The document discusses various measures of central tendency including the mean, median, and mode. It provides definitions and examples of how to calculate each measure. It also discusses how transformations of data and outliers can affect measures of central tendency, with the mean being most impacted by outliers and the mode least impacted. The median and mode are considered more resistant measures of central tendency than the mean.
This document discusses various measures of central tendency including arithmetic mean, median, mode, geometric mean, and harmonic mean. It provides definitions and formulas for calculating each measure, along with examples to illustrate how to calculate the median, mode, geometric mean, and harmonic mean. The document is intended as a guide for understanding and calculating different types of averages and measures of central tendency.
This document discusses statistical inference, which involves drawing conclusions about an unknown population based on a sample. There are two main types of statistical inference: parameter estimation and hypothesis testing. Parameter estimation involves obtaining numerical values of population parameters from a sample, like estimating the percentage of people aware of a product. Hypothesis testing involves making judgments about assumptions regarding population parameters based on sample data. The document also discusses point estimation, interval estimation, standard error, and provides examples of calculating confidence intervals.
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.
Introduction to Statistics - Basic concepts
- How to be a good doctor - A step in Health promotion
- By Ibrahim A. Abdelhaleem - Zagazig Medical Research Society (ZMRS)
Types of Statistics Descriptive and Inferential StatisticsDr. Amjad Ali Arain
Topic: Types of Statistics Descriptive and Inferential Statistics
Student Name: Bushra
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 overview of key concepts in statistics including:
- Statistics involves collecting, organizing, analyzing, and interpreting numerical data.
- There are two main types of statistics: descriptive and inferential.
- Data can be categorical or quantitative. Common measures of central tendency are the mean, median, and mode.
- There are different sampling methods like random, stratified, and cluster sampling.
- Data is often organized and displayed using tables, graphs like histograms, bar charts and pie charts.
This document defines and provides the formula for calculating mean deviation, which is a measure of variation that uses all the scores in a distribution. It is more reliable than range. Mean deviation is calculated by finding the absolute difference between each score and the mean, summing the absolute differences, and dividing by the number of observations. Two examples of calculating mean deviation for sets of data are provided, along with exercises asking students to find the mean deviation of additional data sets and define standard deviation.
The document discusses various measures of central tendency used in statistics. The three most common measures are the mean, median, and mode. The mean is the sum of all values divided by the number of values and is affected by outliers. The median is the middle value when data is arranged from lowest to highest. The mode is the most frequently occurring value in a data set. Each measure has advantages and disadvantages depending on the type of data distribution. The mean is the most reliable while the mode can be undefined. In symmetrical distributions, the mean, median and mode are equal, but the mean is higher than the median for positively skewed data and lower for negatively skewed data.
This document discusses various measures of central tendency including arithmetic mean, median, mode, geometric mean, and harmonic mean. It provides definitions and formulas for calculating each measure, along with examples to illustrate how to calculate the median, mode, geometric mean, and harmonic mean. The document is intended as a guide for understanding and calculating different types of averages and measures of central tendency.
This document discusses statistical inference, which involves drawing conclusions about an unknown population based on a sample. There are two main types of statistical inference: parameter estimation and hypothesis testing. Parameter estimation involves obtaining numerical values of population parameters from a sample, like estimating the percentage of people aware of a product. Hypothesis testing involves making judgments about assumptions regarding population parameters based on sample data. The document also discusses point estimation, interval estimation, standard error, and provides examples of calculating confidence intervals.
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.
Introduction to Statistics - Basic concepts
- How to be a good doctor - A step in Health promotion
- By Ibrahim A. Abdelhaleem - Zagazig Medical Research Society (ZMRS)
Types of Statistics Descriptive and Inferential StatisticsDr. Amjad Ali Arain
Topic: Types of Statistics Descriptive and Inferential Statistics
Student Name: Bushra
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 overview of key concepts in statistics including:
- Statistics involves collecting, organizing, analyzing, and interpreting numerical data.
- There are two main types of statistics: descriptive and inferential.
- Data can be categorical or quantitative. Common measures of central tendency are the mean, median, and mode.
- There are different sampling methods like random, stratified, and cluster sampling.
- Data is often organized and displayed using tables, graphs like histograms, bar charts and pie charts.
This document defines and provides the formula for calculating mean deviation, which is a measure of variation that uses all the scores in a distribution. It is more reliable than range. Mean deviation is calculated by finding the absolute difference between each score and the mean, summing the absolute differences, and dividing by the number of observations. Two examples of calculating mean deviation for sets of data are provided, along with exercises asking students to find the mean deviation of additional data sets and define standard deviation.
The document discusses various measures of central tendency used in statistics. The three most common measures are the mean, median, and mode. The mean is the sum of all values divided by the number of values and is affected by outliers. The median is the middle value when data is arranged from lowest to highest. The mode is the most frequently occurring value in a data set. Each measure has advantages and disadvantages depending on the type of data distribution. The mean is the most reliable while the mode can be undefined. In symmetrical distributions, the mean, median and mode are equal, but the mean is higher than the median for positively skewed data and lower for negatively skewed data.
