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Probability and Statistics
Dr. Saeid Moloudzadeh
Measures of variation
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
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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
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Chapter 1: Descriptive Statistics
Contents0.1 Fundamental Concepts0.2 Frequency table and graphs0.3 Measures of center0.4 Measures of variation
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Chapter 1: Descriptive Statistics
Contents0.1 Fundamental Concepts0.2 Frequency table and graphs0.3 Measures of center0.4 Measures of variation
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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:
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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.
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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.
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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:
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Section 4: Measures of variation
Example 0.16: Consider the example 1-2-5 and calculate the following expressions:a. Rangb. Absolute deviationc. Varianced. Standard deviatione. Coefficient of VariationSolution:f. Range = Max −Min=76-48=28.g. We have briefly necessary calculation in the following table:
