defines statistics and biostatistics, outlines the basic steps of statistical work, and describes different types of variables and methods for collecting data. The document also discusses different types of descriptive and inferential statistics, including measures of central tendency, dispersion, frequency, t-tests, ANOVA, regression, and different types of plots/graphs. It explains how statistics is used in healthcare for areas like disease burden assessment, intervention effectiveness, cost considerations, evaluation frameworks, health care utilization, resource allocation, needs assessment, quality improvement, and product development.
This document provides an overview of biostatistics and data analysis. It defines biostatistics as the application of statistics in health sciences and biology. The fundamental tools of the scientific method like hypothesis formulation, experimental design, data gathering and analysis are discussed. Descriptive statistics, which summarize and describe data through numerical, graphical and mathematical presentations are covered. Common descriptive statistics like mean, median, mode, standard deviation and distribution curves are defined. Inferential statistics which allow generalization from samples to populations through hypothesis testing and significance levels are also introduced.
This document provides an introduction to biostatistics and key concepts. It defines biostatistics as the development and application of statistical techniques to scientific research relating to human life and health. Some key terms discussed include:
- Population, which is the totality of individuals of interest
- Sample, which is a subset of a population
- Variables, which can be qualitative (non-numerical) or quantitative (numerical)
- Levels of measurement for variables, including nominal, ordinal, interval, and ratio scales
- Descriptive methods for qualitative data, including frequency distributions
Biostatistics plays an important role in modern medicine, including determining disease burden, finding new drug treatments, planning resource allocation, and measuring
This document provides an introduction to biostatistics-II, covering various topics:
- Types of variables including qualitative vs. quantitative, and measurement scales such as nominal, ordinal, interval, and ratio.
- Sources of data collection including statistical sources like censuses and surveys, as well as non-statistical sources.
- Key terms are defined, such as variable, qualitative and quantitative variables, and examples are provided.
This document provides an overview of an introductory biostatistics course. The course covers topics such as descriptive statistics, probability, sampling methods, and probability distributions. Lecture 1 introduces biostatistics and discusses its importance in fields like public health and medicine. Biostatistics is applied to analyze biological and health data and help address questions like disease trends, at-risk populations, and health standards. It aids decision-making under uncertainty and helps identify health issues, evaluate programs, and conduct research.
This document provides objectives and details about a biostatistics course. The general objective is to present numerical data in a way that allows for valid interpretations. Specific objectives include designing, organizing, and summarizing data, as well as understanding probabilities, statistical analyses, and how to make inferences about populations based on sample data. The course will cover topics like data collection and presentation, probability distributions, statistical inference through estimation and hypothesis testing, and determining sample sizes. Students will be evaluated based on exercises and a final exam. Computer software will be used to analyze data.
O Biostatistics is the application of statistics to biological and medical data. It plays an integral role in modern medicine by analyzing data to determine treatment efficacy and develop clinical trials. A landmark study in biostatistics was the Framingham Heart Study, which through longitudinal data collection and analysis identified major risk factors for cardiovascular disease and influenced our current understanding of heart disease as a leading cause of death. Biostatistics obtains, analyzes, and interprets quantitative medical data to further human health.
PPT on Sample Size, Importance of Sample Size,Naveen K L
This document discusses factors related to determining sample size for research studies. It defines key terms like sample size, population and importance of sample size. The selection of sample size involves planning the study, specifying parameters, choosing an effect size, and computing the sample size based on those factors. Sample size is influenced by expected effect size, study power, heterogeneity, error risk, and other variables. Dropouts from the sample during a study also impact sample size calculations. Proper determination of sample size is important for obtaining meaningful results and conducting ethical research.
This document discusses statistical analysis using SPSS. It describes descriptive statistics, which present data in a usable form by describing frequency, central tendency, and dispersion. Inferential statistics make broader generalizations from samples to populations using hypothesis testing. Hypothesis testing involves research hypotheses, null hypotheses, levels of significance, and type I and II errors. Choosing an appropriate statistical test depends on the hypothesis and measurement levels of the variables. SPSS is a comprehensive system for statistical analysis that can analyze many file types and generate reports and statistics.
The document provides information about biostatistics and statistical methodology. It begins with definitions of statistics and biostatistics. It then discusses topics like sampling, types of sampling techniques, measures of central tendency, measures of dispersion, and tests of significance. Specifically, it covers [1] the differences between probability and non-probability sampling, [2] common measures of central tendency like mean, median and mode, [3] measures of dispersion like range, mean deviation and standard deviation, and [4] tests of significance like the standard error test and chi-square test.
A sample design is a definite plan for obtaining a sample from a given population. Researcher must select/prepare a sample design which should be reliable and appropriate for his research study.
This document provides an introduction and overview of non-parametric statistical methods, including ranks and the median, Wilcoxon signed rank test, Mann-Whitney test, and Spearman's rank correlation coefficient. It defines what non-parametric tests are, discusses their advantages over parametric tests in situations where data is not normally distributed, and provides examples of calculating and interpreting several non-parametric tests in SPSS.
This document provides an introduction to biostatistics. It outlines several key objectives of a biostatistics course including understanding descriptive statistics, statistical inference, common tests and their assumptions. It defines important statistical concepts like population, sample, parameters, statistics, variables, and types of statistical analysis. Descriptive statistics are used to summarize data, while inferential statistics allow generalizing from samples to populations. Examples of potential statistical abuses are also provided.
This document discusses factors to consider when determining sample size for statistical studies. It notes that sample size is usually based on the study's objective and should be stated in the study protocol. Key factors in determining sample size include estimates of population standard deviation, acceptable sampling error levels, and desired confidence levels. Several methods are described for calculating sample size, including traditional statistical models and Bayesian models. The document also discusses concepts like sampling distributions of means and proportions, and factors that affect sample size calculations for estimating proportions, such as specifying acceptable error levels, confidence levels, and population proportion estimates.
Observational Studies and their Reporting Guidelineskopalsharma85
Observational studies observe individuals and outcomes without influencing them. There are several types including case reports, ecological studies, cross-sectional studies, case-control studies, and cohort studies. Guidelines like STROBE were created to improve reporting transparency. Upcoming developments include using clinical registries as cohort studies and propensity score matching to assemble comparable groups. Observational studies provide important descriptive data and can study long-term outcomes unlike clinical trials.
This document provides an introduction to basic biostatistics. It defines biostatistics as the application of statistical methods to biological and health sciences. Biostatistics has two main branches - descriptive biostatistics which summarizes data, and inferential biostatistics which draws conclusions from samples. The document discusses different types of data, variables, and measurement scales including nominal, ordinal, interval and ratio scales. It also covers topics like population and sample, parameters and statistics, and the stages of a statistical investigation.
This document provides an introduction to biostatistics for health science students at Debre Tabor University in Ethiopia. It defines biostatistics as the application of statistical methods to medical and public health problems. The introduction outlines topics that will be covered, including defining key statistical concepts, classifying variables, and discussing the importance and limitations of biostatistics. Contact information is provided for the lecturer, Asaye Alamneh.
Epidemiological statistics and study designs were discussed. The key points are:
1. Epidemiology deals with disease patterns in populations and epidemiological statistics uses sampling and statistical methods for research.
2. The stages of epidemiological investigations are diagnostic, descriptive, investigative, experimental, analytical, intervention, decision-making, and monitoring phases.
3. Major types of epidemiological studies include descriptive epidemiology, observational studies like cohort, case-control, and cross-sectional studies, and experimental randomized controlled trials. Each type has advantages and disadvantages for investigating different research questions.
The document discusses key concepts in public health methodologies and biostatistics. It defines data as facts that can be processed by computers. Statistics is described as the study of collecting, summarizing, analyzing and interpreting data. Biostatistics applies statistical techniques to health-related fields like medicine. Descriptive statistics refers to methods used to describe data, while inferential statistics are used to draw conclusions from numeric data. Variables, grouped vs. ungrouped data, and types of variables are also outlined.
This document provides an introduction to biostatistics, including definitions, types of data, scales of measurement, and methods of data collection. It defines biostatistics as statistics applied to biological and health problems. Descriptive statistics are used to collect, organize, summarize and present data, while inferential statistics generalize from samples to populations using probabilities. Data can be qualitative, involving categories without numerical values, or quantitative, having numerical values. Methods of collecting data include observation, interviews, questionnaires, and reviewing documentation. The appropriate method depends on the type of data needed and available resources.
Clinical Research Statistics for Non-StatisticiansBrook White, PMP
Through real-world examples, this presentation teaches strategies for choosing appropriate outcome measures, methods for analysis and randomization, and sample sizes as well as tips for collecting the right data to answer your scientific questions.
Topic: Types of Data
Student Name: Duwa
Class: B.Ed. 2.5
Project Name: “Young Teachers' Professional Development (TPD)"
"Project Founder: Prof. Dr. Amjad Ali Arain
Faculty of Education, University of Sindh, Pakistan
This document provides an introduction to statistics and biostatistics in healthcare. It defines statistics and biostatistics, outlines the basic steps of statistical work, and describes different types of variables and methods for collecting data. The document also discusses different types of descriptive and inferential statistics, including measures of central tendency, dispersion, frequency, t-tests, ANOVA, regression, and different types of plots/graphs. It explains how statistics is used in healthcare for areas like disease burden assessment, intervention effectiveness, cost considerations, evaluation frameworks, health care utilization, resource allocation, needs assessment, quality improvement, and product development.
This document provides an overview of biostatistics and data analysis. It defines biostatistics as the application of statistics in health sciences and biology. The fundamental tools of the scientific method like hypothesis formulation, experimental design, data gathering and analysis are discussed. Descriptive statistics, which summarize and describe data through numerical, graphical and mathematical presentations are covered. Common descriptive statistics like mean, median, mode, standard deviation and distribution curves are defined. Inferential statistics which allow generalization from samples to populations through hypothesis testing and significance levels are also introduced.
This document provides an introduction to biostatistics and key concepts. It defines biostatistics as the development and application of statistical techniques to scientific research relating to human life and health. Some key terms discussed include:
- Population, which is the totality of individuals of interest
- Sample, which is a subset of a population
- Variables, which can be qualitative (non-numerical) or quantitative (numerical)
- Levels of measurement for variables, including nominal, ordinal, interval, and ratio scales
- Descriptive methods for qualitative data, including frequency distributions
Biostatistics plays an important role in modern medicine, including determining disease burden, finding new drug treatments, planning resource allocation, and measuring
This document provides an introduction to biostatistics-II, covering various topics:
- Types of variables including qualitative vs. quantitative, and measurement scales such as nominal, ordinal, interval, and ratio.
- Sources of data collection including statistical sources like censuses and surveys, as well as non-statistical sources.
- Key terms are defined, such as variable, qualitative and quantitative variables, and examples are provided.
This document provides an overview of an introductory biostatistics course. The course covers topics such as descriptive statistics, probability, sampling methods, and probability distributions. Lecture 1 introduces biostatistics and discusses its importance in fields like public health and medicine. Biostatistics is applied to analyze biological and health data and help address questions like disease trends, at-risk populations, and health standards. It aids decision-making under uncertainty and helps identify health issues, evaluate programs, and conduct research.
This document provides objectives and details about a biostatistics course. The general objective is to present numerical data in a way that allows for valid interpretations. Specific objectives include designing, organizing, and summarizing data, as well as understanding probabilities, statistical analyses, and how to make inferences about populations based on sample data. The course will cover topics like data collection and presentation, probability distributions, statistical inference through estimation and hypothesis testing, and determining sample sizes. Students will be evaluated based on exercises and a final exam. Computer software will be used to analyze data.
O Biostatistics is the application of statistics to biological and medical data. It plays an integral role in modern medicine by analyzing data to determine treatment efficacy and develop clinical trials. A landmark study in biostatistics was the Framingham Heart Study, which through longitudinal data collection and analysis identified major risk factors for cardiovascular disease and influenced our current understanding of heart disease as a leading cause of death. Biostatistics obtains, analyzes, and interprets quantitative medical data to further human health.
PPT on Sample Size, Importance of Sample Size,Naveen K L
This document discusses factors related to determining sample size for research studies. It defines key terms like sample size, population and importance of sample size. The selection of sample size involves planning the study, specifying parameters, choosing an effect size, and computing the sample size based on those factors. Sample size is influenced by expected effect size, study power, heterogeneity, error risk, and other variables. Dropouts from the sample during a study also impact sample size calculations. Proper determination of sample size is important for obtaining meaningful results and conducting ethical research.