Descriptive statistics is used to describe and summarize key characteristics of a data set. Commonly used measures include central tendency, such as the mean, median, and mode, and measures of dispersion like range, interquartile range, standard deviation, and variance. The mean is the average value calculated by summing all values and dividing by the number of values. The median is the middle value when data is arranged in order. The mode is the most frequently occurring value. Measures of dispersion describe how spread out the data is, such as the difference between highest and lowest values (range) or how close values are to the average (standard deviation).
This document discusses measures of central tendency, including the mean, median, and mode. It provides examples of calculating each measure using sample data sets. The mean is the average value calculated by summing all values and dividing by the number of data points. The median is the middle value when data is ordered from lowest to highest. The mode is the most frequently occurring value. Examples are given to demonstrate calculating the mean, median, and mode from sets of numeric data.
This document discusses variance and standard deviation. It defines variance as the average squared deviation from the mean of a data set. Standard deviation measures how spread out numbers are from the mean and is calculated by taking the square root of the variance. The document provides step-by-step instructions for calculating both variance and standard deviation, including examples using test score data.
This document discusses multivariate analysis (MVA), which involves observing and analyzing multiple outcome variables simultaneously. It describes key components of MVA like variates, measurement scales, and statistical significance. Various MVA techniques are explained, including cross correlations, single-equation models, vector autoregressions, and cointegration. An example using crime rate data from US states is provided. Applications of MVA in fields like marketing, quality control, process optimization, and research are also mentioned.
Descriptive statistics are methods of describing the characteristics of a data set. It includes calculating things such as the average of the data, its spread and the shape it produces.
Descriptive statistics are used to describe and summarize the basic features of data through measures of central tendency like the mean, median, and mode, and measures of variability like range, variance and standard deviation. The mean is the average value and is best for continuous, non-skewed data. The median is less affected by outliers and is best for skewed or ordinal data. The mode is the most frequent value and is used for categorical data. Measures of variability describe how spread out the data is, with higher values indicating more dispersion.
This document provides an overview of data analysis and statistics concepts for a training session. It begins with an agenda outlining topics like descriptive statistics, inferential statistics, and independent vs dependent samples. Descriptive statistics concepts covered include measures of central tendency (mean, median, mode), measures of variability (range, standard deviation), and charts. Inferential statistics discusses estimating population parameters, hypothesis testing, and statistical tests like t-tests, ANOVA, and chi-squared. The document provides examples and online simulation tools. It concludes with some practical tips for data analysis like checking for errors, reviewing findings early, and consulting a statistician on analysis plans.
The document provides an overview of inferential statistics. It defines inferential statistics as making generalizations about a larger population based on a sample. Key topics covered include hypothesis testing, types of hypotheses, significance tests, critical values, p-values, confidence intervals, z-tests, t-tests, ANOVA, chi-square tests, correlation, and linear regression. The document aims to explain these statistical concepts and techniques at a high level.
Measures of dispersion
Absolute measure, relative measures
Range of Coe. of Range
Mean deviation and coe. of mean deviation
Quartile deviation IQR, coefficient of QD
Standard deviation and coefficient of variation
This document defines key terms and concepts related to standard deviation and variance. It provides formulas for calculating range, deviation, variance, and standard deviation for both ungrouped and grouped data. Examples are given to demonstrate calculating these metrics from raw data sets and grouped data tables. Interpreting skewness is also discussed.
This presentation discusses frequency distribution graphs. It provides examples of constructing frequency distribution tables and calculating class midpoints and cumulative frequencies. The key graphs for representing frequency distributions are described as histograms, frequency polygons, and cumulative frequency curves for continuous or quantitative data. Bar charts, line graphs and pie charts are also introduced as options to display grouped or ungrouped categorical data. Examples of each type of graph are included.
The document defines properties of arithmetic mean and provides examples to illustrate these properties. The key properties are: 1) The sum of deviations from the mean is zero. 2) The sum of squares of deviations from the mean is minimum. 3) The mean is unaffected by a constant change of scale or origin. 4) The mean of multiple data sets can be calculated as a weighted average based on sample sizes. The examples demonstrate calculating means and verifying properties for various data sets.
This document introduces key concepts in probability:
- Probability is the likelihood of an event occurring, which can be measured numerically or described qualitatively.
- Events can be classified as exhaustive, favorable, mutually exclusive, equally likely, complementary, and independent.
- There are three approaches to defining probability: classical, frequency, and axiomatic. The classical approach defines probability as the number of favorable outcomes over the total number of possible outcomes. The frequency approach defines probability as the limit of the ratio of favorable outcomes to the total number of trials. The axiomatic approach defines probability based on axioms or statements assumed to be true.
- Key properties of probability include that the probability of an event is between 0
Analysis of covariance (ANCOVA) is a statistical test that assesses whether the means of a dependent variable are equal across levels of a categorical independent variable while statistically controlling for the effects of other continuous variables known as covariates. ANCOVA works by adjusting the sums of squares for the independent variable to remove the influence of the covariate. This allows ANCOVA to test for differences between groups while controlling for the influence of other continuous variables. The assumptions of ANCOVA include those of ANOVA as well as the assumptions that the relationship between the dependent variable and covariate is linear and the same across all groups.
