This document defines and explains various measures of central tendency including the mean, median, and mode. It provides definitions and formulas for calculating the arithmetic mean, geometric mean, and harmonic mean for individual, discrete, and continuous data. The key properties and uses of averages as measures of central tendency are that they provide a single representative value for a dataset, allow for brief descriptions and comparisons of data, and help inform economic policies and other statistical analyses.
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 measures of central tendency and dispersion. It defines mean, median and mode as measures of central tendency, which describe the central location of data. The mean is the average value, median is the middle value, and mode is the most frequent value. It also defines measures of dispersion like range, interquartile range, variance and standard deviation, which describe how spread out the data are. Standard deviation in particular measures how far data values are from the mean. Approximately 68%, 95% and 99.7% of observations in a normal distribution fall within 1, 2 and 3 standard deviations of the mean respectively.
The document discusses different measures of central tendency including the mean, median, and mode. It provides definitions and formulas for calculating each. The mean is the average and is calculated by adding all values and dividing by the total count. The median is the middle value when data is arranged in order. The mode is the most frequently occurring value. Examples are given to demonstrate calculating and interpreting each measure of central tendency.
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.
The document discusses measures of central tendency, specifically the mean. It defines the mean as the average of all values in a data set, found by adding all values and dividing by the total number of data points. The mean represents the balance point of a distribution and feels like the center because it is the value where the data balances on either side when represented visually in a histogram. The mean is unique in that a data set only has one mean, and it is influenced by all observations in the data set.
This document discusses various measures of central tendency used in statistics. It defines central tendency as a typical or average value for a probability distribution. The three most common measures are the mean, median, and mode. The mean is the average value calculated by adding all values and dividing by the total number. The median is the middle value when values are arranged in order. The mode is the most frequently occurring value. Other measures discussed include the geometric mean, harmonic mean, weighted mean, and truncated mean. Factors like the range, type of variable, and data distribution impact which measure is most appropriate.
Includes solved numerical problems on Median, in three series individual, discrete and continuous. Moreover, it consists some unsolved problems for practice after learning.
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 measures of central tendency and dispersion. It defines mean, median and mode as measures of central tendency, which describe the central location of data. The mean is the average value, median is the middle value, and mode is the most frequent value. It also defines measures of dispersion like range, interquartile range, variance and standard deviation, which describe how spread out the data are. Standard deviation in particular measures how far data values are from the mean. Approximately 68%, 95% and 99.7% of observations in a normal distribution fall within 1, 2 and 3 standard deviations of the mean respectively.
The document discusses different measures of central tendency including the mean, median, and mode. It provides definitions and formulas for calculating each. The mean is the average and is calculated by adding all values and dividing by the total count. The median is the middle value when data is arranged in order. The mode is the most frequently occurring value. Examples are given to demonstrate calculating and interpreting each measure of central tendency.
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.
The document discusses measures of central tendency, specifically the mean. It defines the mean as the average of all values in a data set, found by adding all values and dividing by the total number of data points. The mean represents the balance point of a distribution and feels like the center because it is the value where the data balances on either side when represented visually in a histogram. The mean is unique in that a data set only has one mean, and it is influenced by all observations in the data set.
This document discusses various measures of central tendency used in statistics. It defines central tendency as a typical or average value for a probability distribution. The three most common measures are the mean, median, and mode. The mean is the average value calculated by adding all values and dividing by the total number. The median is the middle value when values are arranged in order. The mode is the most frequently occurring value. Other measures discussed include the geometric mean, harmonic mean, weighted mean, and truncated mean. Factors like the range, type of variable, and data distribution impact which measure is most appropriate.
Includes solved numerical problems on Median, in three series individual, discrete and continuous. Moreover, it consists some unsolved problems for practice after learning.
This document discusses different measures of central tendency - mean, median, and mode. It defines each measure and provides examples of how to calculate them. 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 data is arranged in order. The mode is the value that occurs most frequently. Examples are given to demonstrate calculating the mean, median, and mode of sample data sets.
This document discusses measures of central tendency including the mean, median, and mode. It defines each measure and explains their properties and appropriate uses. The mean is the average value obtained 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 most frequently occurring value. Each measure can be affected differently by outliers, with the median being least affected. The appropriate measure depends on the scale of measurement and distribution of the data.
Measures of central tendency provide a single number to describe a set of scores and the performance of a group. The three main measures are the mean, median, and mode. The mean is the average and is calculated using the sum of all scores divided by the total number of scores. The median divides the scores into two equal parts, and the mode is the score that occurs most frequently. Formulas are provided for calculating each measure using grouped or ungrouped data.
This document defines and provides examples of measures of central tendency, including the mean, median, and mode. The mean is the average value and is calculated by summing all values and dividing by the number of values. The median indicates the middle value of an ordered data set. The mode is the most frequently occurring value. These measures can be used to describe sample data or entire populations. The appropriate measure depends on features of the data such as outliers or symmetry.
