This document provides an overview of key concepts in statistics for quantitative analysis, including:
- Statistics are mathematical tools used to describe and make judgments about data. The type of statistics discussed assumes data has a normal (bell-shaped) distribution.
- The normal distribution is characterized by a mean (μ) and standard deviation (σ or s). Standard deviation quantifies the spread of data around the mean.
- Common statistical tests covered include confidence intervals, comparing a measured value to a known value using a t-test, and comparing means of two data sets using an F-test and t-test.
- The F-test determines if the standard deviations of two data sets are significantly different before using
This document provides an overview of key concepts in statistics that will be covered in the CHM 235 course, including:
- The normal distribution and how it relates to sampling from populations. Parameters like the mean, standard deviation, and normal curve shape sampling distributions.
- Common statistical tests like confidence intervals, comparing a measured value to a known value, and comparing means of two data sets using t-tests. These tests rely on assumptions of normal distributions and comparing calculated t values to statistical tables.
- Additional concepts like variance, relative standard deviation, average deviation, and F-tests to compare standard deviations before applying t-tests. An example takes the reader through each of these statistical calculations and tests.
This document discusses determining appropriate sample sizes for quality control studies. It explains that sample sizes are determined based on acceptable levels of risk and variability in test results. Methods are presented for calculating sample sizes to estimate population means and standard deviations within defined confidence levels. Equations show how to determine the number of samples needed based on factors like the desired confidence interval, known standard deviation, and confidence level. Iterative calculations may be needed to precisely determine the sample size.
The document provides objectives and instructions for calculating standard deviation, variance, and student's t-test. It defines standard deviation as the positive square root of the arithmetic mean of the squared deviations from the mean. Standard deviation is considered the most reliable measure of variability. Variance is defined as the square of the standard deviation. Student's t-test is used to compare means of two samples and determine if they are statistically different. The document provides examples of calculating standard deviation, variance, and performing matched pairs and independent samples t-tests on sets of data.
This document discusses various statistical measures of dispersion and variation in data, including range, interquartile range, mean deviation, and standard deviation. It provides formulas and examples for calculating each measure. The range is defined as the difference between the highest and lowest values. The interquartile range is the difference between the first and third quartiles. Mean deviation measures the average distance from the mean. Standard deviation uses the differences from the mean to calculate the average deviation and is the most common measure of variation.
The partitioning of an ordered prognostic factor is important in order to obtain several groups having heterogeneous survivals in medical research. For this purpose, a binary split has often been used once or recursively. We propose the use of a multi-way split in order to afford an optimal set of cut-off points. In practice, the number of groups ($K$) may not be specified in advance. Thus, we also suggest finding an optimal $K$ by a resampling technique. The algorithm was implemented into an \proglang{R} package that we called \pkg{kaps}, which can be used conveniently and freely. It was illustrated with a toy dataset, and was also applied to a real data set of colorectal cancer cases from the Surveillance Epidemiology and End Results.
Common evaluation measures in NLP and IRRushdi Shams
This document discusses various evaluation measures used in information retrieval and natural language processing. It describes precision, recall, and the F1 score as fundamental measures for unranked retrieval sets. It also covers averaged precision and recall, accuracy, novelty and coverage ratios. For ranked retrieval sets, it discusses recall-precision graphs, interpolated recall-precision, precision at k, R-precision, ROC curves, and normalized discounted cumulative gain (NDCG). The document also discusses agreement measures like Kappa statistics and parses evaluation measures like Parseval and attachment scores.
This document provides an overview of key concepts in statistics that will be covered in the CHM 235 course, including:
- The normal distribution and how it relates to sampling from populations. Parameters like the mean, standard deviation, and normal curve shape sampling distributions.
- Common statistical tests like confidence intervals, comparing a measured value to a known value, and comparing means of two data sets using t-tests. These tests rely on assumptions of normal distributions and comparing calculated t values to statistical tables.
- Additional concepts like variance, relative standard deviation, average deviation, and F-tests to compare standard deviations before applying t-tests. An example takes the reader through each of these statistical calculations and tests.
This document discusses determining appropriate sample sizes for quality control studies. It explains that sample sizes are determined based on acceptable levels of risk and variability in test results. Methods are presented for calculating sample sizes to estimate population means and standard deviations within defined confidence levels. Equations show how to determine the number of samples needed based on factors like the desired confidence interval, known standard deviation, and confidence level. Iterative calculations may be needed to precisely determine the sample size.
The document provides objectives and instructions for calculating standard deviation, variance, and student's t-test. It defines standard deviation as the positive square root of the arithmetic mean of the squared deviations from the mean. Standard deviation is considered the most reliable measure of variability. Variance is defined as the square of the standard deviation. Student's t-test is used to compare means of two samples and determine if they are statistically different. The document provides examples of calculating standard deviation, variance, and performing matched pairs and independent samples t-tests on sets of data.