01 parametric and non parametric statisticsVasant Kothari
Definition of Parametric and Non-parametric Statistics
Assumptions of Parametric and Non-parametric Statistics
Assumptions of Parametric Statistics
Assumptions of Non-parametric Statistics
Advantages of Non-parametric Statistics
Disadvantages of Non-parametric Statistical Tests
Parametric Statistical Tests for Different Samples
Parametric Statistical Measures for Calculating the Difference Between Means
Significance of Difference Between the Means of Two Independent Large and
Small Samples
Significance of the Difference Between the Means of Two Dependent Samples
Significance of the Difference Between the Means of Three or More Samples
Parametric Statistics Measures Related to Pearson’s ‘r’
Non-parametric Tests Used for Inference
This document provides an overview of measures of dispersion, including range, quartile deviation, mean deviation, standard deviation, and variance. It defines dispersion as a measure of how scattered data values are around a central value like the mean. Different measures of dispersion are described and formulas are provided. The standard deviation is identified as the most useful measure as it considers all data values and is not overly influenced by outliers. Examples are included to demonstrate calculating measures of dispersion.
This document discusses different types of statistics used in research. Descriptive statistics are used to organize and summarize data using tables, graphs, and measures. Inferential statistics allow inferences about populations based on samples through techniques like surveys and polls. The key difference is that descriptive statistics describe samples while inferential statistics allow conclusions about populations beyond the current data.
Topic: Frequency Distribution
Student Name: Abdul Hafeez
Class: B.Ed. (Hons) Elementary
Project Name: “Young Teachers' Professional Development (TPD)"
"Project Founder: Prof. Dr. Amjad Ali Arain
Faculty of Education, University of Sindh, Pakistan
This document discusses measures of central tendency including the mean, median, and mode. It provides examples of calculating these values for different data sets. It also discusses using the mean to balance a seesaw with blocks and the concept of deviations from the mean. The homework is to finish the class notes on these topics.
The document discusses various measures of central tendency and dispersion. It defines the arithmetic mean, weighted mean, geometric mean, median, and mode as measures of central tendency. It also discusses calculating these measures from grouped data. For measures of dispersion, it covers range, interquartile range, variance, and standard deviation. It provides formulas to calculate these statistics for both population and sample data.
Descriptive statistics is used to describe and summarize key characteristics of a data set. Commonly used measures include central tendency, such as the mean, median, and mode, and measures of dispersion like range, interquartile range, standard deviation, and variance. The mean is the average value calculated by summing all values and dividing by the number of values. The median is the middle value when data is arranged in order. The mode is the most frequently occurring value. Measures of dispersion describe how spread out the data is, such as the difference between highest and lowest values (range) or how close values are to the average (standard deviation).
This document discusses measures of central tendency, including the mean, median, and mode. It provides examples of calculating each measure using sample data sets. The mean is the average value calculated by summing all values and dividing by the number of data points. The median is the middle value when data is ordered from lowest to highest. The mode is the most frequently occurring value. Examples are given to demonstrate calculating the mean, median, and mode from sets of numeric data.
This document discusses variance and standard deviation. It defines variance as the average squared deviation from the mean of a data set. Standard deviation measures how spread out numbers are from the mean and is calculated by taking the square root of the variance. The document provides step-by-step instructions for calculating both variance and standard deviation, including examples using test score data.
This document discusses multivariate analysis (MVA), which involves observing and analyzing multiple outcome variables simultaneously. It describes key components of MVA like variates, measurement scales, and statistical significance. Various MVA techniques are explained, including cross correlations, single-equation models, vector autoregressions, and cointegration. An example using crime rate data from US states is provided. Applications of MVA in fields like marketing, quality control, process optimization, and research are also mentioned.
Descriptive statistics are methods of describing the characteristics of a data set. It includes calculating things such as the average of the data, its spread and the shape it produces.
Descriptive statistics are used to describe and summarize the basic features of data through measures of central tendency like the mean, median, and mode, and measures of variability like range, variance and standard deviation. The mean is the average value and is best for continuous, non-skewed data. The median is less affected by outliers and is best for skewed or ordinal data. The mode is the most frequent value and is used for categorical data. Measures of variability describe how spread out the data is, with higher values indicating more dispersion.
This document provides an overview of data analysis and statistics concepts for a training session. It begins with an agenda outlining topics like descriptive statistics, inferential statistics, and independent vs dependent samples. Descriptive statistics concepts covered include measures of central tendency (mean, median, mode), measures of variability (range, standard deviation), and charts. Inferential statistics discusses estimating population parameters, hypothesis testing, and statistical tests like t-tests, ANOVA, and chi-squared. The document provides examples and online simulation tools. It concludes with some practical tips for data analysis like checking for errors, reviewing findings early, and consulting a statistician on analysis plans.