This document discusses statistical analysis using SPSS. It describes descriptive statistics, which present data in a usable form by describing frequency, central tendency, and dispersion. Inferential statistics make broader generalizations from samples to populations using hypothesis testing. Hypothesis testing involves research hypotheses, null hypotheses, levels of significance, and type I and II errors. Choosing an appropriate statistical test depends on the hypothesis and measurement levels of the variables. SPSS is a comprehensive system for statistical analysis that can analyze many file types and generate reports and statistics.
The document provides information about biostatistics and statistical methodology. It begins with definitions of statistics and biostatistics. It then discusses topics like sampling, types of sampling techniques, measures of central tendency, measures of dispersion, and tests of significance. Specifically, it covers [1] the differences between probability and non-probability sampling, [2] common measures of central tendency like mean, median and mode, [3] measures of dispersion like range, mean deviation and standard deviation, and [4] tests of significance like the standard error test and chi-square test.
A sample design is a definite plan for obtaining a sample from a given population. Researcher must select/prepare a sample design which should be reliable and appropriate for his research study.
This document provides an introduction and overview of non-parametric statistical methods, including ranks and the median, Wilcoxon signed rank test, Mann-Whitney test, and Spearman's rank correlation coefficient. It defines what non-parametric tests are, discusses their advantages over parametric tests in situations where data is not normally distributed, and provides examples of calculating and interpreting several non-parametric tests in SPSS.
This document provides an introduction to biostatistics. It outlines several key objectives of a biostatistics course including understanding descriptive statistics, statistical inference, common tests and their assumptions. It defines important statistical concepts like population, sample, parameters, statistics, variables, and types of statistical analysis. Descriptive statistics are used to summarize data, while inferential statistics allow generalizing from samples to populations. Examples of potential statistical abuses are also provided.
This document discusses factors to consider when determining sample size for statistical studies. It notes that sample size is usually based on the study's objective and should be stated in the study protocol. Key factors in determining sample size include estimates of population standard deviation, acceptable sampling error levels, and desired confidence levels. Several methods are described for calculating sample size, including traditional statistical models and Bayesian models. The document also discusses concepts like sampling distributions of means and proportions, and factors that affect sample size calculations for estimating proportions, such as specifying acceptable error levels, confidence levels, and population proportion estimates.
Observational Studies and their Reporting Guidelineskopalsharma85
Observational studies observe individuals and outcomes without influencing them. There are several types including case reports, ecological studies, cross-sectional studies, case-control studies, and cohort studies. Guidelines like STROBE were created to improve reporting transparency. Upcoming developments include using clinical registries as cohort studies and propensity score matching to assemble comparable groups. Observational studies provide important descriptive data and can study long-term outcomes unlike clinical trials.
This document provides an introduction to basic biostatistics. It defines biostatistics as the application of statistical methods to biological and health sciences. Biostatistics has two main branches - descriptive biostatistics which summarizes data, and inferential biostatistics which draws conclusions from samples. The document discusses different types of data, variables, and measurement scales including nominal, ordinal, interval and ratio scales. It also covers topics like population and sample, parameters and statistics, and the stages of a statistical investigation.
This document provides an introduction to biostatistics for health science students at Debre Tabor University in Ethiopia. It defines biostatistics as the application of statistical methods to medical and public health problems. The introduction outlines topics that will be covered, including defining key statistical concepts, classifying variables, and discussing the importance and limitations of biostatistics. Contact information is provided for the lecturer, Asaye Alamneh.
This document provides an overview of basic health statistics and survey methods for health extension workers. It covers key learning objectives like preparing for and undertaking data collection, compiling and interpreting health data, and preparing reports. Various health data types, variables, and measurement scales are defined. Methods for collecting, analyzing and using health statistics like rates, ratios, proportions are explained. Measures of health like fertility, mortality and morbidity are also described. The document aims to equip health extension workers with essential statistical knowledge and skills for their work.
This document provides an introduction to biostatistics. It defines biostatistics as the development and application of statistical techniques to scientific research relating to human, plant, and animal life, with a focus on human life and health. It discusses the collection, organization, presentation, analysis, and interpretation of numerical data, which are the key components of statistics. Finally, it describes different types and measurement scales of data.
This document provides an overview of basic statistical concepts including descriptive and inferential statistics, variables and levels of measurement, and methods of data collection and presentation. Descriptive statistics summarize and organize data, while inferential statistics make conclusions about a population based on a sample. There are various methods used to collect both primary and secondary data, including observation, surveys, and existing records. Data is typically presented through tables, diagrams, and graphs. Frequency distributions group and summarize data into classes to aid in analysis and interpretation.
This document provides an overview of basic statistical concepts including descriptive and inferential statistics, variables and levels of measurement, and methods of data collection and presentation. Descriptive statistics summarize and organize data, while inferential statistics make conclusions about a population based on a sample. There are various methods used to collect both primary and secondary data, including observation, surveys, and existing records. Data is typically presented through tables, diagrams, and graphs. Frequency distributions group and summarize data into classes to aid in analysis and interpretation.
1. The document discusses a lecture on biostatistics including topics like introduction to statistics, exploratory tools for univariate data, probabilities and distribution curves, and sampling distribution of estimates.
2. It provides examples of different types of data like qualitative vs quantitative and discrete vs continuous data. It also discusses different scales of measurement.
3. Biostatistics is defined as the application of statistical methods to biological and health-related studies and it is widely used in areas like epidemiology, public health, and clinical research.
Statistics is the study of collecting, organizing, summarizing, and interpreting data. Medical statistics applies statistical methods to medical data and research. Biostatistics specifically applies statistical methods to biological data. Statistics is essential for medical research, updating medical knowledge, data management, describing research findings, and evaluating health programs. It allows comparison of populations, risks, treatments, and more.
BIOSTATISTICS FUNDAMENTALS FOR BIOTECHNOLOGYGauravBoruah
1. This document discusses various methods for representing and summarizing data in statistics, including frequency distribution tables, graphical representations, and measures of central tendency and dispersion.
2. Frequency distribution tables organize raw data into a table by counting the frequency of observations in categories. Graphical representations like charts further simplify the data for analysis and interpretation.
3. Measures of central tendency and dispersion numerically summarize data by indicating typical values and the spread or variation in the data set. Together, these descriptive statistics techniques reduce large data sets into more manageable forms for analysis, interpretation, and drawing conclusions.
This document provides an introduction to biostatistics in health. It discusses:
- How data is collected through instruments which have limitations and human biases. Statistics help extract meaningful information from large amounts of raw data.
- Key concepts including populations, samples, variables, and different measurement scales. Variables can be qualitative taking categories like gender, or quantitative measured on interval/ratio scales.
- Descriptive statistics help summarize and present data through tables, graphs, and measures of central tendency and spread. Inferential statistics are used to draw conclusions beyond the sample studied.
- The importance of biostatistics in health fields like understanding diagnostic tests, clinical trials, epidemiology, and evidence-based practice. Statistics under
This document provides an introduction to statistics and biostatistics. It discusses what statistics and biostatistics are, their uses, and what they cover. Specifically, it explains that biostatistics applies statistical methods to biological and medical data. It also discusses different types of data, variables, coding data, and strategies for describing data, including tables, diagrams, frequency distributions, and numerical measures. Graphs and charts discussed include bar charts, pie charts, histograms, scatter plots, box plots, and stem-and-leaf plots. The document provides examples and illustrations of these concepts and techniques.
This document discusses descriptive and inferential statistics. Descriptive statistics involves collecting, organizing and summarizing data, while inferential statistics involves generalizing from samples to populations and making predictions. Examples of descriptive statistics include mean sales and discount rates, while examples of inferential statistics include predicting life satisfaction and the relationship between awareness and resiliency. The document also covers variables, levels of measurement, and types of data.
1Measurements of health and disease_Introduction.pdfAmanuelDina
This document provides an overview of key concepts in biostatistics and measurements of health and disease. It defines biostatistics as the application of statistical methods to biological and health data. The document outlines different types of variables that can be measured, including quantitative, qualitative, discrete and continuous variables. It also describes different scales of measurement for variables, such as nominal, ordinal, interval and ratio scales. Finally, the roles and importance of biostatistical analysis for research, diagnosis and evaluation in public health and medicine are discussed.
This document provides an introduction to statistics, including defining what statistics is, the different types of variables and scales of measurement, and why statistics is important in dentistry. It discusses how statistics can be used for research, understanding medical literature, and informing clinical decision making. Descriptive statistics are used to summarize and describe data, while inferential statistics allow generalizing beyond the sample data to the overall population. Nominal, ordinal, interval, and ratio scales of measurement are explained along with examples. The importance of understanding the scale of measurement is that it determines which statistical tests can appropriately be used for analysis.
This document provides an introduction to statistics, including defining what statistics is, the different types of variables and scales of measurement, and why statistics is important in dentistry. It discusses how statistics can be used for research, understanding medical literature, and informing clinical decision making. Descriptive statistics are used to summarize and describe data, while inferential statistics allow generalizing beyond the sample data to the overall population. Nominal scales name categories, ordinal scales rank order items, interval scales have equal intervals but an arbitrary zero point, and ratio scales have a true zero point where the absence of a trait can be measured.
This document provides an introduction to statistics, including defining what statistics is, the different types of variables and scales of measurement, and why statistics is important in dentistry. It discusses how statistics can be used for research, understanding medical literature, and informing clinical decision making. Descriptive statistics are used to summarize and describe data, while inferential statistics allow generalizing beyond the sample data to the overall population. Nominal, ordinal, interval, and ratio scales of measurement are explained along with examples. The importance of understanding the scale of measurement is that it determines which statistical tests can appropriately be used for analysis.
This document provides an overview of biostatistics and medical statistics. It defines key terms like statistics, biostatistics, medical statistics, and vital statistics. It describes sources of data for vital events and the uses of biostatistics in collecting and analyzing medical data to inform decisions. The document outlines different types of data, variables, scales of measurement, and methods for data collection and presentation. It also defines important statistical measures like mean, median, mode, range, and standard deviation used to summarize central tendencies and dispersion in data.
Mapping Elections1. Identify at least three types of maps (e.docxinfantsuk
Mapping Elections
1. Identify at least three types of maps (e.g., choropleth map, cartogram, proportional symbol map, etc) used to represent information relevant to Presidential elections (these uses might be during each party’s primary competition, after candidates are picked but prior to the election, on the day of the election, or for post-election analysis). Find a specific online example of each and provide a URL for the map. Refer to your text for the kinds of maps that are possible and if you identify a map type not covered in your text, be sure to note that is a type not covered, and briefly explain the characteristic of this map type.
2. Identify the kinds of data that are used for each map type. What is the data source in each case and was that an appropriate source (why)? Are there any better data sources and if so what are they and what makes them better?
3. For each map type, identify at least one key decision that the author made about how to portray the data. Again, refer to your text and other sources for ideas on what the key decisions might be.
4. Produce a critical evaluation of the example(s) you pick to represent each of the three map types you identify. Focus on any geographic information collection, processing, representation issues that are appropriate as well as on how the author uses the map. For the former, things to consider include whether the map type is appropriate to the data type (e.g., nominal, ordinal, interval, ratio; derived versus raw data; etc.); whether any data classification method applied is appropriate and if so whether it is done correctly; whether the specific symbolization choices (colors, sizes, etc) are reasonable; etc. Consider how the map is described by the author and what (if anything) they claim it shows. If they do not make any explicit claims, consider the context in which the map appears and discuss what you believe the motivation for using the maps was and whether this map use was both appropriate and supportable based on ideas you have learned this term. If any of the maps are dynamic (animated or interactive), consider the effectiveness or usability of the dynamic map components. Be sure to include screen shots of each map discussed in your report as well as URLs to the original.
Producing the Project Report
· You may prepare your report in one of two formats: Microsoft Word (.doc) or Adobe Acrobat (.pdf).
· Project reports should include a title page and be formatted so that images are legible.
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Scenario of NCLEX Memorial Hospital where records of the patients with infectious disease are studied in this paper, client number, in ...
This document provides an introduction to biostatistics. It defines biostatistics as the application of statistical tools and concepts to data from biological sciences and medicine. The two main branches of statistics are described as descriptive statistics, which involves organizing and summarizing sample data, and inferential statistics, which involves generalizing from samples to populations. Several key statistical concepts are also defined, including populations, samples, variables, data types, levels of measurement, and common sampling methods. The objectives are to demonstrate knowledge of these fundamental statistical terms and concepts.
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Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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In an education system, it is understood that assessment is only for the students, but on the other hand, the Assessment of teachers is also an important aspect of the education system that ensures teachers are providing high-quality instruction to students. The assessment process can be used to provide feedback and support for professional development, to inform decisions about teacher retention or promotion, or to evaluate teacher effectiveness for accountability purposes.