This document defines and compares various measures of central tendency, including the mean, median, and mode. It provides formulas and examples for calculating the arithmetic mean, geometric mean, harmonic mean, weighted mean, median, and mode. The document also discusses the relationships between the arithmetic mean, geometric mean, and harmonic mean, showing that the harmonic mean will always be less than or equal to the geometric mean, which will always be less than or equal to the arithmetic mean.
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.
The document discusses measures of central tendency including the mean, median, and mode for ungrouped data sets.
The mean is the average and is calculated by adding all values in the data set and dividing by the number of values. The median is the middle number when data is arranged in order. For even data sets, the median is the average of the two middle numbers. The mode is the number that occurs most frequently in the data set. Sometimes there are multiple modes or no mode. The range is the difference between the highest and lowest values in the data set.
This document discusses different types of measures of central tendency used in biostatistics:
- The mean is the average value calculated by adding all data points and dividing by the total number of points.
- The median is the middle value when all data points are ordered.
- The mode is the most frequently occurring value.
It also explains what the geometric mean is, calculated as the nth root of the product of n numbers. Several merits and limitations of the geological mean are provided.
The document discusses various statistical concepts including types of statistics, measures of central tendency, and their definitions and examples. It then uses the sales data of two salesmen to demonstrate calculating the mean, median, and mode in order to help a sales manager determine which salesman to promote based on their sales performance over the past 7 months. Based on the measures of central tendency, Salesman A should be promoted as they have a higher mean and median number of cars sold compared to Salesman B.
Measures of central tendency include the mode, median, mean, geometric mean, and harmonic mean. The arithmetic mean is the sum of all values divided by the sample size. The geometric mean is the nth root of the product of n values. The harmonic mean is equal to the sample size divided by the sum of the reciprocals of the values. The median is the middle value when observations are ranked from smallest to largest. The mode is the value that occurs most frequently in the sample.
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 discusses measures of central tendency, which summarize a data set with a single value representing the center or typical value. There are three main types of measures: the arithmetic mean, which is the sum of all values divided by the number of values; the median, which is the middle value when data is ordered from lowest to highest; and the mode, which is the most frequently occurring value. The document provides formulas for calculating each of these measures and explains their properties and uses for summarizing and comparing data sets.
This document discusses measures of central tendency, which are statistical values that describe the center of a data set. The three main measures are the mean, median, and mode.
The mean is the average value found by dividing the total of all values by the number of values. The median is the middle value when data is arranged from lowest to highest. The mode is the most frequently occurring value.
While the mean is most commonly used, the median and mode are better in some situations, such as when outliers are present or data is categorical. The geometric mean measures rate of change over time. Choosing the appropriate measure depends on the data type and distribution.
The document discusses different measures of central tendency:
- The mode is the most frequent score in a data set and is used for nominal data.
- The median is the middle score when data is arranged from lowest to highest and is not influenced by outliers.
- The mean is the average of all scores, calculated by summing all scores and dividing by the total number, and is preferred for interval/ratio scaled and non-skewed data.
This document provides an introduction to measures of central tendency in statistics. It defines measures of central tendency as statistical measures that describe the center of a data distribution. The three most commonly used measures are the mean, median, and mode. The document focuses on explaining the arithmetic mean in detail, including how to calculate the mean from individual data series, discrete data series, and continuous data series using different methods. It also discusses weighted means, combined means, and the relationship between the mean, median and mode. The objectives and advantages and disadvantages of the mean are provided.
This document discusses various measures of central tendency including the mean, median, and mode. It provides definitions and formulas for calculating each measure. The mean is the average and is calculated by summing all values and dividing by the total number of data points. The median is the middle value when data is arranged in order. The mode is the value that occurs most frequently in the data set. Examples are given to demonstrate calculating each measure. The document also discusses advantages and limitations of each central tendency measure.
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.
The document discusses different measures of central tendency including the mean, median, and mode. The mean is the average value and considers all scores, but can be impacted by outliers. The median is the middle value when scores are arranged in order and is less impacted by outliers. The mode is the most frequently occurring value and can be used for both numerical and categorical data, but may not always indicate the true central tendency. Each measure has advantages and limitations, and different measures are better suited for certain types of data distributions.
This document discusses different measures of central tendency, including mathematical averages, averages of position, and measures of partition values. It describes three types of mathematical averages: arithmetic mean, geometric mean, and harmonic mean. It provides the definitions and formulas for calculating each type of mean. The document also discusses two types of averages of position: median and mode. It defines median as the middlemost value when data are arranged in order, and mode as the value with the highest frequency. For each measure of central tendency, the document outlines merits and demerits. It concludes by describing relationships between the mean, median, and mode for symmetrical versus asymmetrical distributions.