The document provides an overview of inferential statistics. It defines inferential statistics as making generalizations about a larger population based on a sample. Key topics covered include hypothesis testing, types of hypotheses, significance tests, critical values, p-values, confidence intervals, z-tests, t-tests, ANOVA, chi-square tests, correlation, and linear regression. The document aims to explain these statistical concepts and techniques at a high level.
Measures of dispersion
Absolute measure, relative measures
Range of Coe. of Range
Mean deviation and coe. of mean deviation
Quartile deviation IQR, coefficient of QD
Standard deviation and coefficient of variation
This document defines key terms and concepts related to standard deviation and variance. It provides formulas for calculating range, deviation, variance, and standard deviation for both ungrouped and grouped data. Examples are given to demonstrate calculating these metrics from raw data sets and grouped data tables. Interpreting skewness is also discussed.
This presentation discusses frequency distribution graphs. It provides examples of constructing frequency distribution tables and calculating class midpoints and cumulative frequencies. The key graphs for representing frequency distributions are described as histograms, frequency polygons, and cumulative frequency curves for continuous or quantitative data. Bar charts, line graphs and pie charts are also introduced as options to display grouped or ungrouped categorical data. Examples of each type of graph are included.
The document defines properties of arithmetic mean and provides examples to illustrate these properties. The key properties are: 1) The sum of deviations from the mean is zero. 2) The sum of squares of deviations from the mean is minimum. 3) The mean is unaffected by a constant change of scale or origin. 4) The mean of multiple data sets can be calculated as a weighted average based on sample sizes. The examples demonstrate calculating means and verifying properties for various data sets.
This document introduces key concepts in probability:
- Probability is the likelihood of an event occurring, which can be measured numerically or described qualitatively.
- Events can be classified as exhaustive, favorable, mutually exclusive, equally likely, complementary, and independent.
- There are three approaches to defining probability: classical, frequency, and axiomatic. The classical approach defines probability as the number of favorable outcomes over the total number of possible outcomes. The frequency approach defines probability as the limit of the ratio of favorable outcomes to the total number of trials. The axiomatic approach defines probability based on axioms or statements assumed to be true.
- Key properties of probability include that the probability of an event is between 0
Analysis of covariance (ANCOVA) is a statistical test that assesses whether the means of a dependent variable are equal across levels of a categorical independent variable while statistically controlling for the effects of other continuous variables known as covariates. ANCOVA works by adjusting the sums of squares for the independent variable to remove the influence of the covariate. This allows ANCOVA to test for differences between groups while controlling for the influence of other continuous variables. The assumptions of ANCOVA include those of ANOVA as well as the assumptions that the relationship between the dependent variable and covariate is linear and the same across all groups.
01 parametric and non parametric statisticsVasant Kothari
Definition of Parametric and Non-parametric Statistics
Assumptions of Parametric and Non-parametric Statistics
Assumptions of Parametric Statistics
Assumptions of Non-parametric Statistics
Advantages of Non-parametric Statistics
Disadvantages of Non-parametric Statistical Tests
Parametric Statistical Tests for Different Samples
Parametric Statistical Measures for Calculating the Difference Between Means
Significance of Difference Between the Means of Two Independent Large and
Small Samples
Significance of the Difference Between the Means of Two Dependent Samples
Significance of the Difference Between the Means of Three or More Samples
Parametric Statistics Measures Related to Pearson’s ‘r’
Non-parametric Tests Used for Inference
This document provides an overview of measures of dispersion, including range, quartile deviation, mean deviation, standard deviation, and variance. It defines dispersion as a measure of how scattered data values are around a central value like the mean. Different measures of dispersion are described and formulas are provided. The standard deviation is identified as the most useful measure as it considers all data values and is not overly influenced by outliers. Examples are included to demonstrate calculating measures of dispersion.
This document discusses different types of statistics used in research. Descriptive statistics are used to organize and summarize data using tables, graphs, and measures. Inferential statistics allow inferences about populations based on samples through techniques like surveys and polls. The key difference is that descriptive statistics describe samples while inferential statistics allow conclusions about populations beyond the current data.
Topic: Frequency Distribution
Student Name: Abdul Hafeez
Class: B.Ed. (Hons) Elementary
Project Name: “Young Teachers' Professional Development (TPD)"
"Project Founder: Prof. Dr. Amjad Ali Arain
Faculty of Education, University of Sindh, Pakistan
This document discusses measures of central tendency including the mean, median, and mode. It provides examples of calculating these values for different data sets. It also discusses using the mean to balance a seesaw with blocks and the concept of deviations from the mean. The homework is to finish the class notes on these topics.
The document discusses various measures of central tendency and dispersion. It defines the arithmetic mean, weighted mean, geometric mean, median, and mode as measures of central tendency. It also discusses calculating these measures from grouped data. For measures of dispersion, it covers range, interquartile range, variance, and standard deviation. It provides formulas to calculate these statistics for both population and sample data.