Assessment and Planning in Educational technology.pptx
Biostatistics ppt.pptx
1. University of Gondar
College of medicine and health science
Department of Epidemiology and Biostatistics
Basic Biostatistics
Wullo S. (BSc, MPH, Assistant professor)
Tuesday, February 7, 2023 1
2. Chapter One
1.1 Introduction to Biostatistics
Objectives of the chapter
After completing this chapter, the student will be able to:
– Define Statistics and Biostatistics
– Identify the Branch of biostatistics
– Enumerate the importance and limitations of biostatistics
– Define and Identify the different types of data and
understand why we need to classify variables
2
2/7/2023
3. Definition and classification of Biostatistics
Statistics is the science of
collecting
organizing
Presenting
analysing and drawing conclusion (inferences) from
data for the purpose of making decision.
Biostatistics: The application of statistical methods
to the fields of biological and health sciences.
3
2/7/2023
4. Classification of Biostatistics
Descriptive biostatistics
A statistical method that is concerned with the collection,
organization, summarization, and analysis of data from a
sample of population.
Inferential biostatistics
A statistical method that is concerned with the drawing
conclusions/inference about a particular population by
selecting and measuring a random sample from the population.
4
2/7/2023
6. Descriptive Biostatistics
• Some statistical summaries which are especially common in
descriptive analyses are:
Measures of central tendency
Measures of dispersion
Measures of association
Cross-tabulation, contingency table
Histogram
Quantile, Q-Q plot
Scatter plot
Box plot
2/7/2023 6
8. 1.2 Stages in statistical investigation
There are five stages or steps in any statistical investigation.
1. Collection of data
The process of obtaining measurements or counts.
2. Organization of data
Includes editing, classifying, and tabulating the data
collected.
3. Presentation of data:
overall view of what the data actually looks like.
facilitate further statistical analysis.
Can be done in the form of tables and graphs or diagrams.
8
2/7/2023
9. Cont…
4. Analysis of data
To dig out useful information for decision making
It involves extracting relevant information from the data
(like mean, median, mode, range, variance…),
5. Interpretation of data
Concerned with drawing conclusions from the data
collected and analyzed; and giving meaning to analysis
results.
A difficult task and requires a high degree of skill and
experience.
9
2/7/2023
10. 1.3 Definition of Some Basic terms
Population: is the complete set of possible measurements for which
inferences are to be made.
Census: a complete enumeration of the population. But in most real
problems it cannot be realized, hence we take sample.
Sample: A sample from a population is the set of measurements that are
actually collected in the course of an investigation.
Parameter: Characteristic or measure obtained from a population.
Statistic: A statistic (rather than the filed of Statistics) refers to a
numerical quantity computed from sample data (e.g. the mean, the
median, the maximum). 10
2/7/2023
12. Cont...
Sampling: The process or method of sample selection from the
population.
Sample size: The number of elements or observation to be
included in the sample.
variable is a characteristic or attribute that can assume different
values in different persons, places, or things.
Some examples of variables include:
Diastolic blood pressure,
heart rate, heights,
The weights
Data: Refers to a collection of facts, values, observations, or
measurements that the variables can assume.
12
2/7/2023
13. Uses of statistics:
The main function of statistics is to enlarge our knowledge of
complex phenomena. The following are some uses of statistics:
Estimating unknown population characteristics.
Testing and formulating of hypothesis.
Studying the relationship between two or more variable.
Forecasting future events.
Measuring the magnitude of variations in data.
Furnishes a technique of comparison.
13
2/7/2023
14. Limitations of statistics
As a science statistics has its own limitations. The following are
some of the limitations:
Deals with only quantitative information.
Deals with only aggregate of facts and not with individual data
items.
Statistical data are only approximately and not mathematical
correct.
Statistics can be easily misused and therefore should be used
be experts.
14
2/7/2023
15. 1.5 Types of Variables and Measurement Scales
A variable is a characteristic or attribute that can assume
different values in different persons, places, or things.
Examples :
age,
diastolic blood pressure,
heart rate,
the height of adult males,
the weights of preschool children,
gender of Biostatistics students,
marital status of instructors at University of Gondar,
ethnic group of patients
15
2/7/2023
16. A. Depending on the characteristic of the measurement, variable can be:
Qualitative(Categorical) variable
A variable or characteristic which cannot be measured in quantitative
form but can only be identified by name or categories,
for instance place of birth, ethnic group, type of drug, stages of
breast cancer (I, II, III, or IV), degree of pain (minimal, moderate,
sever or unbearable).
The categories should be clear cut, not overlapping, and cover all the
possibilities. For example, sex (male or female), vital status (alive or
dead), disease stage (depends on disease), ever smoked (yes or no).
16
2/7/2023
17. Quantitative(Numerical) variable:
is one that can be measured and expressed numerically.
Example: survival time, systolic blood pressure, number of
children in a family, height, age, body mass index.
they can be of two types
Discrete Variables
Have a set of possible values that is either finite or countabl
infinite.
The values of a discrete variable are usually whole
numbers.
Numerical discrete data occur when the observations are
integers that correspond with a count of some sort.
17
2/7/2023
18. Some common examples are:
Number of pregnancies,
The number of bacteria colonies on a plate,
The number of cells within a prescribed area upon microscopic
examination,
The number of heart beats within a specified time interval,
A mother’s history of numbers of births ( parity) and pregnancies
The number of episode of illness a patient experiences during
some time period, etc.
18
2/7/2023
19. Continuous Variables
A continuous variable has a set of possible values including
all values in an interval of the real line.
No gaps between possible values.
Each observation theoretically falls somewhere along a
continuum.
Example: body mass index, height, blood pressure, serum
cholesterol level, weight, age etc.
19
2/7/2023
20. Con…
Observations are not restricted to take on certain numerical
values: Often measurements (e.g., height, weight, age).
Continuous data are used to report a measurement of the
individual that can take on any value within an acceptable
range.
20
2/7/2023
21. Level of measurement which classifies data into mutually exclusive, all
inclusive categories in which no order or ranking can be imposed on
the data.
Assign subjects to groups or categories
No order or distance relationship
No arithmetic origin
Only count numbers in categories
Only present percentages of categories
Chi-square most often used test of statistical significance
Nominal Scale
2/7/2023 21
22. Sex Social status
Marital status Days of the week (months)
Geographic location Seasons
Ethnic group Types of restaurants
Brand choice Religion
Job type : executive, technical, clerical
Other Examples
Coded as “0” Coded as “1”
Nominal Scale
2/7/2023 22
23. Classifies data according to some order or rank
With ordinal data, it is fair to say that one
response is greater or less than another.
E.g. if people were asked to rate the hotness of 3 chili
peppers, a scale of "hot", "hotter" and "hottest"
could be used. Values of "1" for "hot", "2" for
"hotter" and "3" for "hottest" could be assigned.
Ordinal Scale
Level of measurement which classifies data into categories that can be
ranked. Differences between the ranks do not exist.
2/7/2023 23
24. Ordinal Scales
• Arithmetic operations are not applicable but relational
operations are applicable.
• Ordering is the sole property of ordinal scale.
Examples:
Letter grades (A, B, C, D, F).
Rating scales (Excellent, Very good, Good, Fair, poor).
Military status.
2/7/2023 24
25. Interval Scales
• Level of measurement which classifies data that can be ranked
and differences are meaningful. However, there is no meaningful
zero, so ratios are meaningless.
• All arithmetic operations except division are applicable.
• Relational operations are also possible.
Examples:
IQ
Temperature in oF.
2/7/2023 25
26. assumes that the measurements are made in equal units.
i.e. gaps between whole numbers on the scale are equal.
e.g. Fahrenheit and Celsius temperature scales
an interval scale does not have a true zero.
e.g. A temperature of "zero" does not mean that
there is no temperature...it is just an arbitrary zero
point.
permissible statistics: count/frequencies, mode, median,
mean, standard deviation
Interval Scale
Numerically equal distances on the scale represent equal values in
the characteristic being measured. An interval scale contains all the
information of an ordinal scale, but it also allows you to compare the
differences between objects.
2/7/2023 26
27. Ratio Scales
• Level of measurement which classifies data that can be ranked,
differences are meaningful, and there is a true zero. True ratios
exist between the different units of measure.
• All arithmetic and relational operations are applicable.
Examples: Weight
Height
Number of students
Age
2/7/2023 27
28. Primary Scales of Measurement
4 81 9
Nominal Numbers
assigned to
runners
Ordinal Rank order of
winners
Third
Place
Second
Place
First
Place
Interval Performance
rating on a 0 to
10 Scale
8.2 9.1 9.6
Ratio Time to finish in
20 seconds 15.2 14.1 13.4
2/7/2023 28
29. STATISTICS
SCALE DESCRIPTIVE INFERENTIAL
Nominal Percentages, Mode Chi-square, Binomial test
Ordinal Percentile, Median Rank-order, Correlation,
ANOVA
Interval Range, Mean, SD Correlations, t-tests, ANOVA
Regression, Factor Analysis
Ratio Geometric Mean, Coefficient of Variation (CV)
Harmonic Mean
2/7/2023 29
30. Excercise
Categorize the following variables into nominal, ordinal,
interval or ratio
Gender
Grade(A, B, C, D and F )
Rating scale(poor, good, excelent)
Eye colour
Political affilation
Religious affilation
Ranking of tennis players
Majour field
Nationality
30
Height
Weight
Time
Age
IQ
Temprature
Salary
2/7/2023
31. Chapter 2
Organization and Presentation of data
• Having collected and edited the data, the next important step
is to organize it.
• The process of arranging data in to classes or categories
according to similarities is called classification
• Classification is a preliminary and it prepares the ground for
proper presentation of data.
• The presentation of data is broadly classified in to the
following two categories:
• Tabular presentation
• Diagrammatic and Graphic presentation.
Tuesday, February 7, 2023 Wullo S. 31
32. Tabular presentation of data
• Frequency distribution: is the organization of raw
data in table form using classes and frequencies.
• There are three basic types of frequency
distributions
Categorical frequency distribution
Ungrouped frequency distribution
Grouped frequency distribution
Tuesday, February 7, 2023 Wullo S. 32
33. Categorical frequency Distribution:
Used for data that can be place in specific categories such
as nominal, or ordinal.
Example1: a social worker collected the following data on
marital status for 25 persons.(M=married, S=single,
W=widowed, D=divorced)
M S D W D S S M M M W D S M M W D D S S S W W D D
Class (1) Frequency (3) Percent
(4) M 6
24 S 7
28 D 7
28 W 5
20
Tuesday, February 7, 2023 Wullo S. 33
34. Example 2
• Consider for example, the variable birth weight with levels ‘Very
low ’, ‘Low’, ‘Normal’ and ‘Big’. The frequency distribution for
newborns is obtained simply by counting by the number of
newborns in each birth weight category.
Table 2. Distribution of birth weight of newborns between 1976-1996 at TAH.
BWT Freq. Rel.Freq(%) Cum. FreCum.rel.freq.(%)
Very low 43 0.4 43 0.4
Low 793 8.0 836 8.4
Normal 8870 88.9 9706 97.3
Big 268 2.7 9974 100
Total 9974 100
Tuesday, February 7, 2023 Wullo S. 34
35. 2. Ungrouped frequency Distribution
• Is a table of all the potential raw score values that could possible occur in
the data along with the number of times each actually occurred.
• Example: The following data represent the mark of 20 students. 80 ,76,
90 ,85 ,80 ,70 ,60 ,62 ,70 ,85, 65 ,60 ,63 ,74 ,75 ,76 ,70 ,70 ,80, 85 Construct
ungrouped frequency distribution
• Mark Frequency 60
2
62 1
63 1
65 1
70 4
74 1
75 2
76 1
80 3
85 3
90 1
Tuesday, February 7, 2023 Wullo S. 35
36. 3.Grouped frequency distribution
• When the range of the data is large, the data must be grouped in
to classes that are more than one unit in width.
Definitions of same terms:
• Grouped Frequency Distribution: a frequency distribution when
several numbers are grouped in one class.
• Class limits: Separates one class in a grouped frequency
distribution from another. The limits could actually appear in the
data and have gaps between the upper limits of one class and
lower limit of the next.
• Units of measurement (U): the distance between two possible
consecutive measures. It is usually taken as 1, 0.1, 0.01, 0.001, --
---.
Tuesday, February 7, 2023 Wullo S. 36
37. Cont…
• Class boundaries: Separates one class in a grouped frequency
distribution from another. The boundaries have one more
decimal places than the row data and therefore do not appear in
the data. There is no gap between the upper boundary of one
class and lower boundary of the next class. The lower class
boundary is found by subtracting U/2 from the corresponding
lower class limit and the upper class boundary is found by
adding U/2 to the corresponding upper class limit.