The document discusses different measures of central tendency including the mean, median, and mode. The mean is the average value calculated by adding all values and dividing by the total number of values. The median is the middle value when values are arranged from lowest to highest. The mode is the most frequently occurring value in the data set. The document provides examples of calculating each measure and discusses their advantages and disadvantages.
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.
This document provides an overview of measures of central tendency in statistics, focusing on the arithmetic mean. It defines the arithmetic mean as the sum of all values divided by the total number of items. It discusses different types of series (individual, discrete, continuous) and methods to calculate the arithmetic mean for each type, including the direct method, short-cut method, and step deviation method. The document provides examples of calculating the arithmetic mean using these different methods for an individual series of student marks.
This document discusses different measures of central tendency - mean, median, and mode. It defines each measure and provides examples of how to calculate them. 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 data is arranged in order. The mode is the value that occurs most frequently. Examples are given to demonstrate calculating the mean, median, and mode of sample data sets.
This document discusses measures of central tendency including the mean, median, and mode. It defines each measure and explains their properties and appropriate uses. The mean is the average value obtained 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 most frequently occurring value. Each measure can be affected differently by outliers, with the median being least affected. The appropriate measure depends on the scale of measurement and distribution of the data.
Measures of central tendency provide a single number to describe a set of scores and the performance of a group. The three main measures are the mean, median, and mode. The mean is the average and is calculated using the sum of all scores divided by the total number of scores. The median divides the scores into two equal parts, and the mode is the score that occurs most frequently. Formulas are provided for calculating each measure using grouped or ungrouped data.
This document defines and provides examples of measures of central tendency, including the mean, median, and mode. The mean is the average value and is calculated by summing all values and dividing by the number of values. The median indicates the middle value of an ordered data set. The mode is the most frequently occurring value. These measures can be used to describe sample data or entire populations. The appropriate measure depends on features of the data such as outliers or symmetry.
This document defines and compares various measures of central tendency, including the mean, median, and mode. It provides formulas and examples for calculating the arithmetic mean, geometric mean, harmonic mean, weighted mean, median, and mode. The document also discusses the relationships between the arithmetic mean, geometric mean, and harmonic mean, showing that the harmonic mean will always be less than or equal to the geometric mean, which will always be less than or equal to the arithmetic mean.
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.
The document discusses measures of central tendency including the mean, median, and mode for ungrouped data sets.
The mean is the average and is calculated by adding all values in the data set and dividing by the number of values. The median is the middle number when data is arranged in order. For even data sets, the median is the average of the two middle numbers. The mode is the number that occurs most frequently in the data set. Sometimes there are multiple modes or no mode. The range is the difference between the highest and lowest values in the data set.
This document discusses different types of measures of central tendency used in biostatistics:
- The mean is the average value calculated by adding all data points and dividing by the total number of points.
- The median is the middle value when all data points are ordered.
- The mode is the most frequently occurring value.
It also explains what the geometric mean is, calculated as the nth root of the product of n numbers. Several merits and limitations of the geological mean are provided.
The document discusses various statistical concepts including types of statistics, measures of central tendency, and their definitions and examples. It then uses the sales data of two salesmen to demonstrate calculating the mean, median, and mode in order to help a sales manager determine which salesman to promote based on their sales performance over the past 7 months. Based on the measures of central tendency, Salesman A should be promoted as they have a higher mean and median number of cars sold compared to Salesman B.
Measures of central tendency include the mode, median, mean, geometric mean, and harmonic mean. The arithmetic mean is the sum of all values divided by the sample size. The geometric mean is the nth root of the product of n values. The harmonic mean is equal to the sample size divided by the sum of the reciprocals of the values. The median is the middle value when observations are ranked from smallest to largest. The mode is the value that occurs most frequently in the sample.
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 discusses measures of central tendency, which summarize a data set with a single value representing the center or typical value. There are three main types of measures: the arithmetic mean, which is the sum of all values divided by the number of values; the median, which is the middle value when data is ordered from lowest to highest; and the mode, which is the most frequently occurring value. The document provides formulas for calculating each of these measures and explains their properties and uses for summarizing and comparing data sets.
This document discusses measures of central tendency, which are statistical values that describe the center of a data set. The three main measures are the mean, median, and mode.
The mean is the average value found by dividing the total of all values by the number of values. The median is the middle value when data is arranged from lowest to highest. The mode is the most frequently occurring value.
While the mean is most commonly used, the median and mode are better in some situations, such as when outliers are present or data is categorical. The geometric mean measures rate of change over time. Choosing the appropriate measure depends on the data type and distribution.
The document discusses different measures of central tendency:
- The mode is the most frequent score in a data set and is used for nominal data.
- The median is the middle score when data is arranged from lowest to highest and is not influenced by outliers.