This document discusses measures of central tendency, specifically mean, median, and mode. It begins by defining measures of central tendency as averages that represent central or typical values within a data set. The document then outlines different methods for calculating the mean, or arithmetic average, of both raw (ungrouped) and grouped data sets. It provides examples of calculating the mean of raw data sets directly using the formula for mean, and through the assumed mean method which uses deviations from an assumed mean to simplify calculations for large data sets. The document emphasizes that the mean is the sum of all values divided by the number of values. It also discusses how mean is calculated for grouped data by assigning values to class intervals based on their midpoints.
The lesson plan discusses measures of central tendency for ungrouped data. It defines the three measures - mean, median, and mode. The lesson explains how to calculate each measure through examples and formulas. Students will practice finding the mean, median, and mode of various data sets.
MEASURES OF CENTRAL TENDENCY AND MEASURES OF DISPERSION Tanya Singla
Central tendency refers to typical or average values in a data set or probability distribution. The three most common measures of central tendency are the mean, median, and mode. The mean is the average calculated by summing all values and dividing by the total number. The median is the middle value when data is arranged in order. The mode is the most frequently occurring value. Other measures discussed include range, which is the difference between highest and lowest values, and quartiles, which divide a data set into four equal parts based on the distribution of values.
This document defines and provides examples of measures of central tendency including the mean, median, and mode. It also discusses the range. The mean is the average and is calculated by summing all values and dividing by the total number. The median is the middle value when data is arranged from lowest to highest. The mode is the most frequently occurring value. The range is the difference between the highest and lowest values. Examples of how these are used include calculating the average age or income and determining vaccination guidelines based on age range.
Frequency distribution, central tendency, measures of dispersionDhwani Shah
The presentation explains the theory of what is Frequency distribution, central tendency, measures of dispersion. It also has numericals on how to find CT for grouped and ungrouped data.
This document discusses measures of central tendency, including the mean, median, and mode. It defines each measure and describes their characteristics and how to calculate them. The mean is the average value and is affected by outliers. The median is the middle value and is not affected by outliers. The mode is the most frequently occurring value. The document provides examples of calculating each measure from data sets and discusses their advantages and disadvantages.
Mean, Median, Mode: Measures of Central Tendency Jan Nah
There are three common measures of central tendency: mean, median, and mode. The mean is the average value found by dividing the sum of all values by the total number of values. The median is the middle value when values are arranged from lowest to highest. The mode is the value that occurs most frequently. Each measure provides a single number to represent the central or typical value in a data set.
Here are the modes for the three examples:
1. The mode is 3. This value occurs most frequently among the number of errors committed by the typists.
2. The mode is 82. This value occurs most frequently among the number of fruits yielded by the mango trees.
3. The mode is 12 and 15. These values occur most frequently among the students' quiz scores.
The three main measures of central tendency are the mean, median, and mode. The mean is the average and is calculated by adding all values and dividing by the total number. The median is the middle value when values are arranged from lowest to highest. The mode is the most frequently occurring value. For symmetric distributions, the mean, median, and mode will be equal. However, for skewed distributions the mean will be pulled higher or lower than the median depending on the direction of skew.
This document provides an introduction to statistics. It discusses what statistics is, the two main branches of statistics (descriptive and inferential), and the different types of data. It then describes several key measures used in statistics, including measures of central tendency (mean, median, mode) and measures of dispersion (range, mean deviation, standard deviation). The mean is the average value, the median is the middle value, and the mode is the most frequent value. The range is the difference between highest and lowest values, the mean deviation is the average distance from the mean, and the standard deviation measures how spread out values are from the mean. Examples are provided to demonstrate how to calculate each measure.
This document provides information on measures of central tendency, including the median, mode, and mean. It defines these terms, explains how to calculate them, and discusses their advantages and disadvantages. Specifically, it explains that the median is the middle value when values are arranged in order, and the mode is the most frequently occurring value. Formulas are provided for calculating the median and mode from both individual and grouped data sets. The document also discusses different types of averages and provides examples of calculating the median and mode from various data distributions.
A measure of central tendency (also referred to as measures of centre or central location) is a summary measure that attempts to describe a whole set of data with a single value that represents the middle or centre of its distribution.
This document discusses measures of central tendency and dispersion used to analyze and summarize data. It defines key terms like mean, median, mode, range, variance, and standard deviation. It explains how to calculate these measures both mathematically and using grouped or sample data, and the importance of understanding the central tendency and dispersion of data distributions.
This document discusses measures of central tendency and dispersion used to analyze and summarize data. It defines key terms like mean, median, mode, range, variance, and standard deviation. It explains how to calculate these measures both mathematically and using grouped or sample data, and the importance of understanding the distribution, central tendency and dispersion of data.
Measures of Central Tendency, Variability and ShapesScholarsPoint1
The PPT describes the Measures of Central Tendency in detail such as Mean, Median, Mode, Percentile, Quartile, Arthemetic mean. Measures of Variability: Range, Mean Absolute deviation, Standard Deviation, Z-Score, Variance, Coefficient of Variance as well as Measures of Shape such as kurtosis and skewness in the grouped and normal data.