This document discusses various statistical measures of dispersion and variation in data, including range, interquartile range, mean deviation, and standard deviation. It provides formulas and examples for calculating each measure. The range is defined as the difference between the highest and lowest values. The interquartile range is the difference between the first and third quartiles. Mean deviation measures the average distance from the mean. Standard deviation uses the differences from the mean to calculate the average deviation and is the most common measure of variation.
The partitioning of an ordered prognostic factor is important in order to obtain several groups having heterogeneous survivals in medical research. For this purpose, a binary split has often been used once or recursively. We propose the use of a multi-way split in order to afford an optimal set of cut-off points. In practice, the number of groups ($K$) may not be specified in advance. Thus, we also suggest finding an optimal $K$ by a resampling technique. The algorithm was implemented into an \proglang{R} package that we called \pkg{kaps}, which can be used conveniently and freely. It was illustrated with a toy dataset, and was also applied to a real data set of colorectal cancer cases from the Surveillance Epidemiology and End Results.
Common evaluation measures in NLP and IRRushdi Shams
This document discusses various evaluation measures used in information retrieval and natural language processing. It describes precision, recall, and the F1 score as fundamental measures for unranked retrieval sets. It also covers averaged precision and recall, accuracy, novelty and coverage ratios. For ranked retrieval sets, it discusses recall-precision graphs, interpolated recall-precision, precision at k, R-precision, ROC curves, and normalized discounted cumulative gain (NDCG). The document also discusses agreement measures like Kappa statistics and parses evaluation measures like Parseval and attachment scores.
This document summarizes chapter 3 section 2 of an elementary statistics textbook. It discusses measures of variation, including range, variance, and standard deviation. The standard deviation describes how spread out data values are from the mean and is used to determine consistency and predictability within a specified interval. Several examples demonstrate calculating range, variance, and standard deviation for data sets. Chebyshev's theorem and the empirical rule relate standard deviations to the percentage of values that fall within certain intervals of the mean.
The document discusses factorial analysis of variance (ANOVA) and provides an example to illustrate the steps. It analyzes the flavor acceptability of luncheon meat from different sources. The null hypothesis is that there is no significant difference between the sources. The two-way ANOVA calculations show that the computed F-values are greater than the critical values, so the null hypothesis is rejected, indicating there are significant differences between the sources of luncheon meat.
Solution manual for design and analysis of experiments 9th edition douglas ...Salehkhanovic
Solution Manual for Design and Analysis of Experiments - 9th Edition
Author(s): Douglas C Montgomery
Solution manual for 9th edition include chapters 1 to 15. There is one PDF file for each of chapters.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 3: Describing, Exploring, and Comparing Data
3.1: Measures of Center
Applied statistics and probability for engineers solution montgomery && rungerAnkit Katiyar
This document is the copyright page and preface for the book "Applied Statistics and Probability for Engineers" by Douglas C. Montgomery and George C. Runger. The copyright is held by John Wiley & Sons, Inc. in 2003. This book was edited, designed, and produced by various teams at John Wiley & Sons and printed by Donnelley/Willard. The preface states that the purpose of the included Student Solutions Manual is to provide additional help for students in understanding the problem-solving processes presented in the main text.
This document discusses various measures of dispersion in statistics including range, mean deviation, variance, and standard deviation. It provides definitions and formulas for calculating each measure along with examples using both ungrouped and grouped frequency distribution data. Box-and-whisker plots are also introduced as a graphical method to display the five number summary of a data set including minimum, quartiles, and maximum values.
The document summarizes key concepts in describing data with numerical measures from a statistics textbook chapter. It covers measures of center including mean, median, and mode. It also covers measures of variability such as range, variance, and standard deviation. It provides examples of calculating these measures and interpreting them, as well as using them to construct box plots.
Measure of dispersion by Neeraj Bhandari ( Surkhet.Nepal )Neeraj Bhandari
This document discusses measures of dispersion used in statistics. It defines measures such as range, quartile deviation, mean deviation, variance, and standard deviation. It provides formulas to calculate these measures and examples showing how to apply the formulas. The key points are:
- Measures of dispersion quantify how spread out or varied the values in a data set are. They help identify variation, compare data sets, and enable other statistical techniques.
- Common absolute measures include range, quartile deviation, and mean deviation. Common relative measures include coefficient of range, coefficient of quartile deviation, and coefficient of variation.
- Variance and standard deviation are calculated using all data points. Variance is the average of squared deviations
This document provides an overview of measures of central tendency including the mean, median, and mode. It discusses how to calculate and interpret each measure using examples with data sets. The mean is calculated by adding all values and dividing by the total number. The median is the middle value when data is arranged in order. The mode is the value that occurs most frequently. Other measures discussed include the midrange and calculating the mean from a frequency distribution. Proper rounding of measures is also covered.
This document discusses six measures of variation used to determine how values are distributed in a data set: range, quartile deviation, mean deviation, variance, standard deviation, and coefficient of variation. It provides definitions and examples of calculating each measure. The range is defined as the difference between the highest and lowest values. Quartile deviation uses the interquartile range (Q3-Q1). Mean deviation is the average of the absolute deviations from the mean. Variance and standard deviation measure how spread out values are from the mean, with variance using sums of squares and standard deviation taking the square root of variance.