• Class width: the difference between the upper and lower class
boundaries of any class. It is also the difference between the
lower limits of any two consecutive classes or the difference
between any two consecutive class marks.
Tuesday, February 7, 2023 Wullo S. 37
38. Cont…
• Class mark (Mid points): it is the average of the lower and
upper class limits or the average of upper and lower class
boundary.
• Cumulative frequency: is the number of observations less
than/more than or equal to a specific value.
• Cumulative frequency above: it is the total frequency of all
values greater than or equal to the lower class boundary of a
given class.
• Cumulative frequency blow: it is the total frequency of all
values less than or equal to the upper class boundary of a
given class.
Tuesday, February 7, 2023 Wullo S. 38
39. Cont…
• Cumulative Frequency Distribution (CFD): it is the
tabular arrangement of class interval together with
their corresponding cumulative frequencies. It can be
more than or less than type, depending on the type
of cumulative frequency used.
• Relative frequency (rf): it is the frequency divided by
the total frequency.
• Relative cumulative frequency (rcf): it is the
cumulative frequency divided by the total frequency.
Tuesday, February 7, 2023 Wullo S. 39
40. Steps for constructing Grouped
frequency Distribution
1. Find the largest and smallest values
2. Compute the Range(R) = Maximum - Minimum
3. Select the number of classes desired, usually
between 5 and 20 or use Sturges rule
where k is number of classes desired and n is total
number of observation.
4. Find the class width by dividing the range by the
number of classes and rounding up, not off.
Tuesday, February 7, 2023 Wullo S. 40
41. Cont…
5. Pick a suitable starting point less than or equal to the minimum
value. The starting point is called the lower limit of the first class.
Continue to add the class width to this lower limit to get the rest
of the lower limits.
6. To find the upper limit of the first class, subtract U from the
lower limit of the second class. Then continue to add the class
width to this upper limit to find the rest of the upper limits.
7. Find the boundaries by subtracting U/2 units from the lower
limits and adding U/2 units from the upper limits. The
boundaries are also half-way between the upper limit of one
class and the lower limit of the next class. !may not be necessary
to find the boundaries.
Tuesday, February 7, 2023 Wullo S. 41
42. Cont…
8. Tally the data.
9. Find the frequencies.
10. Find the cumulative frequencies. Depending on
what you're trying to accomplish, it may not be
necessary to find the cumulative frequencies.
11. If necessary, find the relative frequencies and/or
relative cumulative frequencies
Tuesday, February 7, 2023 Wullo S. 42
43. Example*:
• Construct a frequency distribution for the following data.
11, 29 , 6 ,33 , 14 ,31, 22 , 27 , 19 ,20 ,18 ,17 ,22 ,38 ,23 ,21 ,26
,34 ,39 ,27
Solutions:
Step 1: Find the highest and the lowest value H=39, L=6
Step 2: Find the range; R=H-L=39-6=33
Step 3: Select the number of classes desired using Sturges
formula;
Tuesday, February 7, 2023 Wullo S. 43
44. • Step 4: Find the class width; w=R/k=33/6=5.5=6
(rounding up)
• Step 5: Select the starting point, let it be the minimum
observation.
6, 12, 18, 24, 30, 36 are the lower class limits.
Step 6: Find the upper class limit; e.g. the first upper
class=12-U=12-1=11
11, 17, 23, 29, 35, 41 are the upper class limits.
So combining step 5 and step 6, one can construct the
following classes.
Tuesday, February 7, 2023 Wullo S. 44
45. • Class limits
6 – 11
12 – 17
18 – 23
24 – 29
30 – 35
36 – 41
• Step 7: Find the class boundaries;
• E.g. for class 1 Lower class boundary=6-U/2=5.5 Upper class
boundary =11+U/2=11.5
Tuesday, February 7, 2023 Wullo S. 45
46. • Then continue adding w on both boundaries to obtain the rest
boundaries. By doing so one can obtain the following classes.
• Class boundary
5.5 – 11.5
11.5 – 17.5
17.5 – 23.5
23.5 – 29.5
29.5 – 35.5
35.5 – 41.5
Step 8: tally the data.
Step 9: Write the numeric values for the tallies in the frequency
column
Tuesday, February 7, 2023 Wullo S. 46
47. Cont…
• . Step 10: Find cumulative frequency.
• Step 11: Find relative frequency or/and relative
cumulative frequency.
• The complete frequency distribution follows:
Tuesday, February 7, 2023 Wullo S. 47
48. Diagrammatic and Graphic
presentation of data.
• These are techniques for presenting data in visual
displays using geometric and pictures for a
categorical / qualitative types of data.
Importance:
• They have greater attraction.
• They facilitate comparison.
• They are easily understandable.
Diagrams are appropriate for presenting discrete data.
Tuesday, February 7, 2023 Wullo S. 48
49. Cont…
• The three most commonly used diagrammatic presentation
for discrete as well as qualitative data are:
• Pie charts
• Pictogram
• Bar charts
Pie chart
A pie chart is a circle that is divided in to sections or wedges
according to the percentage of frequencies in each category
of the distribution. The angle of the sector is obtained using:
angle of the sector =RF*360
Tuesday, February 7, 2023 Wullo S. 49
50. Cont…
• Example: Draw a suitable diagram to represent the following
population in a town.
Men Women Girls Boys
2500 2000 4000 1500
Solutions:
Step 1: Find the percentage.
Step 2: Find the number of degrees for each class.
Step 3: Using a protractor and compass, graph each section and
write its name corresponding percentage.
Tuesday, February 7, 2023 Wullo S. 50
52. Bar Charts
The frequency distribution of a categorical variable is often
presented graphically as a bar chart or pie chart.
Bar charts: display the frequency distribution for nominal
or ordinal data.
In a bar chart the various categories into which the
observation fall are represented along horizontal axis and a
vertical bar is drawn above each category such that the
height of the bar represents either the frequency or the
relative frequency of observation within the class.
The vertical axis should always start from 0 but the
horizontal can start from any where.
52
Wullo S.
Tuesday, February 7, 2023
53. Cont…
• There are different types of bar charts. The most common
being :
• Simple bar chart
• Component or sub divided bar chart.
• Multiple bar charts.
Simple bar chart:
Are used to display data on one variable.
They are thick lines (narrow rectangles) having the same
breadth.
The magnitude of a quantity is represented by the height /length
of the bar.
Tuesday, February 7, 2023 Wullo S. 53
54. Cont…
• Example: The following data represent sale
by product, in 1957 a given company for
three products A, B, C.
Product In 1957 Sales($)
A 12
B 24
C 24
Tuesday, February 7, 2023 Wullo S. 54
55. Cont…
• Component Bar chart
-When there is a desire to show how a total (or aggregate)
is divided in to its component parts, we use component
bar chart.
• Example: Example: The following data represent sale by
product, 1957- 1959 of a given company for three
products A, B, C.
Product In 1957 Sales($) In 1958 Sales($) In
1959
A 12 14
18
B 24 28
18
C 24 30
36
Tuesday, February 7, 2023 Wullo S. 55
56. Cont…
• Multiple Bar charts
- These are used to display data on more than
one variable.
- - They are used for comparing different
variables at the same time. Example: Draw a
component bar chart to represent the sales
by product from 1957 to 1959.
Tuesday, February 7, 2023 Wullo S. 56
57. Graphical Presentation of data
The histogram, frequency polygon and cumulative frequency
graph or ogive are most commonly applied graphical
representation for continuous data.
Procedures for constructing statistical graphs:
• Draw and label the X and Y axes.
• Choose a suitable scale for the frequencies or cumulative
frequencies and label it on the Y axes.
• Represent the class boundaries for the histogram or ogive or
the mid points for the frequency polygon on the X axes.
• Plot the points.
• Draw the bars or lines to connect the points.
Tuesday, February 7, 2023 Wullo S. 57
58. Stem-and-Leaf
Represents data by separating each value into
two parts: the stem (the left most digit) and
the leaf (such as the rightmost digit).
• Are most effective with relatively small data
sets
• Are not suitable for reports and other
communications,
• Help researchers to understand the nature of
their data
Example: 43, 28, 34, 61, 77, 82, 22, 47, 49, 51, 29,
36, 66, 72, 41
Tuesday, February 7, 2023 Wullo S. 58
59. Histogram
• A graph which displays the data by using vertical bars of various heights
to represent frequencies.
• Class boundaries are placed along the horizontal axes.
• Class marks and class limits are some times used as quantity on the X
axes.
Table 3: Distribution of the age of women at the time of marriage
Age group No. of women
15-19 11
20-24 36
25-29 28
30-34 13
35-39 7
40-44 3
45-49 2
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60. Cont…
• A histogram of the age of women at the time of marriage
Tuesday, February 7, 2023 Wullo S. 60
Age of women at the time of marriage
0
5
10
15
20
25
30
35
40
14.5-19.5 19.5-24.5 24.5-29.5 29.5-34.5 34.5-39.5 39.5-44.5 44.5-49.5
Age group
No
of
women
61. Frequency Polygon
• Frequency Polygon : - A line graph. The
frequency is placed along the vertical axis
and classes mid points are placed along the
horizontal axis.
• It is customer to the next higher and lower
class interval with corresponding frequency
of zero, this is to make it a complete
polygon. Example: Draw a frequency
polygon for the above data
Tuesday, February 7, 2023 Wullo S. 61
62. Ogive (cumulative frequency polygon)
- A graph showing the cumulative frequency (less than
or more than type) plotted against upper or lower
class boundaries respectively.
- That is class boundaries are plotted along the
horizontal axis and the corresponding cumulative
frequencies are plotted along the vertical axis.
- The points are joined by a free hand curve.
- Example: Draw an ogive curve(less than type) for the
above data.
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63. Chapter 3
Numerical summary measures
A. Measures of location
• It is often useful to summarize, in a single number or
statistic, the general location of the data or the point at
which the data tend to cluster.
• Such statistics are called measures of location or
measures of central tendency.
• We describe them mean, median and mode.
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64. Cont…
Arithmetic mean
• The arithmetic mean, usually abbreviated to
‘mean’ is the sum of the observations divided
by the number of observations.
• We use the following data set of 10 numbers
to illustrate the computations:
65. Arithmetic Mean
19 21 20 20 34 22 24 27
27 27
• Then, mean = (19 + 21 + … +27) = 24.1
10
• General formula
a) Ungrouped mean
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Estimation of the mean from a grouped frequency distribution
In calculating the mean from grouped data, we assume that all
values falling into a particular class interval are located at the
mid-point of the interval.
It is calculated as follow:
66. Cont…
Tuesday, February 7, 2023 Wullo S. 66
where,
k = the number of class intervals
xi = the mid-point of the ith class interval
fi = the frequency of the ith class interval
67. Properties of the arithmetic mean
• The mean can be used as a summary measure for both
discrete and continuous data, in general however, it is not
appropriate for either nominal or ordinal data.
• For given set of data there is one and only one arithmetic
mean.
• The arithmetic mean is easily understood and easy to
compute.
• Algebraic sum of the deviations of the given values from their
arithmetic mean is always zero.
• The arithmetic mean is greatly affected by the extreme values.
• In grouped data if any class interval is open, arithmetic mean
can not be calculated.
67
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68. Reading Assignment
A. Weighted mean
B. Correct and wrong mean
C. Combined mean
D. Geometric mean
E. Harmonic
Tuesday, February 7, 2023 Wullo S. 68
69. Median
• With the observations arranged in increasing or decreasing
order, the median is defined as the middle observation.
• If the number of observations is odd, the median is defined as
the [(n+1)/2]th observation.
• If the number of observations is even the median is the
average of the two middle
{(n/2)th +[(n/2)+1]th }/2 values i.e
• If the number of observations is even, so that there is no
middle observation, the median is defined as the average of
the two middle observations.
• Example : 19 20 20 21 22 24 27
27 27 34
• Then, the median = (22 + 24)/2 = 23
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70. Median for Grouped data
In calculating the median from grouped data, we assume that the
values within a class-interval are evenly distributed through the
interval.
– The first step is to locate the class interval in which it is located.
– Find n/2 and see a class interval with a minimum cumulative
frequency which contains n/2.
• Whereas:
• LCB= lower class boundary of the median class
• Fc= cumulative frequency just before the median class
• fc=frequency of the median class
• W =class width and n=number of observations.
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71. Properties of median
• The median can be used as a summary measure for discrete
and continuous data, in general however, it is not
appropriate for nominal data.
• There is only one median for a given set of data
• The median is easy to calculate
• Median is a positional average and hence it is not drastically
affected by extreme values
• Median can be calculated even in the case of open end
intervals
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72. Mode
• Any observation of a variable at which the distribution reaches
a peak is called a mode.