Descriptions of data statistics for researchHarve Abella
This document defines and describes various measures of central tendency and variation that are used to summarize and describe sets of data. It discusses the mean, median, mode, midrange, percentiles, quartiles, range, variance, standard deviation, interquartile range, coefficient of variation, measures of skewness and kurtosis. Examples are provided to demonstrate how to compute and interpret these statistical measures.
This document provides an overview of key concepts in descriptive statistics, including measures of center, variation, and relative standing. It discusses the mean, median, mode, range, standard deviation, z-scores, percentiles, quartiles, interquartile range, and boxplots. Formulas and properties of these statistical concepts are presented along with guidelines for interpreting and applying them to describe data distributions.
1) The document provides information about a statistical methods course, including the title, teacher details, and topics to be covered including measures of central tendency.
2) It defines different measures of central tendency - the mean, median, and mode. It provides formulas and examples of calculating each measure for both raw and grouped data.
3) The characteristics of an ideal average and merits and demerits of each measure are discussed. Applications of each measure are also mentioned.
This document summarizes various statistical measures used to describe and analyze numerical data, including measures of central tendency (mean, median, mode), measures of variation (range, interquartile range, variance, standard deviation, coefficient of variation), and ways to describe the shape of distributions (symmetric vs. skewed using box-and-whisker plots). It provides definitions and formulas for calculating these common statistical concepts.
Measures of central tendency describe the middle or center of a data set and include the mean, median, and mode. The mean is the average value calculated by adding all values and dividing by the number of values. The median is the middle number in a data set arranged in order. The mode is the value that occurs most frequently. These measures are used to understand the typical or common values in a data set.
This document discusses various statistical measures for summarizing and describing numerical data, including measures of central tendency (mean, median, mode, midrange, quartiles), measures of variation (range, interquartile range, variance, standard deviation, coefficient of variation), and shape of distributions (symmetric vs. skewed). It provides definitions and formulas for calculating each measure and describes how to interpret them. Box-and-whisker plots are introduced as a graphical way to display data using the median, quartiles, and range.
The document defines and provides examples of various statistical measures used to summarize data, including measures of central tendency (mean, median, mode), measures of variation (variance, standard deviation, coefficient of variation), and shape of data distribution. It explains how to calculate and interpret these measures and when each is most appropriate to use. Examples are provided to demonstrate calculating various measures for different datasets.
This document discusses measures of central tendency, including the mean, median, and mode. It provides definitions and formulas for calculating each measure for both grouped and ungrouped data. For the mean, it addresses how outliers can influence the value and introduces the trimmed mean. The median is described as the middle value of a data set and is not impacted by outliers. The mode is defined as the most frequent observation. Examples are given to demonstrate calculating each measure. Key differences between the measures are summarized.
This document provides an outline and summary of key concepts related to data analysis, including measures of central tendency (mean, median, mode), spread of distribution (range, variance, standard deviation), and experimental designs (paired t-test, ANOVA). It explains how to calculate and interpret the mean, median, mode, range, variance, and standard deviation. It also provides brief definitions and examples of paired t-tests and ANOVA.
This document provides an overview of key concepts in statistics including measures of central tendency (mean, median, mode), measures of dispersion (variance, standard deviation), and central moments (skewness, kurtosis). It discusses calculating and comparing the mean, median, mode, and how they each describe the central position of a data distribution. It also explains how variance and standard deviation measure how spread out the data is from the mean. The document is intended as a textbook for students and general readers to learn basic statistical concepts.
Measures of central tendency and dispersionAbhinav yadav
This document discusses various measures of central tendency and dispersion. It defines the mean, median, and mode as measures of central tendency, and describes how to calculate the arithmetic mean, geometric mean, harmonic mean, median, and mode. It also discusses measures of dispersion such as variance and standard deviation, and classifies them as either measures of absolute dispersion or relative dispersion.
1. Measures of Central Tendency
Back to Top
The three important measures of central tendency are
1. The Mean
2. The Median
3. The Mode
Measures of Central Tendency Definition
Measure of central tendency can be the term which defines the centre of data. There are three
parameters by which we can measure central tendency - Mean, median and mode.
Central Tendency of Data
Mean:
Mean of data is a set of numerical values is the arithmetic average of the data values in the set. It is found
by adding all the values in the data set and dividing the sum by the total number of values in the set.
Mean of a data set = Sum of the Data ValuesTotal Number of Data Values
Median:
For an ordered data set, median is the value in the middle of the data distribution. If there are even
number of data values in the set, then there will be two middle values and the median is the average of
these two middle values.
Mode
Mode is the most frequently occurring value in the data set.
In addition to these three important measures of central tendency, another measure is also defined.
Midrange:
Midrange is an estimated measure of the average. It is the average of the lowest and highest values in
the data set.
Midrange = Lowest Value + Highest Value2
Midrange is only a rough estimate of the central value. As it uses only the lowest and highest values of
the data set, it is highly affected when one of them is very high or very low.
2. Central Tendency Definition
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The term central tendency refers to the middle value of the data, and is measured using the mean,
median, or mode. It is the tendency of the values of a random variable to cluster around the mean,
median, and mode. And a measure of central tendency for a data distribution is a measure of centralness
of data and it is used to summarize the data set.