Measures of Dispersion: Standard Deviation and Co- efficient of Variation RekhaChoudhary24
This document discusses measures of dispersion, specifically standard deviation and coefficient of variation. It begins by defining standard deviation as a measure of how spread out numbers are from the mean. It then provides the formula for calculating standard deviation and discusses its properties. Several examples are shown to demonstrate calculating standard deviation for individual data series using both the direct and shortcut methods. The document also discusses calculating standard deviation for discrete and continuous data series. It concludes by defining variance and coefficient of variation, and providing an example to calculate coefficient of variation and determine which of two company's share prices is more stable.
Reweighting and Boosting to uniforimty in HEParogozhnikov
This document discusses using machine learning boosting techniques to achieve uniformity in particle physics applications. It introduces the uBoost and uGB+FL (gradient boosting with flatness loss) approaches, which aim to produce flat predictions along features of interest, like particle mass. This provides advantages over standard boosting by reducing non-uniformities that could create false signals. The document also proposes a non-uniformity measure and minimizing this with a flatness loss term during gradient boosting training. Examples applying these techniques to rare decay analysis, particle identification, and triggering are shown to achieve more uniform efficiencies than standard boosting.
Randomization tests provide an alternative to t-tests and F-tests that does not rely on assumptions of normality or random sampling. However, randomization tests can be too liberal or conservative depending on differences in sample sizes. Bootstrapping and Gill's algorithm can help address these issues. Bootstrapping resamples the larger sample to match the size of the smaller sample, controlling for liberal bias. Gill's algorithm uses Fourier expansion to efficiently calculate p-values from all permutations, reducing computational cost compared to full enumeration. However, conservative bias remains a challenge without a known solution.
Applied Statistics and Probability for Engineers 6th Edition Montgomery Solut...qyjewyvu
Full download : http://alibabadownload.com/product/applied-statistics-and-probability-for-engineers-6th-edition-montgomery-solutions-manual/
Applied Statistics and Probability for Engineers 6th Edition Montgomery Solutions Manual
Measure of dispersion has two types Absolute measure and Graphical measure. There are other different types in there.
In this slide the discussed points are:
1. Dispersion & it's types
2. Definition
3. Use
4. Merits
5. Demerits
6. Formula & math
7. Graph and pictures
8. Real life application.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.3: Estimating a Population Standard Deviation or Variance
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.3: Estimating a Population Standard Deviation or Variance
This document discusses various measures of dispersion used to quantify how spread out or varied values in a data set are. It defines dispersion as the difference or deviation of values from the central value. Measures of dispersion described include range, standard deviation, quartile deviation, mean deviation, variance, and coefficient of variation. Both absolute measures, which use numerical variations, and relative measures, which use statistical variations based on percentages, are examined. Relative measures allow for comparison between different data sets.
This document summarizes chapter 3 section 2 of an elementary statistics textbook. It discusses measures of variation, including range, variance, and standard deviation. The standard deviation describes how spread out data values are from the mean and is used to determine consistency and predictability within a specified interval. Several examples demonstrate calculating range, variance, and standard deviation for data sets. Chebyshev's theorem and the empirical rule relate standard deviations to the percentage of values that fall within certain intervals of the mean.
The document discusses factorial analysis of variance (ANOVA) and provides an example to illustrate the steps. It analyzes the flavor acceptability of luncheon meat from different sources. The null hypothesis is that there is no significant difference between the sources. The two-way ANOVA calculations show that the computed F-values are greater than the critical values, so the null hypothesis is rejected, indicating there are significant differences between the sources of luncheon meat.
Solution manual for design and analysis of experiments 9th edition douglas ...Salehkhanovic
Solution Manual for Design and Analysis of Experiments - 9th Edition
Author(s): Douglas C Montgomery
Solution manual for 9th edition include chapters 1 to 15. There is one PDF file for each of chapters.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 3: Describing, Exploring, and Comparing Data
3.1: Measures of Center
Applied statistics and probability for engineers solution montgomery && rungerAnkit Katiyar
This document is the copyright page and preface for the book "Applied Statistics and Probability for Engineers" by Douglas C. Montgomery and George C. Runger. The copyright is held by John Wiley & Sons, Inc. in 2003. This book was edited, designed, and produced by various teams at John Wiley & Sons and printed by Donnelley/Willard. The preface states that the purpose of the included Student Solutions Manual is to provide additional help for students in understanding the problem-solving processes presented in the main text.
This document discusses various measures of dispersion in statistics including range, mean deviation, variance, and standard deviation. It provides definitions and formulas for calculating each measure along with examples using both ungrouped and grouped frequency distribution data. Box-and-whisker plots are also introduced as a graphical method to display the five number summary of a data set including minimum, quartiles, and maximum values.