- The mean is the average of all scores, calculated by summing all scores and dividing by the total number, and is preferred for interval/ratio scaled and non-skewed data.
This document provides an introduction to measures of central tendency in statistics. It defines measures of central tendency as statistical measures that describe the center of a data distribution. The three most commonly used measures are the mean, median, and mode. The document focuses on explaining the arithmetic mean in detail, including how to calculate the mean from individual data series, discrete data series, and continuous data series using different methods. It also discusses weighted means, combined means, and the relationship between the mean, median and mode. The objectives and advantages and disadvantages of the mean are provided.
This document discusses various measures of central tendency including the mean, median, and mode. It provides definitions and formulas for calculating each measure. The mean is the average and is calculated by summing all values and dividing by the total number of data points. The median is the middle value when data is arranged in order. The mode is the value that occurs most frequently in the data set. Examples are given to demonstrate calculating each measure. The document also discusses advantages and limitations of each central tendency measure.
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.
The document discusses different measures of central tendency including the mean, median, and mode. The mean is the average value and considers all scores, but can be impacted by outliers. The median is the middle value when scores are arranged in order and is less impacted by outliers. The mode is the most frequently occurring value and can be used for both numerical and categorical data, but may not always indicate the true central tendency. Each measure has advantages and limitations, and different measures are better suited for certain types of data distributions.
This document discusses different measures of central tendency, including mathematical averages, averages of position, and measures of partition values. It describes three types of mathematical averages: arithmetic mean, geometric mean, and harmonic mean. It provides the definitions and formulas for calculating each type of mean. The document also discusses two types of averages of position: median and mode. It defines median as the middlemost value when data are arranged in order, and mode as the value with the highest frequency. For each measure of central tendency, the document outlines merits and demerits. It concludes by describing relationships between the mean, median, and mode for symmetrical versus asymmetrical distributions.
The document discusses different measures of central tendency including the mean, median, and mode. The mean is the average value calculated by adding all values and dividing by the total number of values. The median is the middle value when values are arranged from lowest to highest. The mode is the most frequently occurring value in the data set. The document provides examples of calculating each measure and discusses their advantages and disadvantages.
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.
This document provides an overview of measures of central tendency in statistics, focusing on the arithmetic mean. It defines the arithmetic mean as the sum of all values divided by the total number of items. It discusses different types of series (individual, discrete, continuous) and methods to calculate the arithmetic mean for each type, including the direct method, short-cut method, and step deviation method. The document provides examples of calculating the arithmetic mean using these different methods for an individual series of student marks.
This document discusses measures of central tendency and different methods for calculating averages. It begins by defining central tendency as a single value that represents the characteristics of an entire data set. Three common measures of central tendency are introduced: the mean, median, and mode. The document then focuses on explaining how to calculate the arithmetic mean, or average, including the direct method, shortcut method, and how it applies to discrete and continuous data series. Weighted averages are also covered. In summary, the document provides an overview of key concepts in measures of central tendency and how to calculate various types of averages.
This document discusses the concept and calculation of the mean as a measure of central tendency. It defines the mean as the sum of all values divided by the total number of items. It provides the formula for calculating the mean from both ungrouped and grouped data, using frequency tables. It gives an example of calculating the mean from an ungrouped data set and from a grouped frequency table using midpoints. It also describes a shortcut formula that can be used to calculate the mean from grouped data. Finally, it discusses when the mean is most appropriate to use, noting that it is the most reliable measure when accuracy is needed and when further statistical analysis will be done.
This document discusses various measures of central tendency used in statistics including the mean, median, and mode. It provides definitions and formulas for calculating each measure. The mean is the average value found by summing all values and dividing by the total count. The median is the middle value when values are arranged from lowest to highest. The mode is the most frequently occurring value. The document also discusses weighted mean, geometric mean, harmonic mean, and compares the properties of each central tendency measure.
The document discusses various measures of central tendency (averages) and dispersion that are used to summarize and describe data in statistics. It defines common averages like the arithmetic mean, median, mode, harmonic mean, and geometric mean. It also covers measures of dispersion such as the range, quartile deviation, mean deviation, and standard deviation. As an example, it analyzes test score data from 5 students using the arithmetic mean to find the average score.
This document discusses various measures of central tendency including the mean, median, and mode. It provides definitions and formulas for calculating each measure. The mean is the average value and is calculated by summing all values and dividing by the total number of items. The median is the middle value when data is arranged in order. The mode is the value that occurs most frequently in a data set. Examples are provided to demonstrate calculating each measure for both grouped and ungrouped data. The advantages and disadvantages of each measure are also briefly discussed.
The document discusses measures of central tendency including the mean, median, and mode. It provides definitions and formulas for calculating each measure. For the mean, formulas are given for both raw data and frequency data. The relationships between the mean, median, and mode are explored, including an empirical relationship that can be used to find one value if the other two are known for symmetrical data. The importance of each measure is discussed for different business applications depending on the characteristics of the 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.