Mean
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The mean of a sample data is denoted by x¯ and the population mean by μ. The mean of a small number
of data set can be found by adding all the data values and dividing the sum by total number of values.
Characteristics of Mean
1. Mean is computed using all the values in the data set.
2. Mean varies less for samples taken from the same population when compared to the median or
mode.
3. The Mean is unique for a data set. The mean may not be one of the data values in the
distribution.
4. Other statistics such as variance are computed using mean.
5. Mean is affected the most by the outliers present in the data set. Hence mean is not to be used
for data sets containing outliers.
Mean for the grouped data is also computed applying above methods, the mid point of the class is used
as x.
Solved Examples
Question 1: The following data set is the worth(in billions of dollars) of 10 hypothetical wealthy men. Find
the mean worth of these top 10 rich men.
12.6, 13.7, 18.0, 18.0, 18.0, 20.0, 20.0, 41.2, 48.0, 60.0
Solution:
Given data,
12.6, 13.7, 18.0, 18.0, 18.0, 20.0, 20.0, 41.2, 48.0, 60.0
Mean of the data set,
x¯ = 12.6+13.7+18+18+18+20+20+41.2+48+6010
3. = 269.510
= 26.95
Question 2: Compute the mean for the distribution given below
Value
x
Frequency
f
20 2
29 4
30 4
39 3
44 2
Solution:
The frequency table is redone adding one more column f * x
Value
x
Frequency
f
f * x
20 2 40
29 4 116
30 4 120
39 3 117
44 2 88
∑f = 15 ∑fx = 481
Mean of the distribution x¯=∑fx∑f
= 48115
= 32.1 (Answer rounded to the tenth).
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Median
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4. When we say the median value of earnings of Actuarial experts is 60,000 dollars, we mean that 50% of
these experts earn less than 60,000 dollars and 50% earn more than this. Thus median is the balancing
point in an ordered data set. As median represents the 50% mark in a distribution, this is a measure of
position as well. Median is much more easier to find than computing the mean.
Uses of Median
1. Median is used if the analysis requires the middle value of the distribution.
2. Median is used to determine whether the given data value/s fall in the upper or lower half of the
distribution.
3. Medan can be used even if the classes in the frequency distribution are open ended.
4. Median is generally used as the central value, when the data is likely to contain outliers.
Solved Examples
Question 1: The number of rooms in 11 hotels in a city is as follows:
380, 220, 555, 678, 756, 823, 432, 367, 546, 402, 347.
Solution:
The data is first arranged starting from the lowest as follows:
220, 347, 367, 380, 402, 432, 546, 555, 678, 756, 823.
As the number of data elements 11 is an odd number, there is only one middle value in the data array,
which is the 6th.
=> The value of data in 6th position = 432.
Hence the mean number of Hotel rooms in the city = 432.
Question 2: Find the median of the given data
Value
X
Frequency
f
20 2
29 4
30 4
39 3
44 2
Solution:
Value
x
Frequency
f
Cumulative
frequency
5. 20 2 2
29 4 2 + 4 = 6
30 4 6 + 4 = 10
39 3 10 + 3 = 13
44 2 13 + 2 = 15
∑f = 15 ∑fx = 481
=> ∑f = 15 items,
The 8th item in the ordered data array will be the median. The 8 item will be included in the cumulative
frequency 10. Hence the median of the distribution is the x value corresponding to cumulative frequency
10 which reads as 30.
=> Median of the data = 30.
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Mode
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Mode is the value or category that occurs most in a data set.
If all the elements in the data set have the same frequency of occurrence, then distribution does
not have a mode.
In a unimodal distribution, one value occurs most frequently in comparison to other values.
A bimodal distribution has two elements have the highest frequency of occurrence.
Characteristics of Mode:
1. Mode is the easiest average to determine and it is used when the most typical value is required
as the central value.
2. Mode can be found for nominal data set as well.
3. Mode need not be a unique measure. A distribution can have more than one mode or no mode at
all.
Solved Example
Question: Find the mode of a numerical data set
109 112 109 110 109 107 104 104 104 111 111 109 109 104 104
Solution:
6. Given data,
109 112 109 110 109 107 104 104 104 111 111 109 109 104 104
Total number of element = 15
Among the 15 data elements the values 104 and 109 both occur five times which are hence the modes of
the data set.
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Effect of Transformations on Central Tendency
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If all the data values in a data distribution is subjected to some common transformation, what would be
the effect of this on the measures of central tendency?
If each element in a data set is increased by a constant, the mean, median and mode of the
resulting data set can be obtained by adding the same constant to the corresponding values of
the original data set.
When each element of a data set is multiplied by a constant, then the mean, median and mode of
the new data set is obtained by multiplying the corresponding values of the original data set.
Central Tendency and Dispersion
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Two kinds of statistics are frequently used to describe data. They are measures of central tendency and
dispersion. These are often called descriptive statistics because they can help us to describe our data.
