1 Review Mean—arithmetic average, sum of all scores divided by the number of scores Median—balance point of the data, exact middle of the distribution, 50 th percentile Mode—highest frequency, can be more than one

2 Review Find the mean, median, mode Xf

3 Review Mean=sum of all scores(∑fX) /number of scores(N) XffX N∑fX

4 Review Find the mean, median, mode Mean=sum of all scores(∑fX) /number of scores(N) Median=middle point (N-1/2) th position XffX

5 Review Find the mean, median, mode Mean=sum of all scores(∑fX) /number of scores(N) Median=middle point (N-1/2) th position Mode=greatest f XffX

Measures of Variability

7 Major Points The general problem Range and related statistics Deviation scores The variance and standard deviation Boxplots Review questions

8 The General Problem Central tendency only deals with the center Dispersion Variability of the data around something Variability of the data around something The spread of the points The spread of the points Example: Mice and Music

9 Mice and Music Study by David Merrell Raised some mice in quiet environment Raised some mice listening to Mozart Raised other mice listening to Anthrax Dependent variable is the time to run a straight alley maze after 4 weeks. Borrowed from David Howell, 2000

10 Results Anthrax mice took much longer to run Much greater variability in Anthrax group See following graphs for Anthrax and Mozart See following graphs for Anthrax and Mozart We often see greater variability with larger mean

13 Range and Related Statistics 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) 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

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

15 Deviation Scores 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 meanImportance

16 Variance Definitional formula Example See next slide See next slide

17 Definitional formula Find the mean N=6∑X=3030/6=5X X - X (X - X) ¯¯ Computing the Variance

18 Computing the Variance X X - X (X - X) ¯¯ Calculate the difference between each score and the mean and sum

19 Computing the Variance X X - X (X - X) ¯¯ Calculate the square of the difference between each score and the mean and sum Standard Deviation is the square root

20 Standard Deviation Definitional formula The square root of the variance The square root of the variance Computational formula based on algebraic manipulation Makes it easier to calculate Makes it easier to calculate

21 Computational Formula

22 Try one Xf N

23 Try one XffX N∑fX

24 Try one XffX

25 Try one XffX X2X2X2X2 fX ∑fX 2

26 Try one XffX X2X2X2X2 fX

27 XffX X2X2X2X2 fX

28 Estimators Mean Unbiased estimate of population mean (  ) Unbiased estimate of population mean (  ) Define unbiased Long range average of statistic is equal to the parameter being estimated. Long range average of statistic is equal to the parameter being estimated.Variance Unbiased estimate of  2 Unbiased estimate of  2 Cont.

29 Estimators--cont. Using Using gives biased estimate Standard deviation Standard deviation use square root of unbiased estimate.

30 Merrell’s Music Study SPSS Printout WEEK4 TreatmentMeanNStd. Deviation Quiet Mozart Anthrax Total

31 Boxplots 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) 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

32 Combined Merrell Data

33 Merrell Data by Group

34 Review Questions What do we look for in a measure of dispersion? What role do outliers play? 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? Cont.

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