This document provides an introduction to different types of averages, including arithmetic mean, median, and mode. It discusses:
1) Arithmetic mean is calculated by adding all values and dividing by the total number of items. It can be calculated using direct or shortcut methods for individual and discrete data series.
2) Median is the middle value that divides the data set into equal halves. It is calculated by ordering the values and finding the middle one.
3) Mode is the value that occurs most frequently in the data set. It represents the peak of the frequency distribution.
Unit 1 - Mean Median Mode - 18MAB303T - PPT - Part 1.pdfAravindS199
Sir Francis Galton was a prominent English statistician, anthropologist, eugenicist, and psychometrician in the 19th century. He produced over 340 papers and books, and created the statistical concepts of correlation and regression. As a pioneer in meteorology and differential psychology, he devised early weather maps, proposed theories of weather patterns, and developed questionnaires to study human communities and intelligence. The document discusses Galton's background and contributions to statistics, anthropology, meteorology, and psychometrics.
This document discusses measures of central tendency, which are values used to describe the center or typical value of a data set. There are three main measures: mean, median, and mode. 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 values are arranged from lowest to highest. The mode is the most frequently occurring value. The document provides formulas and examples for calculating each measure, and discusses their relative advantages and disadvantages.
The document discusses different measures of central tendency including the mean, median, mode, geometric mean, harmonic mean. It provides definitions and formulas for calculating each measure. The mean is the average calculated by adding all values and dividing by the 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. The geometric mean uses multiplication and the harmonic mean uses reciprocals of values. Examples are provided to demonstrate calculating each measure.
Central tendency of data is defined as the tendency of data to concentrate around some central value. here all the measures of central tendency have been explained such as mean, arithmetic mean, geometric mean, harmonic mean, mode, and median with examples.
This document discusses measures of central tendency. It defines measures of central tendency as summary statistics that represent the center point of a distribution. The three main measures discussed are the mean, median, and mode. The mean is the sum of all values divided by the total number of values. There are different types of means including the arithmetic mean, weighted mean, and geometric mean. The document provides formulas for calculating each type of mean and discusses their properties and applications.
This document discusses various statistical measures used to summarize data, specifically focusing on measures of central tendency. It defines measures of central tendency as indices that tell us the central or typical value of a data set. The three most popular measures of central tendency discussed are the mean, median, and mode. The mean is the average value and is computed by summing all values and dividing by the total number of observations. The median is the middle value when data is arranged in order. The mode is the most frequent value in the data set. The document provides examples of computing each measure and discusses their advantages and disadvantages. It also briefly introduces other measures like the geometric mean and harmonic mean.
This document proposes a unified approach to refine measures of central tendency and dispersion. It defines a generalized measure of central tendency as the value that minimizes the deviation between a point and a dataset. Various common measures of central tendency like mean, median, mode, geometric mean and harmonic mean are derived as special cases of this generalized definition. The concept is extended to introduce an "interval of central tendency" and methods to estimate it. Simulation studies show the interval of central tendency can capture more observations than a single point estimate, and allow comparison of different measures. The approach is also applied to derive confidence intervals for the population mean and probability of success in Bernoulli trials.
Refining Measure of Central Tendency and DispersionIOSR Journals
A unified approach is attempted to bring the descriptive statistics in to a more refined frame work. Different measure of central tendencies such as arithmetic mean, median, mode, geometric mean and harmonic mean are derived from a generalized notion of a measure of central tendency developed through an optimality criteria. This generalized notion is extended to introduce the concept of an interval of central tendency. Retaining the spirit of this notion, measure of central tendency may be called point of central tendency. The same notion is further extended to obtain confidence interval for population mean in a finite population model and confidence interval for probability of success in Bernoulli population.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
How to Setup Warehouse & Location in Odoo 17 Inventory
Measures of central tendency
1. Prepared by
M.RAJASEKHAR REDDY
Contact Number :8688683936
MEASURES OF CENTRAL TENDENCY
MEANING OF MEASURES OF CENTRAL TENDENCY:
A measure of central tendency is a single value, which
describes a set or group of data by identifying the central position within set or group of
data.
Some times Measures of central tendency are also
called as measures of central location or central value. Here the central value or location is
called average.
Definitions:
According to Croxton & Cowden:
An average value is a single value within the range of the data
that is used to represent all the values in the series.
According to Clark:
An average is a figure (number) that represents the whole group.
According to M.R.Speigal:
An average is a value, which is representative of a set of data.