The document summarizes key concepts in describing data with numerical measures from a statistics textbook chapter. It covers measures of center including mean, median, and mode. It also covers measures of variability such as range, variance, and standard deviation. It provides examples of calculating these measures and interpreting them, as well as using them to construct box plots.
Measure of dispersion by Neeraj Bhandari ( Surkhet.Nepal )Neeraj Bhandari
This document discusses measures of dispersion used in statistics. It defines measures such as range, quartile deviation, mean deviation, variance, and standard deviation. It provides formulas to calculate these measures and examples showing how to apply the formulas. The key points are:
- Measures of dispersion quantify how spread out or varied the values in a data set are. They help identify variation, compare data sets, and enable other statistical techniques.
- Common absolute measures include range, quartile deviation, and mean deviation. Common relative measures include coefficient of range, coefficient of quartile deviation, and coefficient of variation.
- Variance and standard deviation are calculated using all data points. Variance is the average of squared deviations
This document provides an overview of measures of central tendency including the mean, median, and mode. It discusses how to calculate and interpret each measure using examples with data sets. The mean is calculated by adding all values and dividing by the total number. The median is the middle value when data is arranged in order. The mode is the value that occurs most frequently. Other measures discussed include the midrange and calculating the mean from a frequency distribution. Proper rounding of measures is also covered.
This document discusses six measures of variation used to determine how values are distributed in a data set: range, quartile deviation, mean deviation, variance, standard deviation, and coefficient of variation. It provides definitions and examples of calculating each measure. The range is defined as the difference between the highest and lowest values. Quartile deviation uses the interquartile range (Q3-Q1). Mean deviation is the average of the absolute deviations from the mean. Variance and standard deviation measure how spread out values are from the mean, with variance using sums of squares and standard deviation taking the square root of variance.
Measures of Dispersion: Standard Deviation and Co- efficient of Variation RekhaChoudhary24
This document discusses measures of dispersion, specifically standard deviation and coefficient of variation. It begins by defining standard deviation as a measure of how spread out numbers are from the mean. It then provides the formula for calculating standard deviation and discusses its properties. Several examples are shown to demonstrate calculating standard deviation for individual data series using both the direct and shortcut methods. The document also discusses calculating standard deviation for discrete and continuous data series. It concludes by defining variance and coefficient of variation, and providing an example to calculate coefficient of variation and determine which of two company's share prices is more stable.
Reweighting and Boosting to uniforimty in HEParogozhnikov
This document discusses using machine learning boosting techniques to achieve uniformity in particle physics applications. It introduces the uBoost and uGB+FL (gradient boosting with flatness loss) approaches, which aim to produce flat predictions along features of interest, like particle mass. This provides advantages over standard boosting by reducing non-uniformities that could create false signals. The document also proposes a non-uniformity measure and minimizing this with a flatness loss term during gradient boosting training. Examples applying these techniques to rare decay analysis, particle identification, and triggering are shown to achieve more uniform efficiencies than standard boosting.
Randomization tests provide an alternative to t-tests and F-tests that does not rely on assumptions of normality or random sampling. However, randomization tests can be too liberal or conservative depending on differences in sample sizes. Bootstrapping and Gill's algorithm can help address these issues. Bootstrapping resamples the larger sample to match the size of the smaller sample, controlling for liberal bias. Gill's algorithm uses Fourier expansion to efficiently calculate p-values from all permutations, reducing computational cost compared to full enumeration. However, conservative bias remains a challenge without a known solution.
Applied Statistics and Probability for Engineers 6th Edition Montgomery Solut...qyjewyvu
Full download : http://alibabadownload.com/product/applied-statistics-and-probability-for-engineers-6th-edition-montgomery-solutions-manual/
Applied Statistics and Probability for Engineers 6th Edition Montgomery Solutions Manual
Measure of dispersion has two types Absolute measure and Graphical measure. There are other different types in there.
In this slide the discussed points are:
1. Dispersion & it's types
2. Definition
3. Use
4. Merits
5. Demerits
6. Formula & math
7. Graph and pictures
8. Real life application.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.3: Estimating a Population Standard Deviation or Variance
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.3: Estimating a Population Standard Deviation or Variance
This document discusses various measures of dispersion used to quantify how spread out or varied values in a data set are. It defines dispersion as the difference or deviation of values from the central value. Measures of dispersion described include range, standard deviation, quartile deviation, mean deviation, variance, and coefficient of variation. Both absolute measures, which use numerical variations, and relative measures, which use statistical variations based on percentages, are examined. Relative measures allow for comparison between different data sets.
1) Statistics are important in analytical chemistry to objectively analyze experimental data, communicate significance, and optimize experimental design.
2) Key statistical terms include mean, median, population, sample, standard deviation, and accuracy vs precision.
3) Spreadsheet software can be used to calculate statistical values like standard deviation and perform regressions for calibration curves.