• Most distributions encountered in practice have one peak and
are described as uni-modal.
• E.g. Consider the example of ten numbers
19 21 20 20 34 22 24 27
27 27
• In the above data set, the mode is 27, because the value 27
occurs three times (the most frequent).
• The mode of grouped data, usually refers to the modal class,
where the modal class is the class interval with the highest
frequency.
• If a single value for the mode of grouped data must be
specified, it is taken as the mid point of the modal class
interval
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73. Properties of mode
• The mode can be used as a summary measure for nominal,
ordinal, discrete and continuous data, in general however, it is
more appropriate for nominal and ordinal data.
• It is not affected by extreme values
• It can be calculated for distributions with open end classes
• Often its value is not unique
• The main drawback of mode is that often it does not exist
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74. proportion
• As we have seen from the previous section, a variable can be
either categorical or numerical
• Proportion is one of summery measures for categorical variable
and also numerical variable if we are counting the number of
cases under the specific category.
• If we denote “x” as the number of success in an experiment and
“n” is the number of trials then the proportion of success from n
number of trials is given by x/n.
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75. Percentiles, Quartiles and Inter-quartile Range
• The quartiles are sets of values which divide the distribution into
four parts such that there are an equal number of observations
in each part.
– Q1 = [(n+1)/4]th
– Q2 = [2(n+1)/4]th
– Q3 = [3(n+1)/4]th
• The inter-quartile range is the difference between the third and
the first quartiles.
– Q3 - Q1
Example1: We use the data set of 11 numbers:
19 21 20 20 34 22 24 27 27 27
28
– The first quartile is 20 and the third quartile is 27
– The inter quartile range = 27 – 20 = 7.
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76. Percentiles, Quartiles and Inter-quartile Range
• Percentiles divide the data into 100 parts of observations in each
part.
• It follows that the 25th percentile is the first quartile, the 50th
percentile is the median and the 75th percentile is the third
quartile.
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77. B. Measures of Dispersion (Variation)
• The scatter or spread of items of a distribution is known as
dispersion or variation.
• In other words the degree to which numerical data tend to
spread about an average value is called dispersion or
variation of the data.
• The most commonly used measures of dispersions are:
1) Range and relative range
2) Quartile deviation and coefficient of Quartile deviation
3) Mean deviation and coefficient of Mean deviation
4) variance
5) Standard deviation and coefficient of variation.
Tuesday, February 7, 2023 Wullo S. 77
78. Range
• The range is the largest score minus the smallest
score.
• It is a quick and dirty measure of variability
• Because the range is greatly affected by extreme
scores and its only depends on two observations
Relative Range (RR)
• It is also some times called coefficient of range and
given:
Tuesday, February 7, 2023 Wullo S. 78
79. Cont..
• Example:
1. If the range and relative range of a series are 4 and
0.25 respectively. Then what is the value of Smallest
observation and Largest observation
The Quartile Deviation (Semi-inter quartile range)
The inter quartile range is the difference between the
third and the first quartiles of a set of items and
semi-inter quartile range is half of the inter quartile
range.
Tuesday, February 7, 2023 Wullo S. 79
80. Variance
• variance is the "average squared deviation from the mean".
• A good measure of dispersion should make use of all the data.
• For the case of frequency distribution it is expressed as:
the variance is limited as a descriptive statistic because it is not
in the same units as in the observations.
Tuesday, February 7, 2023 Wullo S. 80
82. Coefficient of Variation (C.V)
Is defined as the ratio of standard deviation to
the mean usually expressed as percents.
The distribution having less C.V is said to be less
variable or more consistent.
Tuesday, February 7, 2023 Wullo S. 82
83. Which measures to use?
• When the distribution of the data is symmetric and unimodal (i.e.
the data are approximately normally distributed), it is usual to
summarize the data using means and standard deviations.
• However when the data are skewed, it is preferable to use the
median and quartiles as summary statistics.
• Median and quartiles are not easily influenced by extreme values
in a skewed distribution unlike means and standard deviations.
• Remark:
– The mean and median of symmetric distribution coincide.
– When the distribution is skewed to the right, its mean is larger than its
median.
– When the distribution is skewed to the left, its mean is smaller than its
median. [See Figures 7(a-c)].
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84. Median Mode Mean
Fig. 2(a). Symmetric Distribution
Mean = Median = Mode
Mode Median Mean
Fig. 2(b). Distribution skewed to the right
Mean > Median > Mode
Mean Median Mode
Fig. 2(c). Distribution skewed to the left
Mean < Median < Mode
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85. Chapter 4
Introduction to probability
Objectives
• After completing this chapter, you should be able to
– Determine sample spaces and find the probability of an events
– Understand the different properties of probability
– Explain the various types of probability distributions – emphasis will be
given to the two widely used probability distributions
Tuesday, February 7, 2023 Wullo S. 85
86. Introduction
• Many medical decisions are made on a statistical
basis since individuals differ in their reactions to
medications or surgery in an unpredictable way.
• In that case the treatment applied is based on
getting the best outcome for as many patients as
possible
– The life experienced consists of a series of events
– “Probability” is a very useful concept and are used in
everyday communication.
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87. Introduction con'td…
An understanding of probability is fundamental for
quantifying the uncertainty in the decision-making
process
drawing conclusions about a population of patients based on known
information about a sample of patients drawn from that population.
Probability can be defined as the chance of an event
occurring.
Many people are familiar with probability from
observing or playing games of chance, such as card
games, slot machines, or lotteries.
Probability theory is used in the various fields of area
like insurance, investments, and weather forecasting
and other areas.
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88. Basic concepts
• The following definitions and terms are used in
studying the theory of probability.
– Random experiment: a chance process that leads to
well-defined results called outcomes, that is the result
cannot be predicted. E.g. Tossing of coins, throwing of
dice are some examples of random experiments.
– Trial: Performing a random experiment is called a
trial.
– Outcomes: The results of a random experiment are
called its outcomes. When two coins are tossed the
possible outcomes are HH, HT, TH, TT.
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89. Basic concepts con'td….
– Sample space: Each conceivable outcome of an
experiment is called a sample point. The totality of all
sample points is called a sample space and i s denoted
by S.
– Event: An outcome or a combination of outcomes of a
random experiment is called an event. It is a subset of
the sample space of a random experiment.
– Equally-likely Approach: If an experiment must result in
n equally likely outcomes, then each possible outcome
must have probability 1/n of occurring.
– Mutually exclusive events: when the occurrence of any
one event excludes the occurrence of the other event.
Mutually exclusive events cannot occur simultaneously.
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90. Basic concepts cont’d…..
• Some sample spaces for various probability experiments are.
• Probability attempts to quantify an uncertain situation and relative
tries to make it more concrete the occurrence of events.
• Probability is used to quantify the likelihood, or chance, that an
outcome of a random experiment will occur.
• Probability is a number between 0 and 1 that expresses how likely the
event is occur.
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91. Basic concepts cont’d…..
• Example: Find the sample space for the gender of the children if
a family has three children. Use B for boy and G for girl.
– Solution: There are two genders, male and female, and each
child could be either gender. Hence, there are eight
possibilities, as shown here.
S= {BBB, BBG, BGB, GBB, GGG, GGB, GBG, BGG}
• Note: the way to find all possible outcomes of a probability
experiment (the sample spaces)
– by observation and reasoning;
– use a tree diagram (a device consisting of line segments
emanating from a starting point and also from the outcome
point.)
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92. Tree diagram of the above example
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93. Types of probability
1. Classical Probability
If the number of outcomes belonging to an event E is NE, and the total
number of outcomes is N, then the probability of event E is defined
as P(E)=NE/N
Example: A couple wants to have exactly 3 children. Assume that each
child is either a Boy or a Girl and that there are no duplicate births.
Find the probability that two of them will be boys? List all possible
orderings for the three children.
Solution: S {BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG}
Then the event E={BBG, BGB, GBB}
P(E)=3/8
2. Relative Frequency Probabilities
This approach to probability is well-suited to a wide range of scientific
disciplines. It is based on the idea that the underlying probability of
an event can be measured by repeated trials
Tuesday, February 7, 2023 Wullo S. 93
94. • In this view, probability is treated as a quantifiable level of belief
ranging from 0 (complete disbelief) to 1 (complete belief)
• For instance, an experienced physician may say “this patient has a
50% chance of recovery.”
• An appreciation of the various types of probability are not mutually
exclusive. And fortunately, all obey the same
• mathematical laws, and their methods of calculation are similar.
• All probabilities are a type of relative frequency—the number of
times something can occur divided by the total number of
possibilities or occurrences.
3. Subjective probability
Tuesday, February 7, 2023 Wullo S. 94
95. Rules of Probability
• Any probability assigned must be a nonnegative number.
• The probability of the sample space (the collection of all possible
outcomes) is equal to 1.
• The probability of A or B involves addition.
• P(A or B) = P(A) + P(B) if the two are mutually exclusive.
• The probability of A and B involves multiplication
• P(A and B) = P(A) P(B) if the two are independent
• P( Not A) = 1- P(A)
• P(At least one) = 1- P(none)
• P(none) = P(each event not happening)^number of events
Tuesday, February 7, 2023 Wullo S. 95
96. Addition Rule
• General rule: if two events, A and B, are not mutually exclusive,
then the probability that event A or event B occurs is:
– P(A or B) = P(A) + P(B) – P(A and B)
– P(AuB) = P(A) + P(B) – P(AnB)
• Special case: when two events, A and B, are mutually exclusive,
then the probability that event A or event B occurs is:
– P(A or B) = P(A) + P(B)
– P(AuB) = P(A) + P(B)
• since P(A and B) = 0 for mutually exclusive events
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Tuesday, February 7, 2023
97. Conditional Probabilities
• Sometimes the chance a particular event happens depends on the
outcome of some other event. This applies obviously with many
events that are spread out in time.
• The probability that an event occurs subject to the condition that
another event has already occurred is called conditional probability
• If A and B are events with Pr(A) > 0, the conditional probability of
B given A is
• Example: Drug test
• Given A is independent from B, what is the relationship between
Pr(A|B) and Pr(A)?
Pr( )
Pr( | )
Pr( )
AB
B A
A
97
Women Men
Success 200 1800
Failure 1800 200
A = {Patient is a Women}
B = {Drug fails}
Pr(B|A) = 1800/2000
Pr(A|B) =
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98. Conditional probability:
• The conditional probability of an event A given an
event B is present is:
– P(A | B) =P(AnB)/P(B)
• where P(B)≠0
• Joint probability
• The joint probability of an event A and an event B is
– P(AnB) = P(A and B)
• When events A and B are mutually exclusive, then
– P(A and B) = 0
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Tuesday, February 7, 2023
99. Multiplication rule
• General rule: the multiplication rule specifies the
joint probability as:
• P(AnB)=P(B)P(A/B)
• Special case: When events A and B are independent,
then:
– P(A|B) = P(A)
– P(AnB)=P(A)P(B)
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100. Example
1. Find the Probability of at least one male birth in ten consecutive
births?
Solution
• P(At least one male) = 1- P(all females)
• P(all females) = P(each single birth is a female)10 = (0.5)10
= 9.77 x 10-4
• So P(At least one male) = 1 – 9.77 x 10-4 = 0.999023.
Tuesday, February 7, 2023 Wullo S. 100
101. Exercises
1. 50% of the students in a school weigh more than 140 pounds, but
70% weigh less than 170. If a student is randomly selected, what
is the probability the student will weigh between 140 and 170?
• Solution
• P(w<140)=0.5, P(w>170)=0.3, and P(w<140)+P(140<w<170)+
P(w>170)= 1
• Then
• 0.5+P(140<w<170)+ 0.3= 1 P(140<w<170)=1-0.8=0.2
Tuesday, February 7, 2023 Wullo S. 101
102. Probability Distribution
• A probability distribution is a table or an equation
that links each outcome of a statistical experiment
with its probability of occurrence.
• Probability distribution for discrete variable
• Probability distribution for continuous variable
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103. Probability Distribution for discrete Variable
1. Binomial distribution
• A binomial experiment (also known as a Bernoulli trial) is a statistical
experiment that has the following properties:
• The experiment consists of n repeated fixed number of trials.
• Each trial can result in just two possible outcomes. We call one of these
outcomes a success and the other, a failure.
• The probability of success, denoted by P, is the same on every trial.
• The trials are independent; that is, the outcome on one trial does not
affect the outcome on other trials.
• The probability distribution of this experiment is known as binomial
probability distribution.
• The binomial distribution describes the distribution of "success" in a series
of trials, that is, out of N tries, what is the probability that X of them
succeed.