Properties of Good Measures of Central Tendency:
The ideal measures of central tendency are as follows, which are
1. It should be based on all observations in given data.
2. Its definition should be in the form of a mathematical formula.
3. It should be easy to calculate
4. It should be simple to understand
5. It should be rigidly (accurately) defined.
6. It should be capable further algebraic treatments.
7. It can be found by graphical method also.
8. It should have sampling stability.
Functions (or) objectives of averages:
The main objectives of the measures of central tendency are as
follows. Which are:
1. Representative of the group
2. Brief description
2. Prepared by
M.RAJASEKHAR REDDY
Contact Number :8688683936
3. Comparison
4. Formulation of Economic policies
5. Other statistical analysis.
1. Representative of the group:
The average is the single value, which represent whole group of data.
So this single value will help to identifying entire data with in short period of time
2. Brief description:
The average gives a small or brief description of the whole
data in systematic manner.
3. Comparison:
The measure of central value is helpful in comparison with other
groups.
4. Formulation of economic policies :
The measure of central tendency helps to develop the business in
case of economical activities as well as state and central governments are also widely
using the averages to formulate the economic policies.
5. Other statistical analysis:
The use of averages becomes compulsory for other statistical analysis,
such as index numbers, analysis of time series etc.
Types of averages (or) measures of central tendency:
The measures of central tendency (or) the averages are classified in to
the following types .which are
Types of Averages (or) Different
Measures of Central Tendency:
• Arithmetic Mean
• Geometirc Mean
• Harmonic Mean
• Median
• Mode
3. Prepared by
M.RAJASEKHAR REDDY
Contact Number :8688683936
1. Arithmetic Mean:
The most widely & popular used measure for representing the
entire data by on value is arithmetic mean. It is a scientific method. Here each & every
item taken in to account.
Arithmetic Mean (A.M) means adding all items then the number of
items to be divided. The result is known as A.M (or) X bar ( ).
Sum of All the Items (or) Observations.
A.M= =
Number of items (or) Observations.
X1+X2+X3+…………….Xn
A.M= X =
n
∑ X
A.M= X =
n
Where ∑ X =sum of all observation in the given data
n = number of observation in the given data
Merits of Arithmetic Mean:
1. It is simple to understand and easy to calculate
2. In the calculation of mean each & every item or observation is taken in to account.
3. It is good to comparison
4. It is possible to calculate even some of the details of the data are locking or
unknown.
5. It doesn’t require arranging the data in the order i.e., ascending or descending order.
Demerits of Arithmetic Mean:
1. It can’t find by the simple observation of the given series or data.
2. If any item of the series is ignored, then the accuracy of the mean will be affected.
3. It is not possible to find graphically.
4. It can’t be calculated exactly in case of open-end classed.
Types of Arithmetic Mean:
There are two types of arithmetic mean, which are
4. Prepared by
M.RAJASEKHAR REDDY
Contact Number :8688683936
1. Simple (or) Unweighted Arithmetic Mean
2. Weighted Arithmetic Mean
1. Simple (or) Unweighted Arithmetic Mean:
The simple A.M can be calculated for 3 types of data .which are
1. Calculation of A.M for personal data or individual data
2. Calculation of A.M for discrete data
3. Calculation of A.M for continuous data
Calculation of A.M for personal data or Individual data:
If the given data is individual data then A.M can be
calculated as follows.
Add all the observations(x )in the given data, it gives ∑x
Count the number of observation, it gives n
Apply the following formula
X1+X2+X3+…………….Xn
A.M= X =
n
∑ X
A.M= X =
n
Calculation of A.M for discrete data:
If the given data is discrete data then A.M can be calculated as
follows.
Calculate the sum of frequencies ,it gives N. i,e., N=∑f
Multiply each frequency value(f) with the corresponding observation value(x),it gives
fx
Calculate the sum of all fx values ,it gives ∑fx
Apply the following formula
f1 x1+f2 x2+f3 x3+…………….fn xn
A.M= X =
N
∑f x
A.M= X =
N
Where ∑f x= the sum of multiplication of f and x
∑f = the sum of frequencies
5. Prepared by
M.RAJASEKHAR REDDY
Contact Number :8688683936
Calculation of A.M for continuous data:
If the given data is continuous data then A.M can be calculated as
follows.
Calculate the sum of frequencies ,it gives N. i,e., N=∑f
Find the mid values(x) in a separate column for the given class intervals (C.I).the mid
values can be calculated by using the following formula.
Upper limit +Lower limit
Mid value(x) =
2
Multiply each frequency value(f) with the corresponding mid value(x),it gives fx
Calculate the sum of all fx values ,it gives ∑fx
Apply the following formula
f1 x1+f2 x2+f3 x3+…………….fn xn
A.M= X =
N
∑f x
A.M=X =
N
Where ∑f x= the sum of multiplication of f and x
∑f = the sum of frequencies
2. Weighted Arithmetic Mean:
The weighted A.M calculated as follows
Multiply weights(w) by the variables(x) and add up the wx values, it gives ∑wx
Calculate the sum of weights ,it gives ∑w
Apply the following formula
∑wx
XW =
∑w
Where XW = weighted arithmetic mean
W=weights
X =variables
2. Geometric Mean :
6. Prepared by
M.RAJASEKHAR REDDY
Contact Number :8688683936
Geometric mean is obtained by multiplying the values of the items &
extracting the root of the product corresponding to the number of items.
