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Review

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. Review. Find the mean, median, mode. Review. Find the mean, median, mode

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Review

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  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, 50th percentile • Mode—highest frequency, can be more than one

  2. Review • Find the mean, median, mode

  3. Review • Find the mean, median, mode • Mean=sum of all scores(∑fX) /number of scores(N)

  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

  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

  6. 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 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 • We often see greater variability with larger mean

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

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

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

  14. Variance Definitional formula Example See next slide

  15. Computing the Variance ¯ ¯ • Definitional formula • Find the mean • N=6 • ∑X=30 • 30/6=5

  16. Computing the Variance ¯ ¯ • Calculate the difference between each score and the mean and sum

  17. Computing the Variance ¯ ¯ • Calculate the square of the difference between each score and the mean and sum • Standard Deviation is the square root

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

  19. Computational Formula

  20. Try one

  21. Try one

  22. Try one

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

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

  27. Merrell’s Music Study SPSS Printout • WEEK4 • Treatment Mean N Std. Deviation • Quiet 307.2319 23 71.8267 • Mozart 114.5833 24 36.1017 • Anthrax 1825.8889 24 103.1392 • Total 755.4601 71 777.9646

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

  29. Combined Merrell Data

  30. Merrell Data by Group

  31. 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.

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

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