This document discusses key statistical concepts used in analytical chemistry, including accuracy, precision, standard deviation, probability distributions, and significance testing. It explains how statistics are applied to evaluate experimental data quality and validate analytical methods. Spreadsheets and linear regression are also summarized as tools for statistical data analysis.
1. Rentang
2. Rentang antarkuartil
3. Rentang semikuartil
4. Rentang a-b persentil
5. Simpangan baku
6. Variansi
7. Ukuran penyebaaran relatif
8. Bilangan baku
This document provides an overview of measures of relative standing and boxplots. It defines key terms like percentiles, quartiles, and outliers. Percentiles and quartiles divide a data set into groups based on the number of data points that fall below each value. The document also provides examples of calculating percentiles and quartiles for a data set of cell phone data speeds. Boxplots use the five-number summary (minimum, Q1, Q2, Q3, maximum) to visually depict a data set's center and spread through its quartiles and outliers.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 9: Inferences from Two Samples
9.2: Two Means, Independent Samples
This document discusses statistical analysis of experimental data. It outlines concepts like accuracy, precision, error, and data rejection. Accuracy refers to how close a measurement is to the true value, while precision refers to the agreement between multiple measurements. There are different types of errors like systematic and random errors. Various statistical metrics are presented for evaluating accuracy and precision, including mean, standard deviation, percent error and Gaussian distributions. Guidelines for identifying and rejecting outlier data points using the Q-test are also provided.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 3: Describing, Exploring, and Comparing Data
3.2: Measures of Variation
This document discusses estimating parameters and determining sample sizes from populations. It covers estimating population proportions, means, standard deviations, and variances. For each parameter, it describes how to construct confidence intervals and determine the necessary sample size. Formulas are provided for margin of error, t-scores, z-scores and the chi-square distribution, which is used for estimating variances and standard deviations. Examples show how to apply the concepts to find confidence intervals and critical values for specific population problems.
This document discusses parametric and non-parametric statistical tests. It begins by defining different types of data and the standard normal distribution curve. It then covers hypothesis testing, including the different types of errors. Both parametric and non-parametric tests are examined. Parametric tests discussed include z-tests, t-tests, and ANOVA, while non-parametric tests include chi-square, sign tests, McNemar's test, and Fischer's exact test. Examples are provided to illustrate several of the tests.
The document discusses different methods of estimating population parameters from sample data, including point and interval estimation. It provides formulas for calculating confidence intervals for a population mean when the standard deviation is known or unknown, and when the sample size is small or large. Examples are given to demonstrate how to construct 95% confidence intervals for a mean in different scenarios. The final example shows how to calculate the necessary sample size needed to estimate a population mean within a specified level of precision at a given confidence level.
This document discusses various statistical tests used to analyze dental research data, including parametric and non-parametric tests. It provides information on tests of significance such as the t-test, Z-test, analysis of variance (ANOVA), and non-parametric equivalents. Key points covered include the differences between parametric and non-parametric tests, assumptions and applications of the t-test, Z-test, ANOVA, and non-parametric alternatives like the Mann-Whitney U test and Kruskal-Wallis test. Examples are provided to illustrate how to perform and interpret common statistical analyses used in dental research.
This document provides information on chi-square tests and other statistical tests for qualitative data analysis. It discusses the chi-square test for goodness of fit and independence. It also covers Fisher's exact test and McNemar's test. Examples are provided to illustrate chi-square calculations and how to determine statistical significance based on degrees of freedom and critical values. Assumptions and criteria for applying different tests are outlined.
Dispersion- It is a statistical term that describes the size of the distribution of values expected for a particular variable and can be measured by several different statistics, such as Range, Variance and standard deviation.
Method of Dispersion-A measure of dispersion indicates the scattering of data. It explains the disparity of data from one another, delivering a precise view of their distribution.
Methods of Dispersion.
1.Relative Dispersion
a. Coefficient of Mean Deviation
b. Coefficient of Quartile Deviation
c. Coefficient of Range
d. Coefficient of Variation
2. Absolute Dispersion
a. Range
b. Quartile range
c. Standard deviation
d. Mean Deviation
Range- It is the difference between smallest & largest values in the dataset. Also the relative measure of range is known as Coefficient of Range.
Advantages and disadvantages of Range.
Calculation of Range by different Methods.
b. Quartile Range- The interquartile range of a group of observations is the interval between the values of upper quartile and the lower quartile for that group.
Advantages and Disadvantages of Quartile Range.
Calculation of Quartile Range by different Methods.
c. Standard Deviation- It measures the absolute dispersion (or) variability of a distribution. A small standard deviation means a high degree of uniformity of the observations as well as homogeneity in the series.
Advantages and Disadvantages of Quartile Range.
Calculation of Standard Deviation using.
i) Direct Method
ii) Short-cut Method
iii) Step Deviation Method.
This document discusses various measures of dispersion and variation in statistical data. It defines absolute and relative measures of dispersion, and describes common measures like range, interquartile range, mean deviation, variance, and standard deviation. These measures express the degree of variation or spread in a data set through a single number. The standard deviation is particularly useful because it is measured in the same units as the original data. The coefficient of variation allows comparison of variability between data sets with different units or scales by expressing variation relative to the mean. Understanding measures of dispersion is important as they provide a more complete picture of a data set than the mean alone.