G.M is defined as the nth root of the products of n items or values. If
there are two items we can take the square root, if there are three items we can take cube
root so on…….if there are n items then we can take the nth root so on….
G.M=n√x1 x2 x3 ……..xn
G.M=( x1 x2 x3 ……..xn)1/n
Take log on both sides
Log (G.M) =Log (x1 x2 x3 ……..xn)1/n
Log (G.M) = (1/n) [Log (x1 x2 x3 ……..xn)]
Log (G.M) = (1/n) [log x1+log x2+log x3+…….log xn]
Log (G.M) = (1/n) [∑logx]
Merits of Geometric Mean:
It is rigidly defined
It is based on all observations
As compared with mean, geometric mean is affected to a lesser by extreme
observations
It is not much affected much by fluctuations of sampling
It is useful for construction of index numbers
It can be calculated with mathematical precision provided all the values are positive
Demerits of Geometric Mean:
It can’t be used when any observation is zero or negative value
It is not easily understand
It is very difficult to calculate
It can’t be computed if any value is missing
Calculation of G.M:
The geometric mean can be calculated for 3 types of data .which are
1. Calculation of G.M for personal data or individual data
2. Calculation of G.M for discrete data
3. Calculation of G.M for continuous data
Calculation of G.M for personal data or Individual data:
G.M= Antilog [(∑logx)/n]
7. Prepared by
M.RAJASEKHAR REDDY
Contact Number :8688683936
If the given data is individual data then G.M can be
calculated as follows.
Find out the logarithm of each value (x),it gives log(x) values
Add all the values of log(x),it gives ∑log(x)
Count the number of observation, it gives n
Apply the following formula
Calculation of G.M for discrete data:
If the given data is discrete data then G.M can be calculated
as follows.
Calculate the sum of frequencies ,it gives N. I,e., N=∑f
Find out the logarithm of each value, it gives log(x)
Multiply each log(x) value with its corresponding frequency value(f),it gives flog(x)
Add all the flog(x) values, it gives ∑ flog(x)
Apply the formula
Where ∑f log(x) = the sum of multiplication of f and log(x)
N=∑f = the sum of frequencies
Calculation of G.M for continuous data:
If the given data is continuous data then G.M can be calculated as
follows.
Calculate the sum of frequencies ,it gives N. i,e., N=∑f
Find the mid values(x) in a separate column for the given class intervals (C.I).the mid
values can be calculated by using the following formula.
Upper limit +Lower limit
Mid value(x) =
2
Find out the logarithm of each mid value(x), it gives log(x)
Multiply each log(x) value with its corresponding frequency value(f),it gives flog(x)
Add all the flog(x) values, it gives ∑ flog(x)
Apply the formula
G.M= Antilog [(∑logx)/n]
G.M= Antilog [(∑flogx)/N]
G.M= Antilog [(∑flogx)/N]
8. Prepared by
M.RAJASEKHAR REDDY
Contact Number :8688683936
Where ∑f log(x) = the sum of multiplication of f and log(x)
N=∑f = the sum of frequencies
3. Harmonic Mean :
The harmonic mean is based on the reciprocal of numbers averaged. It is defined as the
reciprocal of the A.M of the reciprocal of the individual observations.
If X1,X2,X3,……………,Xn are n observations then the harmonic mean of these
observations will be as follows.
n
H.M=
(1/ X1)+ (1/ X2)+ (1/ X3)+……. (1/ Xn)
Where n= number of observations
Merits of Harmonic Mean:
H.M is rigidly defined
It is based on all the observations
It is suitable for further mathematical treatment
It is not affected by fluctuations of sampling
It gives greater importance to small items and also it is useful only when small items
have to be given a greater weight
Demerits of Harmonic Mean:
It is not easy to understand
It is difficult to calculate
It gives greater importance to small items
Calculation of H.M:
The harmonic mean can be calculated for 3 types of data .which are
1. Calculation of H.M for personal data or individual data
2. Calculation of H.M for discrete data
3. Calculation of H.M for continuous data
Calculation of H.M for personal data or Individual data:
H.M=n/[∑(1/x)]
9. Prepared by
M.RAJASEKHAR REDDY
Contact Number :8688683936
If the given data is individual data then
H.M can be calculated as follows.
Find the reciprocal of each observation ,it gives (1/x)
Add all the reciprocals ,it gives ∑(1/x)
Count the number of observations, it gives n
Apply the formula
Calculation of H.M for discrete data:
If the given data is discrete data then H.M can be calculated as
follows.