Measures of Central Tendency and Dispersion
Mean, median and mode are all measures of central tendency whereas range, variance and standard
deviation are all measures of dispersion. The measures used to describe the data set are measures of
central tendency and measures of dispersion or variability.
Central Tendency Dispersion
If different sets of numbers can have the same mean. Then we will study two measures of dispersion,
which give you an idea of how much the numbers in a set differ from the mean of the set. These two
measures are called thevariance of the set and the standard deviation of the set.
Formula for variance and Standard Deviation:
7. For the set of numbers {x1,x2,.............,xn} with a mean of x¯.
The variance of the set is
=> V = (x1−x¯)2+(x2−x¯)2+.........+(xn−x¯)2n
and the standard deviation is,
=> σ=V−−√.
Standard Deviation can be represented as;
σ = x21+x22+...............+x2nn−x¯2−−−−−−−−−−−−−−−−−√
Resistant Measures of Central Tendency
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A resistant measure is one that is less influenced by extreme data values. The mean is less resistant than
the median, that is the mean is more influenced by extreme data values. Resistant measure of central
tendency can resist the influence of extreme observations or outliers.
Let us see the effect of outlier with the help of example:
Solved Example
Question: Consider the data set, 5, 19, 19, 20, 21, 23, 23, 23, 24 , 25.
Solution:
The value 5 is an outlier of the data as it is too less than the other values in the distribution.
Let us calculate the the central values for the data set either by including and excluding 5.
Step 1:
The data set excluding 5 is 19, 19, 20, 21, 23, 23, 23, 24 , 25
Mean = x¯=19+19+20+21+23+23+23+24+259=1979 = 21.89
=> Mean = 21.89
Median = 23
Mode = 23
Step 2:
For the data including the outlier 5, 19, 19, 20, 21, 23, 23, 23, 24 , 25
8. Mean = x¯=5+19+19+20+21+23+23+23+24+259=20210 = 20.2
=> Mean = 20.2
Median = 21+232 = 22
Mode = 23
Step 3:
Comparing the values of mean, median and mode found in step 1 and step 2, the mean is most affected
and mode is least affected by the inclusion of the outlier value 5.
Central Tendency and Variability
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Central tendency is a statistical measure that represents a central entry of a data set. The problem is that
there is no single measure that will always produce a central, representative value in every situation.
There are three main measures of central tendency, mean, median and mode.
Variability is the important feature of a frequency distribution. Range, variance and standard deviation
are all measures of variability. Range, variance and standard deviation are all measures of variability.
Range - The simplest measure of variability is the range, which is the difference between the highest and
the lowest scores.
Standard Deviation - The standard deviation is the average amount by which the scores differ from the
mean.
Variance - The variance is another measure of variability. It is just the mean of the squared differences,
before we takethe square root to get the standard deviation.
Central Tendency Theorem
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A more formal and mathematical statement of the Central Limit Theorem is stated as follows:
Suppose that x1,x2,x3,...................,xn are independent and identically distributed with mean μ and finite
variance σ2 . Then the random variable Un is defined as,
Un = X¯−μσn√
Where, X¯=1n∑ni=1Xi
Then the distribution function of Un converges to the standard normal distribution function as n increases
without bound.
9. Central Tendency Examples
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A measure of central tendency is a value that represents a central entry of a data set. Central tendency of
the data can be calculated by measuring mean, median and mode of the data.
Below you could see some examples of central tendency:
Solved Examples
Question 1: Find the mean, median and mode of the given data.
10, 12, 34, 34, 45, 23, 42, 36, 34, 22, 20, 27, 33.
Solution:
Given Data,
X = 10, 12, 34, 34, 45, 23, 42, 36, 34, 22, 20, 27, 33.
ΣX = 10 + 12 + 34 + 34 + 45 + 23 + 42 + 36 + 34 + 22 + 20 + 27 + 33
= 372
=> ΣX = 372
Step 1:
Mean = ∑XX
= 37213
[ X = Total number of terms ]
= 28.6
=> Mean = 28.6
Step 2:
For Median,
Arrange the data in ascending order.
10. 10, 12, 20, 22, 23, 27, 33, 34, 34, 34, 36, 42, 45.
The median is 33. Half of the values fall above this number and half fall below.
=> Median = 33
Step 3:
Mode
Mode = 34
Because 34 occur maximum times.
Question 2: The following table shows the sport activities of 2400 students.
Sport
Frequenc
y
Swimming 423
Tennis 368
Gymnastic
s
125
Basket
ball
452
Base ball 380
Athletics 275
None 377
Solution:
From the given table:
For grouped data the class with highest frequency is called the Modal class. The category with the
longest column in the bar graph represents the mode of data set.
Basket ball has the highest frequency of 452. Hence Basket ball is the mode of the sport activities.
11. Question 3: Find the median of the distribution,
223, 227, 240, 211, 212, 209, 211, 213, 240, 229.
Solution:
The ordered data array will be:
209, 211, 211, 212, 213, 223, 227, 229, 240, 240
The number of data values is even. Hence the two central values are those in the 5th and the 6thpositions.
Median = 213+2232 = 4362 = 218
=> Median = 218.
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