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104. Binomial formula
• If the probability of success on an individual trial is P, then the binomial
probability is defined by:
• Where
• K=the number of success
• P=probability of success
• n=the number of experiments
• 1-p=probability of failure
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105. 2. Poisson Distribution
• The Poisson Distribution is a discrete distribution which takes on the values
X = 0, 1, 2, 3, and so on.
• It is often used as a model for the number of events in a specific time
period.
• It is determined by one parameter, lambda. The Poisson random variable
satisfies the following conditions:
• The number of successes in two disjoint time intervals is independent.
• The probability of a success during a small time interval is proportional to
the entire length of the time interval.
• Some of the examples in which Poisson distribution is appropriate are:
birth defects and genetic mutations
rare diseases (like Leukemia, but not AIDS because it is infectious and so
not independent)
car accidents
traffic flow and ideal gap distance, and so on
Tuesday, February 7, 2023 Wullo S. 105
106. Poisson formula
• The probability distribution of a Poisson random variable X representing the
number of successes occurring in a given time interval or a specified region of
space is given by the formula:
Where
• X=Number of successes per unit time
• e=The base of the natural log
• λ= The expected number of successes per unit time
• If λ is the average number of successes occurring in a given time interval or
region in the Poisson distribution, then the mean and the variance of the
Poisson distribution are both equal to λ.
Tuesday, February 7, 2023 Wullo S. 106
107. Probability Distribution for continuous Variables
• If a random variable is a continuous variable, its probability
distribution is called a continuous probability distribution.
• A continuous probability distribution differs from a discrete
probability distribution in several ways by:
• Under different circumstances, the outcome of a random
variable may not be limited to categories or counts.
• Because a continuous random variable X can take on an
uncountable infinite number of values, the probability
associated with any particular one value is almost equal to zero.
• As a result, a continuous probability distribution cannot be
expressed in tabular form.
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108. Normal distribution
• The normal distribution refers to a family of continuous
probability distributions described by the normal equation and
described as follows:
where
• X is a normal random variable,
• μ is the mean
• σ is the standard deviation
• pi is approximately 3.14159, and e is approximately 2.71828.
• The random variable X in the normal equation is called the
normal random variable. 108
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109. Characteristics of Normal Distribution
• It links frequency distribution to probability distribution
• Has a Bell Shape Curve and is Symmetric
• It is Symmetric around the mean: Two halves of the curve are the
same (mirror images)
• Hence Mean = Median
• The total area under the curve is 1 (or 100%)
• Normal Distribution has the same shape as Standard Normal
Distribution.
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110. Normal Curve
• The graph of the normal distribution depends on two factors:
the mean and the standard deviation.
• The mean of the distribution determines the location of the center of the
graph, and the standard deviation determines the height and width of the
graph.
• When the standard deviation is large, the curve is short and wide; when the
standard deviation is small, the curve is tall and narrow.
• All normal distributions look like a symmetric, bell-shaped curve.
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111. Standard Normal Distribution
• It makes life a lot easier for us if we standardize our normal curve, with a mean
of zero and a standard deviation of 1 unit.
• We can transform all the observations of any normal random variable X with
mean μ and variance σ to a new set of observations of another normal random
variable Z with mean 0 and variance 1 using the following transformation:
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112. • About 95% of the area under the curve falls within 2 standard deviations
of the mean.
• About 99.7% of the area under the curve falls within 3 standard deviations
of the mean.
• A graph of this standardized (mean 0 and variance 1) normal curve is given
in Graph:
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113. Table of normal
• Example 1: Suppose we want to compute the area under the
normal curve to the right of 1.45
• This area can be computed by finding the probability under
the normal curve. The probability can be read at the normal
curve by combining the value of 1.4 under the first column
and 0.05 under the first row.
• The green shaded area in the diagram represents the area
that is within 1.45 standard deviations from the mean. The
area of this shaded portion is 0.4265 (or 42.65% of the total
area under the
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115. Example 2
• Assuming the normal heart rate (H.R) in normal healthy individuals is
normally distributed with Mean = 70 and Standard Deviation =10 beats/min
Then:
1) What area under the curve is above 80 beats/min?
Now we know, Z =X-M/SD,
Z=? X=80, M= 70, SD=10 .
So we have to find the value of Z.
For this we need to draw the figure…..and find the area which corresponds to Z.
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116. • Since M=70, then the area under the curve which is above 80 beats
per minute corresponds to above + 1 standard deviation.
•
• The total shaded area corresponding to above 1+ standard
deviation in percentage is 15.9% or Z= 15.9/100 =0.159.
• Or we can find the value of z by substituting the values in the
formula Z= X-M/ standard deviation.
• Therefore, Z= 70-80/10 -10/10= -1.00 is the same as +1.00. The
value of z from the table for 1.00 is 0.159. How do we interpret this?
•
• This means that 15.9% of normal healthy individuals have a heart
rate above one standard deviation (greater than 80 beats per
minute).
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117. 13.6%
2.2%
0.15
-3 -2 -1 μ 1 2 3
Diagram of Exercise
0.159
33.35%
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118. Example …
2) What area of the curve is above 90
beats/min?
3) What area of the curve is between
50-90 beats/min?
4)What area of the curve is above 100
beats/min?
5) What area of the curve is below 40 beats per
min or above 100 beats per min?
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119. solution
2) 2.3% or 0.023
3) 95.4% or 0.954
4) 0.15 % or 0.015
5) 0.3 % or 0.015 (for each tail)
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120. Example 3
• Suppose scores on an IQ test are normally distributed. If the test has a
mean of 100 and a standard deviation of 10, what is the probability that a
person who takes the test will score between 90 and 110?
• Solution:
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121. Application/Uses of Normal Distribution
• It’s application goes beyond describing distributions
• It is used by researchers and modelers.
• The major use of normal distribution is the role it plays
in statistical inference.
• The z score along with the t –score, chi-square and F-
statistics is important in hypothesis testing.
• It helps managers/management make decisions.
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122. Exercises
• Find the probability under the normal curve of the
following:
• The area greater than 1.25
• The area lower than 0.87
• The area greater than -2.36
• The area lower than -1.96
• The area between 0.87 and 1.25
• The area between -1.96 and 1.25
• The value of z which cuts the lower 20%
• The value of z which cuts the upper 10%
• The values of z which cut the middle 80%
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123. Chapter s
Sampling methods and sample size
determination
• Sampling is a procedure by which some members of
the given population are selected as representative
of the entire population
Theoretical population
Target population
Study population
sampling frame
Sampling unit
Study unit
125. Error in sampling
• No sample is the exact mirror image of the population
• Potential Source of Error in research are two.
1. Sampling error is any type of bias that is attributable to
mistakes in either drawing a sample or determining the
sample size
Sampling error (chance ) Can not be avoided or totally
eliminated
The causes are
– One is chance: That is the error that occurs just because of bad
luck
– Design error
– Un representativeness of the sample
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Tuesday, February 7, 2023
126. Error in sampling con’d..
2. Non-sampling error is any error which will be
committed during data collection, coding, entry, and
so on
Observational error
Respondent error
Lack of preciseness of definition
Error in editing and tabulation of the data
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Tuesday, February 7, 2023
128. Advantage of sampling
Feasibility it may be the only feasible method
of collecting data
Reduced cost sampling reduces demands on
resource such as finance, personal and
material
Greater accuracy sampling may lead to better
accuracy of collecting data/ detailed
information
Greater speed data can be collected and
summarized more quickly
Wullo S. 128
Tuesday, February 7, 2023
129. Disadvantage of sampling
• There is always sampling error
• Sampling may create a feeling of discrimination within
the population
• It may be inadvisable where every unit in the
population is legally required to have a record
• Demands more rigid control in undertaking sample
operation
• The presence of bias creates difference between the
parameter and the statistic
• Minority and smallness in number of sub-groups often
render study to be suspected.
• Sample results are good approximations at best.
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Tuesday, February 7, 2023
130. Types of sampling
I. Probability sampling
• Is any method of sampling that utilizes some form of random
selection.
• probability sampling is a procedure for sampling from a
population in which
– The selection of a sample unit is based on chance
– Every element of the population has a known and non-zero
probability of being selected
– Random sampling helps produce representative samples by
eliminating voluntary response bias and guarding against under
coverage bias
Every individual of the target population has equal chance to be
included in the sample.
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Tuesday, February 7, 2023
131. 1. Simple random sample (SRS)
• Objective: To select n units out of N
• If the population is homogenous
• If frame is available
• If the study area is not very wide
– Note: Homogeneity refers to the similarity of the population with regard
to the outcome variable .
• Procedure:
Use a table of random numbers: takes on values
0,1,2,…….,9 with equal probability
a computer random number generator
mechanical device to select the sample.
RAND() function from Excel sheet if frame is available
Lottery method
Wullo S. 131
Tuesday, February 7, 2023
133. 2. Stratified random sampling
• Stratified Random Sampling involves dividing your population
into homogeneous subgroups and then taking a simple random
sample in each subgroup.
• Objective: Divide the population into non-overlapping groups
(i.e., strata) N1, N2, N3, ... Ni, such that N1 + N2 + N3 + ... + Ni = N.
Then do a simple random sample depending on the type of
allocation
Proportional allocation:
Equal allocation
Example:
An agency has clients from three ethnic groups and the agency
wants to asses clients view of quality of service for the last year.
i
i N
N
n
n *
Wullo S. 133
Tuesday, February 7, 2023
135. 3. Systematic random sampling
Here are the steps you need to follow in order to achieve a
systematic random sample:
number the units in the population from 1 to N
decide on the n (sample size) that you want or need
k = N/n = the interval size
randomly select an integer between 1 to k
then take every kth unit
• Assumptions
– Homogeneous population
– Frame is not available
– If the study area is not very wide
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Tuesday, February 7, 2023
139. 4. Cluster (area) random sampling
• The problem with random sampling methods when we have to
sample a population that's disbursed across a wide geographic
region is that you will have to cover a lot of ground geographically
in order to get to each of the units you sampled
• In cluster sampling, we follow these steps:
divide population into clusters (usually along geographic
boundaries)
randomly sample clusters
measure all units within sampled clusters
Wullo S. 139
Tuesday, February 7, 2023
140. Example
• In the figure we see a map of the counties in New York State.
Let's say that we have to do a survey of town governments that
will require us going to the towns personally. If we do a simple
random sample state-wide we'll have to cover the entire state
geographically
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Tuesday, February 7, 2023
141. 5. Multi-stage sampling
• The four methods we've covered so far -- simple, stratified,
systematic and cluster -- are the simplest random sampling
strategies
• When we combine sampling methods, we call this multi-stage
sampling
• Consider the problem of sampling students in grade schools.
We might begin with a national sample of school districts
stratified by educational level. Within selected districts, we
might do a simple random sample of schools. Within schools,
we might do a simple random sample of classes or grades.
And, within classes, we might even do a simple random
sample of students. In this case, we have three or four stages
in the sampling process and we use both stratified and simple
random sampling.
Wullo S. 141
Tuesday, February 7, 2023
143. II. Non probability sampling
• Non probability sampling does not involve random selection
• Does that mean that non probability samples aren‘t representative
of the population?
• It does mean that non probability samples cannot depend upon the
rationale of probability theory
• Most sampling methods are purposive in nature because we usually
approach the sampling problem with a specific plan in mind.
Wullo S. 143
Tuesday, February 7, 2023
144. Convenience sampling
Example
Man in the street (attitude of foreigners about Ethiopia)
College students for psychological study
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145. Purposive sampling
• In purposive sampling, we sample with a purpose in mind
1. Modal Instance Sampling
In statistics, the mode is the most frequently occurring value
in a distribution.
we are sampling the most frequent case, or the "typical"
case
We could say that the modal voter is a person who is of
average age, educational level, and income in the
population
Wullo S. 145
Tuesday, February 7, 2023
146. Purposive sampling…
2. Expert Sampling
Expert sampling involves the assembling of a sample of persons
with known or demonstrable experience and expertise in some
area.
3. Quota Sampling
In quota sampling, you select people non randomly according to
some fixed quota
There are two types of quota sampling: proportional and non
proportional
4. Heterogeneity Sampling
We sample for heterogeneity when we want to include all
opinions or views, and we aren't concerned about representing
these views proportionately.
Another term for this is sampling for diversity.
Wullo S. 146
Tuesday, February 7, 2023
147. Purposive sampling…
5. Snowball sampling
In snowball sampling, you begin by identifying someone who
meets the criteria for inclusion in your study
You then ask them to recommend others who they may know
who also meet the criteria
Snowball sampling is especially useful when you are trying to
reach populations that are inaccessible or hard to find
For instance, if you are studying the homeless, you are not
likely to be able to find good lists of homeless people within a
specific geographical area. However, if you go to that area and
identify one or two, you may find that they know very well
who the other homeless people in their vicinity are and how
you can find them
Wullo S. 147
Tuesday, February 7, 2023
148. Summary
Selecting a sampling method depends on
• Population to be studied
Size and geographic distributions
Heterogeneity with respect to the variable studied
• Resource available
• Level of precision required
• Importance of having a precise estimate of sampling error
Wullo S. 148
Tuesday, February 7, 2023
149. Sample size determination
• Determining the sample size for a study is a crucial
component of study design.