Calculate the sum of frequencies, It gives N. i,e., N=∑f
Find the reciprocal of each observation ,it gives (1/x)
Multiply the reciprocal (1/x) of each observation by its corresponding frequency ,it
gives f(1/x)
Add all the f(1/x) values , it gives ∑f(1/x)
Apply the formula
Where N=∑f = the sum of frequencies
Calculation of H.M for continuous data:
If the given data is continuous data then H.M can be calculated
as follows.
Calculate the sum of frequencies, it gives N. i,e., N=∑f
Find the mid values(x) in a separate column for the given class intervals (C.I).the mid
values can be calculated by using the following formula.
Upper limit +Lower limit
Mid value(x) =
2
Find the reciprocal of each mid value ,it gives (1/x)
Multiply the reciprocal (1/x) of each mid value by its corresponding frequency ,it
gives f(1/x)
Add all the f(1/x) values , it gives ∑f(1/x)
Apply the formula
H.M=n/ [∑ (1/x)]
H.M=N/ [∑f (1/x)]
H.M=N/ [∑f (1/x)]
10. Prepared by
M.RAJASEKHAR REDDY
Contact Number :8688683936
Where N=∑f = the sum of frequencies
4. Median :
Median is the value which divides the given series into two equal parts, when
series arranged in ascending order (or) descending order.
The number of items less than the median value and the number of items more
than the median value will be equal.
Merits of Median:
It is easy to understand and easy to calculate
It can be calculated by graphically
Median can be calculated in case of open-end classes
It can be located by inspection, after arranging the data in to ascending or
descending order.
Median is also use to calculate median deviation & standard deviation.
Demerits of Median:
It may not show correct values, if the series was not arranged in order.
All the items of series are not taken into account
It is not based on all observations. So it is called positional measure also.
Calculation of Median:
The median can be calculated for 3 types of data .which are
1. Calculation of Median for personal data or individual data
2. Calculation of Median for discrete data
3. Calculation of Median for continuous data
Calculation of Median for personal data or Individual data:
If the given data is individual data then Median can be
calculated as follows.
Arrange the observations in ascending /descending order
Locate middle value
If the number of observations(n) are odd, then median is the middle value
Or
If the number observations are odd, then Median =[(n+1)/2 ]thterm
If the number of observations(n) are even, then median is the average of
middle two values
11. Prepared by
M.RAJASEKHAR REDDY
Contact Number :8688683936
Or
If the numbers of observations (n) are even, then
[(n/2)th term +((n/2)+1)th term]
Median =
2
Calculation of Median for discrete data:
If the given data is discrete data then Median can be calculated as
follows.
Arrange the data in ascending /descending order
Calculate the sum of frequencies, it gives N. i,e., N=∑f
Calculate the cumulative frequency by adding the frequency values one by one
Locate the median value based on (N/2)th term value in cumulative frequency ,if it is
not exist in cumulative frequency, then find the greater value of (N/2)th term
apply the formula
Median = (N/2)th term
Where N= sum of frequencies
Calculation of Median for continuous data:
If the given data is continuous data then Median can be calculated as follows.
Calculate the sum of frequencies, it gives N. i,e., N=∑f
Calculate the cumulative frequency by adding the frequency values one by one
Find (N/2)th term
Find (N/2)th term value in cumulative frequency ,if it is not exist in cumulative
frequency, then find the greater value of (N/2)th term
Identify L,f,m values in the table
Apply the formula
[(N/2)-m]
Median=L+ *C
f
Mode:
Mode may be defined as the value that occurs most frequently in a statistical
distribution. It is denoted by Z.
Merits of Mode:
It is easy to understand
It is not affected by the extreme values
It can be calculated for open-end class intervals
12. Prepared by
M.RAJASEKHAR REDDY
Contact Number :8688683936
It can be calculated by the graphical method also
It is usually an actual value as it occurs most frequently in the series.
Demerits of Mode:
it is not based on all observations
it is not capable for further mathematical treatment
As compared with mean, mode is affected much by fluctuations of sampling.
Some time mode can’t be identify clearly
Calculation of Mode:
The mode can be calculated for 3 types of data .which are
1. Calculation of Mode for personal data or individual data
2. Calculation of Mode for discrete data
3. Calculation of Mode for continuous data
Calculation of Mode for personal data or Individual data:
If the given data is individual data then Mode can be
calculated as follows.
Maximum repeated value in the given data is called mode
Calculation of Mode for discrete data:
If the given data is discrete data then Mode can be calculated as
follows.
Mode can be obtained by inspection. The value of the variable having maximum
frequency is known as modal value.
Calculation of Mode for continuous data:
If the given data is discrete data then Mode can be calculated using the
following formula
f - f1
Mode (Z) =L+ *c
2f-f1-f2
Where L=lower limit of modal class
C=class interval
f= frequency of modal class