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MEASURE OF DISPERSION
- 1. Group MembersGroup Members CopyRight and Designed By Students Era .com Hasnan 015 Naeem 078 Shahbaz 002 Zainab 008 Pola 088 Saqib 036
- 2. Measures of dispersion Learning Objectives Calculate common measures of dispersion from grouped and ungrouped data (including the range, interquartile range, mean deviation, and standard deviation) Calculate and interpret the coefficient of variation © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 2 Chapter S4
- 3. The range Simply the difference between the largest and smallest values in a set of data Useful for: daily temperature fluctuations or share price movement Is considered primitive as it considers only the extreme values which may not be useful indicators of the bulk of the population. The formula is: Range = largest observation - smallest observation © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 3
- 4. Interquartile range Measures the range of the middle 50% of the values only Is defined as the difference between the upper and lower quartiles Interquartile range = upper quartile - lower quartile = Q3- Q1 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 4
- 5. The mean deviation Measures the ‘average’ distance of each observation away from the mean of the data Gives an equal weight to each observation Generally more sensitive than the range or interquartile range, since a change in any value will affect it © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 5
- 6. Actual and absolute deviations from mean A set of xvalues has a mean of The residualresidual of a particular x-value is: Residual or deviation = x - The absolute deviationabsolute deviation is: © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 6 x x x-x
- 7. Mean deviation The mean of the absolute deviations © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 7 n xx deviationMean ∑ − =
- 8. To calculate mean deviation 1.Calculate mean of data Find x 2.Subtract mean from each observation Record the differences For each x, find xx − 3.Record absolute value of each residual Find xx − for each x 4.Calculate the mean of the absolute values n xx deviationMean ∑ − = Add up absolute values and divide by n © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 8
- 9. The standard deviation Measures the variation of observations from the mean The most common measure of dispersion Takes into account every observation Measures the ‘average deviation’ of observations from mean Works with squares of residuals not absolute values—easier to use in further calculations © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 9
- 10. Standard deviation of a population δ Every observation in the population is used. The square of the population standard deviation is called the variance. © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 10 ( ) n xx δdeviationdardtanS ∑ − == 2 2 δVariance =
- 11. Standard deviation of a sample s In practice, most populations are very large and it is more common to calculate the sample standard deviation. Where: (n-1) is the number of observations in the sample © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 11 ( ) 1 2 − − == ∑ n xx sdeviationdardtansSample
- 12. To calculate standard deviation © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 12 1. Calculate the mean x 2. Calculate the residual for each x xx − 3. Square the residuals 2 )( xx − 4. Calculate the sum of the squares ( )2 ∑ − xx 5. Divide the sum in Step 4 by (n-1) ( ) 1 2 − ∑ − n xx 6. Take the square root of quantity in Step 5 ( ) 1 2 − ∑ − n xx
- 13. Standard deviations for frequency distributions If data is in a frequency distribution Calculate standard deviation using: © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 13 No. Units n Frequency f 1 85 2 192 3 123 Total 400 Total ( ) ∑ ∫ − ∑ ∫ − = 1 2 xx s
- 14. Coefficient of variation Is a measure of relative variability used to: measure changes that have occurred in a population over time compare variability of two populations that are expressed in different units of measurement expressed as a percentage rather than in terms of the units of the particular data © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 14
- 15. Formula for coefficient of variation Denoted by V © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 15 %100 = x s V where = the mean of the sample s = the standard deviation of the sample x
- 16. Summary Measures of central tendency no ideal measure of dispersion exists standard deviation is the most important measure of central tendency it is the most frequently used the value is affected by the value of every observation in the data extreme values in the population may distort the data © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 16