1. Chapter 5 Measures of Variability

2. 2 Measures of Variability Major Points The general problem The general problem Range and related statistics Range and related statistics Deviation scores Deviation scores The variance and standard deviation The variance and standard deviation Boxplots Boxplots Review questions Review questions

3. 3 Measures of Variability The General Problem Central tendency only deals with the center Central tendency only deals with the center Dispersion Dispersion Variability of the data around some midpoint, usually mean Variability of the data around some midpoint, usually mean The spread of the points The spread of the points Its important to look at dispersion in relation to the central tendency, data sets with similar means may have very different dispersion Its important to look at dispersion in relation to the central tendency, data sets with similar means may have very different dispersion

4. 4 Measures of Variability 15 Measures of Variability Describe the degree to which scores vary around the central tendency indicator; they provide single numbers that summarize the degree to which individual scores in the distribution differ from one another. This is sometimes called spread. When scores are equal there is no spread or variability (equals 0).Describe the degree to which scores vary around the central tendency indicator; they provide single numbers that summarize the degree to which individual scores in the distribution differ from one another. This is sometimes called spread. When scores are equal there is no spread or variability (equals 0). What! I can’t take anymore

5. Peanut Butter Spread Clumped in middle, or spread across your slice of bread Measures of Variability 5

8. 8 16 Measures of Variability Range: The distance or difference between the largest and smallest scores.Range: The distance or difference between the largest and smallest scores. X H-L = Range X Unstable because it relies on only 2 scores Variance & Standard Deviation: take into account every score in the distributionVariance & Standard Deviation: take into account every score in the distribution X Average distance of scores from the mean X More stable than range Con’t

9. 9 Measures of Variability Range and Related Statistics The range The range Distance from lowest to highest score Distance from lowest to highest score Too heavily influenced by extremes Too heavily influenced by extremes The interquartile range (IQR) The interquartile range (IQR) Delete lowest and highest 25% of scores Delete lowest and highest 25% of scores IQR is range of what remains IQR is range of what remains May be too little influenced by extremes May be too little influenced by extremes

10. 10 Measures of Variability Trimmed Samples Delete a fixed (usually small) percentage of extreme scores (outliers) Delete a fixed (usually small) percentage of extreme scores (outliers) Trimmed statistics are statistics computed on trimmed samples. Trimmed statistics are statistics computed on trimmed samples.

11. 11 Measures of Variability Deviation Scores Definition Definition distance between a score and a measure of central tendency distance between a score and a measure of central tendency usually deviation around the mean usually deviation around the mean Importance Importance

12. 12 Measures of Variability Why use variance and standard deviation? While the variance is useful for inferential statistics, the standard deviation is more useful as a descriptive measure because it uses the same units as original scores.

13. 13 Measures of VariabilityVariance Definitional formula Definitional formula Example Example See next slide See next slide

14. 14 Measures of Variability Calculation

15. 15 Measures of Variability Standard Deviation Definitional formula Definitional formula The square root of the variance The square root of the variance

16. 16 Measures of VariabilityEstimators Mean Mean Unbiased estimate of population mean (  ) Unbiased estimate of population mean (  ) Define unbiased Define unbiased  Long range average of statistic is equal to the parameter being estimated. Variance Variance Unbiased estimate of  2 Unbiased estimate of  2 If using N-1 in demoninator Cont.

17. 17 Measures of VariabilityEstimators--cont. Standard deviation Standard deviation use square root of unbiased estimate. use square root of unbiased estimate.

18. 18 Measures of Variability Merrell’s Music Study SPSS Printout WEEK4 TreatmentMeanNStd. Deviation Quiet 307.231923 71.8267 Mozart 114.583324 36.1017 Anthrax 1825.888924 103.1392 Total 755.460171 777.9646

19. 19 Measures of VariabilityBoxplots The general problem The general problem A display that shows dispersion for center and tails of distribution A display that shows dispersion for center and tails of distribution Calculational steps (simple solution) Calculational steps (simple solution) Find median Find median Find top and bottom 25% points (quartiles) Find top and bottom 25% points (quartiles) eliminate top and bottom 2.5% (fences) eliminate top and bottom 2.5% (fences) Draw boxes to quartiles and whiskers to fences, with remaining points as outliers Draw boxes to quartiles and whiskers to fences, with remaining points as outliers Boxplots for comparing groups Boxplots for comparing groups

20. 20 Measures of Variability Combined Merrell Data

21. 21 Measures of Variability Merrell Data by Group

22. 22 Measures of Variability Review Questions What do we look for in a measure of dispersion? What do we look for in a measure of dispersion? What role do outliers play? What role do outliers play? Why do we say that the variance is a measure of average variability around the mean? Why do we say that the variance is a measure of average variability around the mean? Why do we take the square root of the variance to get the standard deviation? Why do we take the square root of the variance to get the standard deviation? Cont.

23. 23 Measures of Variability Review Questions --cont. How does a boxplot reveal dispersion? How does a boxplot reveal dispersion? What do David Merrell’s data tell us about the effect of music on rats? What do David Merrell’s data tell us about the effect of music on rats?