OBJECTIVES:
Run the test of hypothesis for mean difference using paired samples. Construct a confidence interval for the difference in population means using paired samples.
Observation of interest will be the difference in the readings
before and after intervention called paired difference observation.
Paired t test:
A paired t-test is used to compare two means where you have two samples in which observations in one sample can be paired with observations in the other sample.
Examples of where this might occur are:
Before-and-after observations on the same subjects (e.g. students’ test
results before and after a particular module or course).
A comparison of two different methods of measurement or two different treatments where the measurements/treatments are applied to the same subjects (e.g. blood pressure measurements using a sphygmomanometer and a dynamap).
When there is a relationship between the groups, such as identical twins.
This test is concerned with the pair-wise differences
between sets of data.
This means that each data point in one group has a related data point in the other group (groups always have equal numbers).
ASSUMPTIONS:
The sample or samples are randomly selected
The sample data are dependent
The distribution of differences is approximately normally
distributed.
Note: The under root is onto the entire numerator and denominator, so you should take the root after solving it entirely
where “t” has (n-1) degrees of freedom and “n” is
the total number of pairs.
Statistical Process Control (SPC) uses statistical methods like control charts to monitor and control processes by distinguishing between common and assignable causes of variation, with the goal of keeping processes stable and within specification limits through the detection and elimination of assignable causes. SPC analyzes variables and attributes data through techniques such as x-charts, R-charts, and p-charts to measure factors like the central tendency and dispersion of a process. Process capability analysis compares the natural variation in a process to specification limits to determine if the process is capable of consistently meeting customer requirements.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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Training: ISO/IEC 27001 Information Security Management System - EN | PECB
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General Data Protection Regulation (GDPR) - Training Courses - EN | PECB
Webinars: https://pecb.com/webinars
Article: https://pecb.com/article
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Leveraging Generative AI to Drive Nonprofit InnovationTechSoup
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Your Skill Boost Masterclass: Strategies for Effective Upskilling
Statistics
1. Statistics for Quantitative Analysis
• Statistics: Set of mathematical tools used to describe
and make judgments about data
• Type of statistics we will talk about in this class has
important assumption associated with it:
Experimental variation in the population from which samples
are drawn has a normal (Gaussian, bell-shaped) distribution.
- Parametric vs. non-parametric statistics
CHM 235 – Dr. Skrabal
2. Normal distribution
• Infinite members of group:
population
• Characterize population by taking
samples
• The larger the number of samples,
the closer the distribution becomes to
normal
• Equation of normal distribution:
22
2/)(
2
1 σµ
πσ
−−
= x
ey
3. Normal distribution
• Estimate of mean value
of population = µ
• Estimate of mean value
of samples =
Mean =
x
n
x
x i
i∑
=
4. Normal distribution
• Degree of scatter (measure of
central tendency) of population is
quantified by calculating the
standard deviation
• Std. dev. of population = σ
• Std. dev. of sample = s
• Characterize sample by calculating
1
)( 2
−
−∑
=
n
xx
s
i
i
sx ±
5. Standard deviation and the
normal distribution
• Standard deviation
defines the shape of the
normal distribution
(particularly width)
• Larger std. dev. means
more scatter about the
mean, worse precision.
• Smaller std. dev. means
less scatter about the
mean, better precision.
6. Standard deviation and the
normal distribution
• There is a well-defined relationship
between the std. dev. of a population
and the normal distribution of the
population:
∀ µ ± 1σ encompasses 68.3 % of
measurements
∀ µ ± 2σ encompasses 95.5% of
measurements
∀ µ ± 3σ encompasses 99.7% of
measurements
• (May also consider these
percentages of area under the curve)
7. Example of mean and standard
deviation calculation
Consider Cu data: 5.23, 5.79, 6.21, 5.88, 6.02 nM
= 5.826nM 5.82 nM
s = 0.368 nM 0.36 nM
Answer: 5.82 ± 0.36 nM or 5.8 ± 0.4 nM
Learn how to use the statistical functions on your
calculator. Do this example by longhand calculation
once, and also by calculator to verify that you’ll get
exactly the same answer. Then use your calculator
for all future calculations.
x
8. Relative standard deviation (rsd)
or coefficient of variation (CV)
rsd or CV =
From previous example,
rsd = (0.36 nM/5.82 nM) 100 = 6.1% or 6%
100
x
s
9. Standard error
• Tells us that standard deviation of set of samples should
decrease if we take more measurements
• Standard error =
• Take twice as many measurements, s decreases by
• Take 4x as many measurements, s decreases by
• There are several quantitative ways to determine the sample
size required to achieve a desired precision for various statistical
applications. Can consult statistics textbooks for further
information; e.g. J.H. Zar, Biostatistical Analysis
n
s
sx =
4.12 ≈
24 =
10. Variance
Used in many other statistical calculations and tests
Variance = s2
From previous example, s = 0.36
s2
= (0.36)2
= 0. 129 (not rounded because it is usually
used in further calculations)
11. Average deviation
• Another way to express
degree of scatter or
uncertainty in data. Not as
statistically meaningful as
standard deviation, but
useful for small samples.