• The goal is to include sufficient numbers of subjects so that
statistically significant results can be detected.
• Among the questions that a researcher should ask when
planning a survey or study is that "How large a sample do I
need?"
• The answer will depend on the aims, nature and scope of the
study and on the expected result.
• All of which should be carefully considered at the planning
stage
Wullo S. 149
Tuesday, February 7, 2023
150. Sample size determination ….
• In general, sample size depends on
– Objective of the study
– Design of the study
– Plan for statistical analysis
– Accuracy of the measurement to be made
– Degree of precision required for generalization
– Degree of confidence
• We can use three approaches to determine sample size
– Rules of thumb for determining the sample size
– Statistical formula
• Confidence interval approach
• Hypothesis testing approach
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Tuesday, February 7, 2023
151. Rules of thumb
• For smaller samples(N < 100), there is little point in sampling.
Survey the entire population.
• If the population size is around 500 (give or take 100), 50%
should be sampled.
• If the population size is around 1500, 20% should be sampled.
• Beyond a certain point (N = 5000), the population size is
almost irrelevant and a sample size of 400 may be adequate.
• Statistician maximalist: at least 500
• To make generalizations about entire population, need a total
sample size of 200-400 (depending on total population and
confidence level desired)
Wullo S. 151
Tuesday, February 7, 2023
152. confidence interval
• Hence the absolute precision denoted by d is given as
• Where s.e is the standard error of the estimator of the
parameter of interest.
• The margin of error (d) measures the precision of the estimate
– Small value of w indicates high precision
– It lies in the interval (0%; 5%]
– For p close to 50%, w is assumed to be close to 5%
– For smaller value of p, w is assumed to be lower than 5%
e
s
z
proportion
mean .
)
(
2
e
s
z
d .
2
Wullo S. 152
Tuesday, February 7, 2023
153. Estimating a single population mean
Where the standard deviation δ can be estimated by;
From previous study, if there is
From pilot study
From educate guess
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Tuesday, February 7, 2023
154. Single population proportion
• Let p denotes proportion of success, then
Where the standard deviation p can be estimated by;
From previous study, if there is
From pilot study
P=50%
Wullo S. 154
Tuesday, February 7, 2023
155. Point to be considered
Wullo S. 155
Tuesday, February 7, 2023
158. Chapter six
Inferential statistics
• After complete this session you will be able to do
– Parameter estimations
– Point estimate
– Confidence interval
– Hypothesis testing
– Z-test
– T-test
– Testing associations
– Chi-Square test
160. Introduction con'td……..
• Before beginning statistical analyses
– it is essential to examine the distribution of the variable for
skewness (tails),
– kurtosis (peaked or flat distribution), spread (range of the
values) and
– outliers (data values separated from the rest of the data).
• Information about each of these characteristics
determines to choose the statistical analyses and can
be accurately explained and interpreted.
160
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161. Introduction con’td …….
• Statistical tests can be either parametric or non-
parametric
• The path way for the analysis of continuous variables
is shown below
161
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162. Sampling Distribution
• The frequency distribution of all these samples forms the sampling
distribution of the sample statistic
162
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163. Sampling distribution .......
Three characteristics about sampling distribution of a statistic
its mean
its variance
its shape
If we repeatedly take sample of the same size n from a
population the means of the samples form a sampling
distribution of means of size n is equal to population mean.
In practice we do not take repeated samples from a population
i.e. we do not encounter sampling distribution empirically, but it
is necessary to know their properties in order to draw statistical
inferences.
163
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164. The Central Limit Theorem
• Regardless of the shape of the frequency distribution of
a characteristic in the parent population,
• the means of a large number of samples (independent
observations) from the population will follow a normal
distribution (with the mean of means approaches the
population mean μ, and standard deviation of σ/√n ).
• Inferentialstatisticaltechniqueshavevariousassumptionst
hatmustbemetbeforevalid conclusions can be obtained
• Samples must be randomly selected.
• sample size must be greater (n>=30)
• the population must be normally or approximately normally distributed if the
sample size is less than 30.
164
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165. Sampling Distribution......
• E.g. Sampling Distribution of the mean
• Suppose we choose a random sample of size n, the
sampling distribution of the sample mean x posses the
following properties.
– The sample mean x will be an estimate of the population
mean μ.
– The standard deviation of x is σ/√n (called the standard
error of the mean).
– Provided n is large enough the shape of the sampling
distribution of x is normal.
165
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166. Sampling Distribution ..........
• Proportion
Suppose we choose a random sample of size n, the sampling
distribution of the sample means p posses the following
properties.
The sample proportion p will be an estimate of the
population mean p.
________
The standard deviation of p is = √p(1-p) /n called the
standard error of the proportion).
Provided n is large enough the shape of the sampling
distribution of p is normal.
166
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167. Parameter Estimations
• In parameter estimation, we generally assume that the
underlying (unknown) distribution of the variable of interest is
adequately described by one or more (unknown) parameters,
referred as population parameters.
• As it is usually not possible to make measurements on every
individual in a population, parameters cannot usually be
determined exactly.
• Instead we estimate parameters by calculating the
corresponding characteristics from a random sample
estimates .
• the process of estimating the value of a parameter from
information obtained from a sample.
• Point estimation
• Interval estimation
167
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168. Point estimation
• A point estimate is a specific numerical value estimate of a
parameter.
• Sample measures (i.e., statistics) are used to estimate population
measures (i.e., parameters). These statistics are called estimators.
• Point estimate for population mean µ is
• Point estimate for population proportion is given by
• Where x is the total number of success (events) 168
n
x
=
x
n
1
=
i
i
n
=
p
x
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169. Three Properties of a Good Estimator
• The estimator should be an unbiased estimator. That is, the
expected value or the mean of the estimates obtained from
samples of a given size is equal to the parameter being
estimated.
• The estimator should be consistent. For a consistent
estimator, as sample size increases, the value of the estimator
approaches the value of the parameter estimated.
• The estimator should be a relatively efficient estimator. That
is, of all the statistics that can be used to estimate a parameter,
the relatively efficient estimator has the smallest variance.
169
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171. Confidence Interval estimate
• However the value of the sample statistic will vary from
sample to sample therefore, to simply obtain an estimate of
the single value of the parameter is not generally acceptable.
– We need also a measure of how precise our estimate is likely to be
– We need to take into account the sample to sample variation of the
statistic
• A confidence interval defines an interval within which the true
population parameter is like to fall (interval estimate).
171
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172. Confidence intervals…
• Confidence interval therefore takes into account the sample to sample variation of the
statistic and gives the measure of precision.
• The general formula used to calculate a Confidence interval is Estimate ± K × Standard
Error, k is called reliability coefficient
• Confidence intervals express the inherent uncertainty in any medical study by
expressing upper and lower bounds for anticipated true underlying population
parameter.
• The confidence level is the probability that the interval estimate will contain the
parameter, assuming that a large number of samples are selected and that the
estimation process on the same parameter is repeated
• Most commonly the 95% confidence intervals are calculated, however 90% and 99%
confidence intervals are sometimes used.
172
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178. Confidence intervals…
• The 95% confidence interval is calculated in such a way that,
under the conditions assumed for underlying distribution, the
interval will contain true population parameter 95% of the time.
• Loosely speaking, you might interpret a 95% confidence interval
as one which you are 95% confident contains the true parameter.
• 90% CI is narrower than 95% CI since we are only 90% certain that
the interval includes the population parameter.
• On the other hand 99% CI will be wider than 95% CI; the extra
width meaning that we can be more certain that the interval will
contain the population parameter. But to obtain a higher
confidence from the same sample, we must be willing to accept a
larger margin of error (a wider interval).
178
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179. Confidence intervals…
• For a given confidence level (i.e. 90%, 95%, 99%) the
width of the confidence interval depends on the
standard error of the estimate which in turn depends
on the
– 1. Sample size:-The larger the sample size, the narrower the
confidence interval (this is to mean the sample statistic will
approach the population parameter) and the more precise
our estimate. Lack of precision means that in repeated
sampling the values of the sample statistic are spread out or
scattered. The result of sampling is not repeatable.
179
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180. Confidence intervals…
- To increase precision (of an SRS), use a larger sample. You
can make the precision as high as you want by taking a
large enough sample. The margin of error decreases as√n
increases.
• 2. Standard deviation:-The more the variation among the
individual values, the wider the confidence interval and
the less precise the estimate. As sample size increases SD
decreases.
• Z is the value from SND
• 90% CI, z=1.64
• 95% CI, z=1.96
• 99% CI, z=2.58
180
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181. Confidence interval ……
• If the population standard deviation is unknown and
the sample size is small (<30), the formula for the
confidence interval for sample mean is: x ± t (s/√n)
– x is the sample mean
– s is the sample standard deviation
– n is the sample size
– t is the value from the t-distribution with (n-1) degrees of freedom
181
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182. Mean Example
A SRS of 16 apparently healthy subjects yielded the following values
of urine excreted (milligram per day);
0.007, 0.03, 0.025, 0.008, 0.03, 0.038, 0.007, 0.005, 0.032, 0.04,
0.009, 0.014, 0.011, 0.022, 0.009, 0.008
Compute point estimate of the population mean
Construct 90%, 95%, 98% confidence interval for the mean
(0.01844-1.65x0.0123/4, 0.01844+1.65x0.0123/4)=(0.0134, 0.0235)
(0.01844-1.96x0.0123/4, 0.01844+1.96x0.0123/4)=(0.0124, 0.0245)
(0.01844-2.33x0.0123/4, 0.01844+2.33x0.0123/4)=(0.0113, 0.0256)182
01844
.
0
16
295
.
0
n
x
=
x
then
,
values
observed
n
are
x
...,
,
x
,
x
If
n
1
=
i
i
n
2
1
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183. Proportion example
• In a survey of 300 automobile drivers in one city, 123 reported
that they wear seat belts regularly. Estimate the seat belt rate of
the city and 95% confidence interval for true population
proportion.
• Answer : p= 123/300 =0.41=41%
n=300,
Estimate of the seat belt of the city at 95%
CI = p ± z ×(√p(1-p) /n) =(0.35,0.47)
183
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184. Summary
• Students sometimes have difficulty deciding whether to use
Za/2 or t a/2 values when finding confidence intervals
184
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185. HYPOTHESIS TESTING
Introduction
– Researchers are interested in answering many types of questions. For example, A
physician might want to know whether a new medication will lower a person’s
blood pressure.
– These types of questions can be addressed through statistical hypothesis testing,
which is a decision-making process for evaluating claims about a population.
185
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186. Hypothesis Testing
• The formal process of hypothesis testing provides us with a
means of answering research questions.
• Hypothesis is a testable statement that describes the nature of
the proposed relationship between two or more variables of
interest.
• In hypothesis testing, the researcher must defined the
population under study, state the particular hypotheses that will
be investigated, give the significance level, select a sample from
the population, collect the data, perform the calculations
required for the statistical test, and reach a conclusion.
186
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188. type of Hypotheses
• Null hypothesis (represented by HO) is the statement about the value of the
population parameter. That is the null hypothesis postulates that ‘there is no
difference between factor and outcome’ or ‘there is no an intervention effect’.
• Alternative hypothesis (represented by HA) states the ‘opposing’ view that
‘there is a difference between factor and outcome’ or ‘there is an intervention
effect’.
188
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189. Methods of hypothesis testing
• Hypotheses concerning about parameters
which may or may not be true
• The three methods used to test hypotheses
are
• The traditional method
• The P-value method (New approach)
• The confidence interval method
189
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Tuesday, February 7, 2023
190. Steps in hypothesis testing
1
Identify the null hypothesis H0 and
the alternate hypothesis HA.
190
3
Select the test statistic and
determine its value from the sample
data. This value is called the
observed value of the test statistic.
Remember that t statistic is usually
appropriate for a small number of
samples; for larger number of
samples, a z statistic can work well if
data are normally distributed.
4
Compare the observed value of the statistic to
the critical value obtained for the chosen a.
5
Make a decision.
6
Conclusion
2
Choose a. The value should be small, usually less
than 10%. It is important to consider the
consequences of both types of errors.
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191. Test Statistics
Because of random variation, even an unbiased sample may not accurately
represent the population as a whole.
