Probability and Statistics Dr. Saeid Moloudzadeh Measures of variation 1 Contents Descriptive Statistics Axioms of Probability Combinatorial Methods Conditional Probability and Independence Distribution Functions and Discrete Random Variables Special Discrete Distributions Continuous Random Variables Special Continuous Distributions Bivariate Distributions

Probability and Statistics Contents Descriptive Statistics Axioms of Probability Combinatorial Methods Conditional Probability and Independence Distribution Functions and Discrete Random Variables Special Discrete Distributions Continuous Random Variables Special Continuous Distributions Bivariate Distributions 2

Chapter 1: Descriptive Statistics Contents 0.1 Fundamental Concepts 0.2 Frequency table and graphs 0.3 Measures of center 0.4 Measures of variation 3

Chapter 1: Descriptive Statistics Contents 0.1 Fundamental Concepts 0.2 Frequency table and graphs 0.3 Measures of center 0.4 Measures of variation 4

Section 4: Measures of variation Sometimes mean, median and mode may not be able to reflect the true picture of some data. The following example explains the reason. Example 0.14: There were two companies, Company A and Company B. Their salaries profiles given in mean, median and mode were as follow: 5

Section 4: Measures of variation However, their detail salary ($) structures could be completely different as that: Hence it is necessary to have some measures on how data are scattered. That is, we want to know what is the dispersion, or variability in a set of data. 6

Section 4: Measures of variation Definition (Range): The sample range of the variable is the difference between its maximum and minimum values in a data set: Range = Max −Min. The sample range of the variable is quite easy to compute. However, in using the range, a great deal of information is ignored, that is, only the largest and smallest values of the variable are considered. Example 0.15: Two corporations each hired 10 graduates. The starting salaries for each are shown. Find the range of the starting salaries for Corporation A. 7

Section 4: Measures of variation 8

Section 4: Measures of variation 9

Section 4: Measures of variation cancel. Instead, we should be concerned about the individual deviations without regard to their signs. This can be accomplished either by considering the absolute values of the deviations or, as turns out to be more useful, by considering their squares. Definition (Mean absolute deviation): Mean absolute deviation is the mean of the absolute values of all deviations from the mean. Therefore it takes every item into account. Mathematically it is given as: 10

Section 4: Measures of variation 11

Section 4: Measures of variation 12

Section 4: Measures of variation Example 0.16: Consider the example and calculate the following expressions: a.Rang b.Absolute deviation c.Variance d.Standard deviation e.Coefficient of Variation Solution: a.Range = Max −Min=76-48=28. b.We have briefly necessary calculation in the following table: 13

Section 4: Measures of variation 14

Section 4: Measures of variation 15