Using previous data:
n
xx
d i
i∑ −
=
)(
5
8.502.68.588.58.521.68.579.58.523.5 22222 −+−+−+−+−
=d
nMord 2.02.025.0 5→=
nMornMAnswer 2.08.52.08.5: 52 ±±
12. Relative average deviation (RAD)
Using previous data,
RAD = (0. 25/5.82) 100 = 4.2 or 4%
RAD = (0. 25/5.82) 1000 = 42 ppt
4.2 x 101
or 4 x 101
ppt (0
/00)
),(1000
)(100
pptthousandperpartsas
x
d
RAD
percentageas
x
d
RAD
=
=
13. Some useful statistical tests
• To characterize or make judgments about data
• Tests that use the Student’s t distribution
– Confidence intervals
– Comparing a measured result with a “known”
value
– Comparing replicate measurements (comparison
of means of two sets of data)
15. Confidence intervals
• Quantifies how far the true mean (µ) lies from the
measured mean, . Uses the mean and standard
deviation of the sample.
where t is from the t-table and n = number of
measurements.
Degrees of freedom (df) = n - 1 for the CI.
n
ts
x ±=µ
x
16. Example of calculating a
confidence interval
Consider measurement of dissolved Ti
in a standard seawater (NASS-3):
Data: 1.34, 1.15, 1.28, 1.18, 1.33, 1.65,
1.48 nM
DF = n – 1 = 7 – 1 = 6
= 1.34 nM or 1.3 nM
s = 0.17 or 0.2 nM
95% confidence interval
t(df=6,95%) = 2.447
CI95 = 1.3 ± 0.16 or 1.3 ± 0.2 nM
50% confidence interval
t(df=6,50%) = 0.718
CI50 = 1.3 ± 0.05 nM
x
n
ts
x ±=µ
17. Interpreting the confidence interval
• For a 95% CI, there is a 95% probability that the true
mean (µ) lies between the range 1.3 ± 0.2 nM, or
between 1.1 and 1.5 nM
• For a 50% CI, there is a 50% probability that the true
mean lies between the range 1.3± 0.05 nM, or
between 1.25 and 1.35 nM
• Note that CI will decrease as n is increased
• Useful for characterizing data that are regularly
obtained; e.g., quality assurance, quality control
18. Comparing a measured result
with a “known” value
• “Known” value would typically be a certified value
from a standard reference material (SRM)
• Another application of the t statistic
Will compare tcalc to tabulated value of t at appropriate df
and CL.
df = n -1 for this test
n
s
xvalueknown
tcalc
−
=
19. Comparing a measured result
with a “known” value--example
Dissolved Fe analysis verified using NASS-3 seawater SRM
Certified value = 5.85 nM
Experimental results: 5.76 ± 0.17 nM (n = 10)
(Keep 3 decimal places for comparison to table.)
Compare to ttable; df = 10 - 1 = 9, 95% CL
ttable(df=9,95% CL) = 2.262
If |tcalc| < ttable, results are not significantly different at the 95% CL.
If |tcalc| ≥ ttable, results are significantly different at the 95% CL.
For this example, tcalc < ttest, so experimental results are not significantly
different at the 95% CL
674.110
1.0
7.585.5
7
6
=
−
=
−
= n
s
xvalueknown
tcalc
20. Comparing replicate measurements or
comparing means of two sets of data
• Yet another application of the t statistic
• Example: Given the same sample analyzed by two
different methods, do the two methods give the “same”
result?
Will compare tcalc to tabulated value of t at appropriate df
and CL.
df = n1 + n2 – 2 for this test
21
2121
nn
nn
s
xx
t
pooled
calc
+
−
=
2
)1()1(
21
2
2
21
2
1
−+
−+−
=
nn
nsns
spooled
21. Comparing replicate measurements
or comparing means of two sets of
data—example
Method 1: Atomic absorption
spectroscopy
Data: 3.91, 4.02, 3.86, 3.99 mg/g
= 3.945 mg/g
= 0.073 mg/g
= 4
Method 2: Spectrophotometry
Data: 3.52, 3.77, 3.49, 3.59 mg/g
= 3.59 mg/g
= 0.12 mg/g
= 4
1x
Determination of nickel in sewage
sludge
using two different methods
2x
1s 2s
1n 2n
22. Comparing replicate measurements or
comparing means of two sets of data—
example
0993.0
244
)14()1.0()14()07.0(
2
)1()1( 2
2
2
3
21
2
2
21
2
1
=
−+
−+−
=
−+
−+−
=
nn
nsns
spooled
056.5
44
)4)(4(
0993.0
5.394.3 95
21
2121
=
+
−
=
+
−
=
nn
nn
s
xx
t
pooled
calc
Note: Keep 3 decimal places to compare to ttable.