As a result, it is possible that any observed differences or associations may
have occurred by chance.
• A test statistics is a value we can compare with known distribution of what
we expect when the null hypothesis is true.
• The general formula of the test statistics is:
Observed _ Hypothesized
Test statistics = value value .
Standard error
• The known distributions are Normal distribution, student’s distribution , Chi-square
distribution ….
191
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Tuesday, February 7, 2023
192. Critical value
• The critical value separates the critical region from the noncritical region
for a given level of significance
192
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193. Decision making
• Accept or Reject the null hypothesis
• There are 2 types of errors
• Type I error is more serious error and it is the level of significant
• power is the probability of rejecting false null hypothesis and it is
given by 1-β
193
Type of decision H0 true H0 false
Reject H0 Type I error (a)
Correct decision (1-
β)
Accept H0
Correct decision (1-
a)
Type II error (β)
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197. H0: m m0
H1: m < m0
0
0
0
H0: m m0
H1: m > m0
H0: m m0
H1: m m0
/2
Critical
Value(s)
Rejection Regions
One tailed test
Two tailed test
Types of testes
Wullo S. 197
Tuesday, February 7, 2023
199. Steps in hypothesis testing…..
199
If the test statistic falls in the critical
region:
Reject H0 in favour of HA.
If the test statistic does not fall in the
critical region:
Conclude that there is not enough
evidence to reject H0.
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201. The P- Value
• In most applications, the outcome of performing a hypothesis
test is to produce a p-value.
• P-value is the probability that the observed difference is due to
chance.
• A large p-value implies that the probability of the value observed,
occurring just by chance is low, when the null hypothesis is true.
• That is, a small p-value suggests that there might be sufficient
evidence for rejecting the null hypothesis.
• The p value is defined as the probability of observing the
computed significance test value or a larger one, if the H0
hypothesis is true. For example, P[ Z >=Zcal/H0 true].
201
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202. P-value……
• A p-value is the probability of getting the
observed difference, or one more extreme, in
the sample purely by chance from a population
where the true difference is zero.
• If the p-value is greater than 0.05 then, by
convention, we conclude that the observed
difference could have occurred by chance and
there is no statistically significant evidence (at
the 5% level) for a difference between the
groups in the population.
202
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203. How to calculate P-value
o Use statistical software like SPSS, SAS……..
o Hand calculations
—obtained the test statistics (Z Calculated or t-
calculated)
—find the probability of test statistics from
standard normal table
—subtract the probability from 0.5
—the result is P-value
Note if the test two tailed multiply 2 the result.
Wullo S. 203
Tuesday, February 7, 2023
204. P-value and confidence interval
• Confidence intervals and p-values are based upon the same
theory and mathematics and will lead to the same conclusion
about whether a population difference exists.
• Confidence intervals are referable because they give information
about the size of any difference in the population, and they also
(very usefully) indicate the amount of uncertainty remaining
about the size of the difference.
• When the null hypothesis is rejected in a hypothesis-testing
situation, the confidence interval for the mean using the same
level of significance will not contain the hypothesized mean.
204
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205. The P- Value …..
• But for what values of p-value should we reject the null
hypothesis?
– By convention, a p-value of 0.05 or smaller is considered
sufficient evidence for rejecting the null hypothesis.
– By using p-value of 0.05, we are allowing a 5% chance of
wrongly rejecting the null hypothesis when it is in fact
true.
• When the p-value is less than to 0.05, we often say that the
result is statistically significant.
205
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207. Hypothesis testing for single population mean
• EXAMPLE 1: A researcher claims that the mean of the IQ for 16
students is 110 and the expected value for all population is 100
with standard deviation of 10. Test the hypothesis .
• Solution
1. Ho:µ=100 VS HA:µ≠100
2. Assume α=0.05
3. Test statistics: z=(110-100)4/10=4
4. z-critical at 0,025 is equal to 1.96.
5. Decision: reject the null hypothesis since 4 ≥ 1.96
6. Conclusion: the mean of the IQ for all population is different
from 100 at 5% level of significance.
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208. Hypothesis testing for single proportions
• Example: In the study of childhood abuse in psychiatry patients, brown found
that 166 in a sample of 947 patients reported histories of physical or sexual
abuse.
a) constructs 95% confidence interval
b) test the hypothesis that the true population proportion is 30%?
• Solution (a)
• The 95% CI for P is given by
208
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2
.
0
;
151
.
0
[
0124
.
0
96
.
1
175
.
0
947
825
.
0
175
.
0
96
.
1
175
.
0
)
1
(
2
n
p
p
z
p
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Tuesday, February 7, 2023
209. Example……
• To the hypothesis we need to follow the steps
Step 1: State the hypothesis
Ho: P=Po=0.3
Ha: P≠Po ≠0.3
Step 2: Fix the level of significant (α=0.05)
Step 3: Compute the calculated and tabulated value of the test statistic
209
96
.
1
39
.
8
0149
.
0
125
.
0
947
)
7
.
0
(
3
.
0
3
.
0
175
.
0
)
1
(
tab
cal
z
n
p
p
Po
p
z
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Tuesday, February 7, 2023
210. Example……
• Step 4: Comparison of the calculated and tabulated values of the
test statistic
• Since the tabulated value is smaller than the calculated value of
the test the we reject the null hypothesis.
• Step 6: Conclusion
• Hence we concluded that the proportion of childhood abuse in
psychiatry patients is different from 0.3
• If the sample size is small (if np<5 and n(1-p)<5) then use student’s
t- statistic for the tabulated value of the test statistic.
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211. Two sample mean and proportion
• Still now we have seen estimate for only single mean and single
proportion. However it is possible to compute point and interval
estimation for the difference of two sample means.
• let x1, x2, …, xn1 are samples from the first population and y1, y2,
…, yn2 be sample from the second population.
• Sample mean for the first population be
• Sample mean for the second population
• Then the point estimate for the difference of means (µ1-µ2) is
given by
211
)
( Y
X
Y
X
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Tuesday, February 7, 2023
212. Two sample estimation
• Confidence interval estimation
• A (1-α)100% confidence interval for the
difference of means is given by
• If are unknown, then can be estimated
by
212
2
2
2
1
2
1
2
)
(
n
n
z
y
x
2
1,
and
2
1, s
and
s
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Tuesday, February 7, 2023
213. Hypothesis testing for two sample means
• The steps to test the hypothesis for difference of means is the
same with the single mean
Step 1: state the hypothesis
Ho: µ1-µ2 =0
VS
HA: µ1-µ2 ≠0, HA: µ1-µ2 <0, HA: µ1-µ2 >0
Step 2: Significance level (α)
Step 3: Test statistic
213
2
2
2
1
2
1
2
1 )
(
)
(
n
n
y
x
zcal
m
m
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Tuesday, February 7, 2023
215. Small sample size and population variance is not given
• The test statistic will be student’s t-statistic with degree of
freedom equals to n1+n2 -2
• Hence the tabulated value of t is read from the table.
• The decision remains the same
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Tuesday, February 7, 2023
216. Example
• A researchers wish to know if the data they have collected provide
sufficient evidence to indicate a difference in mean serum uric
acid levels between normal individual and individual with down’s
syndrome. The data consists of serum uric acid readings on 12
individuals with down’s syndrome and 15 normal individuals. The
means are 4.5mg/100ml and 3.4 mg/100ml with standard
deviation of 2.9 and 3.5 mg/100ml respectively.
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0
:
0
:
2
1
2
1
m
m
m
m
A
O
H
H
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Tuesday, February 7, 2023
218. Estimation and hypothesis testing for two population proportion
• Let n1 and n2 be the sample size from the two population. If x and
y are the out come of interest then the point estimate for each
population is given by p1=x/n1 and p2=y/n2 respectively.
• The point estimates π1-π2 =p1-p2
• The interval estimate for the difference of proportions is given by
• If the sample size is large and n1p1>5, n1 (1-p1)>5, n2p2>5, then
218
2
2
2
1
1
1
2
2
1
)
1
(
)
1
(
n
p
p
n
p
p
z
p
p
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Tuesday, February 7, 2023
219. Hypothesis testing for two proportions
• To test the hypothesis
Ho: π1-π2 =0
VS
HA: π1-π2 ≠0
The test statistic is given by
219
2
2
2
1
1
1
2
1
2
1
)
1
(
)
1
(
)
(
)
(
n
p
p
n
p
p
p
p
zcal
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Tuesday, February 7, 2023
220. Small sample size
• If the sample size is small and n1p1>5, n2p2<5,
then use student’s t-statistic at n1+ n2-2
degrees of freedom with the given level of
significant.
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Tuesday, February 7, 2023
221. Chi-square test
• In recent years, the use of specialized statistical methods for
categorical data has increased dramatically, particularly for
applications in the biomedical and social sciences.
• Categorical scales occur frequently in the health sciences, for
measuring responses.
• E.g.
• patient survives an operation (yes, no),
• severity of an injury (none, mild, moderate, severe), and
• stage of a disease (initial, advanced).
• Studies often collect data on categorical variables that can be
summarized as a series of counts and commonly arranged in a
tabular format known as a contingency table
221
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222. Chi-square Test Statistic cont’d…
• As with the z and t distributions, there is a different chi-square
distribution for each possible value of degrees of freedom.
Chi-square distributions with a small number of degrees of freedom
are highly skewed; however, this skewness is attenuated as the
number of degrees of freedom increases.
The chi-squared distribution is concentrated over nonnegative
values. It has mean equal to its degrees of freedom (df), and its
standard deviation equals √(2df ). As df increases, the distribution
concentrates around larger values and is more spread out.
The distribution is skewed to the right, but it becomes more bell-
shaped (normal) as df increases.
222
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Tuesday, February 7, 2023
223. The degrees of freedom for tests of hypothesis that involve an rxc
contingency table is equal to (r-1)x(c-1);
223
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Tuesday, February 7, 2023
224. Test of Association
• The chi-squared (2) test statistics is widely used in the analysis
of contingency tables.
• It compares the actual observed frequency in each group with
the expected frequency (the later is based on theory,
experience or comparison groups).
• The chi-squared test (Pearson’s χ2) allows us to test for
association between categorical (nominal!) variables.
• The null hypothesis for this test is there is no association
between the variables. Consequently a significant p-value
implies association.
224
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Tuesday, February 7, 2023
225. Test Statistic: 2-test with d.f. = (r-1)x(c-1)
225
j
i ij
ij
ij
E
E
O
,
2
2
n
C
R
i
E
j
i
th
ij
total
grand
al
column tot
j
total
raw th
Oij=observed frequency, Eij=expected frequency of the cell at the
juncture of I th raw & j th column
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Tuesday, February 7, 2023
226. Procedures of Hypothesis Testing
1. State the hypothesis
2. Fix level of significance
3. Find the critical value (x2 (df, α))
4. Compute the test statistics
5. Decision rules; reject null hypothesis if test statistics > table
value.
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Tuesday, February 7, 2023
227. Test of associations for 2x2 tables
• If we call the frequencies in the four cells of 2x2 table a, b, c and d then
the table is given by
227
Disease status Row total
D ND
Exposur
e Status
E a b a+b
NE c d c+d
Column total a+c b+d N
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Tuesday, February 7, 2023
228. Test of association
• If the contingency table is 2x2
• Is the table is rxc then
228
)
)(
)(
)(
(
2
2
d
c
b
a
d
b
c
a
bc
ad
n
j
i ij
ij
ij
E
E
O
,
2
2
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Tuesday, February 7, 2023
229. Assumptions of the 2 - test
The chi-squared test assumes that
• Data must be categorical
• The data be a frequency data
– the numbers in each cell are ‘not too small’. No expected frequency should be
less than 1, and
– no more than 20% of the expected frequencies should be less than 5.
• If this does not hold row or column variables categories can
sometimes be combined (re-categorized) to make the expected
frequencies larger or use Yates continuity correction.
229
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Tuesday, February 7, 2023
230. Yates correction
• It is a requirement that a chi-squared test be applied to discrete data. Counting
numbers are appropriate, continuous measurements are not. Assuming continuity in
the underlying distribution distorts the p value and may make false positives more
likely.
• Frank Yates proposed a correction to the chi-squared formula. Adding a small negative
term to the argument. This tends to increase the p-value, and makes the test more
conservative, making false positives less likely. However, the test may now be *too*
conservative.
• Additionally, chi squared test should not be used when the observed values in a cell
are <5. It is, at times not inappropriate to pad an empty cell with a small value,
though, as one can only assume the result would be more significant with no value
there.
230
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Tuesday, February 7, 2023
231. Examples
Consider hypothetical example on smoking and symptoms of asthma. The study
involved 150 individuals and the result is given in the following table:
231
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Tuesday, February 7, 2023