Compare to ttable at df = 4 + 4 – 2 = 6 and 95% CL.
ttable(df=6,95% CL) = 2.447
If |tcalc| < ttable, results are not significantly different at the 95%. CL.
If |tcalc| ≥ ttable, results are significantly different at the 95% CL.
Since |tcalc| (5.056) ≥ ttable (2.447), results from the two methods are
significantly different at the 95% CL.
23. Comparing replicate measurements or
comparing means of two sets of data
Wait a minute! There is an important assumption
associated with this t-test:
It is assumed that the standard deviations (i.e., the
precision) of the two sets of data being compared are not
significantly different.
• How do you test to see if the two std. devs. are different?
• How do you compare two sets of data whose std. devs.
are significantly different?
24. F-test to compare standard
deviations
• Used to determine if std. devs. are significantly
different before application of t-test to compare
replicate measurements or compare means of two
sets of data
• Also used as a simple general test to compare the
precision (as measured by the std. devs.) of two sets
of data
• Uses F distribution
25. F-test to compare standard
deviations
Will compute Fcalc and compare to Ftable.
DF = n1 - 1 and n2 - 1 for this test.
Choose confidence level (95% is a typical CL).
212
2
2
1
sswhere
s
s
Fcalc >=
27. F-test to compare standard deviations
From previous example:
Let s1 = 0.12 and s2 = 0.073
Note: Keep 2 or 3 decimal places to compare with Ftable.
Compare Fcalc to Ftable at df = (n1 -1, n2 -1) = 3,3 and 95% CL.
If Fcalc < Ftable, std. devs. are not significantly different at 95% CL.
If Fcalc ≥ Ftable, std. devs. are significantly different at 95% CL.
Ftable(df=3,3;95% CL) = 9.28
Since Fcalc (2.70) < Ftable (9.28), std. devs. of the two sets of data are
not significantly different at the 95% CL. (Precisions are
similar.)
70.2
)07.0(
)1.0(
2
3
2
2
2
2
2
1
===
s
s
Fcalc
28. Comparing replicate measurements or
comparing means of two sets of data--
revisited
The use of the t-test for comparing means was justified
for the previous example because we showed that
standard deviations of the two sets of data were not
significantly different.
If the F-test shows that std. devs. of two sets of data
are significantly different and you need to compare
the means, use a different version of the t-test
29. Comparing replicate measurements or
comparing means from two sets of data
when std. devs. are significantly
different
2
1
)/(
1
)/(
)//(
//
2
2
2
2
2
1
2
1
2
1
2
2
2
21
2
1
2
2
21
2
1
21
−
+
+
+
+
=
+
−
=
n
ns
n
ns
nsns
DF
nsns
xx
tcalc
30. Flowchart for comparing means of two
sets of data or replicate
measurements
Use F-test to see if std.
devs. of the 2 sets of
data are significantly
different or not
Std. devs. are
significantly different
Std. devs. are not
significantly different
Use the 2nd
version
of the t-test (the
beastly version)
Use the 1st
version of the
t-test (see previous, fully
worked-out example)
31. One last comment on the F-test
Note that the F-test can be used to simply test whether
or not two sets of data have statistically similar
precisions or not.
Can use to answer a question such as: Do method one
and method two provide similar precisions for the
analysis of the same analyte?
32. Evaluating questionable data
points using the Q-test
• Need a way to test questionable data points (outliers) in an
unbiased way.
• Q-test is a common method to do this.
• Requires 4 or more data points to apply.
Calculate Qcalc and compare to Qtable
Qcalc = gap/range
Gap = (difference between questionable data pt. and its
nearest neighbor)
Range = (largest data point – smallest data point)
33. Evaluating questionable data
points using the Q-test--example
Consider set of data; Cu values in sewage sample:
9.52, 10.7, 13.1, 9.71, 10.3, 9.99 mg/L
Arrange data in increasing or decreasing order:
9.52, 9.71, 9.99, 10.3, 10.7, 13.1
The questionable data point (outlier) is 13.1
Calculate
Compare Qcalc to Qtable for n observations and desired CL (90% or
95% is typical). It is desirable to keep 2-3 decimal places in Qcalc
so judgment from table can be made.
Qtable (n=6,90% CL) = 0.56
670.0
)52.91.13(
)7.101.13(
=
−
−
==
range
gap
Qcalc
35. Evaluating questionable data points
using the Q-test--example
If Qcalc < Qtable, do not reject questionable data point at stated CL.
If Qcalc ≥ Qtable, reject questionable data point at stated CL.
From previous example,
Qcalc (0.670) > Qtable (0.56), so reject data point at 90% CL.
Subsequent calculations (e.g., mean and standard deviation)
should then exclude the rejected point.
Mean and std. dev. of remaining data: 10.04 ± 0.47 mg/L