Descriptive Statistics: Overview

1. Measures of Spread Measures of Center Range Inter-quartile Range VarianceStandard deviation Mode Median Mean * * * Measures of Symmetry Measures of Position Skewness Percentile Deviation ScoreZ-score * * Descriptive Statistics: Overview

2. Central tendency • Seeks to provide a single value that best represents a distribution

3. Central tendency

4. Central tendency

5. Central tendency

6. Central tendency • Seeks to provide a single value that best represents a distribution • Typical measures are • mode • median • mean

7. For a given sample N=16: 33 35 36 37 38 38 38 39 39 39 39 40 40 41 41 45 The mode = 39 Mode • the most frequently occurring score value • corresponds to the highest point on the frequency distribution

8. For a given sample N=16: 33 35 36 37 38 38 38 39 39 39 39 40 40 41 41 50 The mode = 39 Mode • The mode is not sensitive to extreme scores.

9. For a given sample N=16: 34 34 35 35 35 35 36 37 38 38 39 39 39 39 40 40 The modes = 35 and 39 Mode • a distribution may have more than one mode

10. For a given sample N=16: 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 No unique mode Mode • there may be no unique mode, as in the case of a rectangular distribution

11. Median • the score value that cuts the distribution in half (the “middle” score) • 50th percentile For N = 15 the median is the eighth score = 37

12. Median For N = 16 the median is the average of the eighth and ninth scores = 37.5

13. For a sample: For a population: Mean • this is what people usually have in mind when they say “average” • the sum of the scores divided by the number of scores Changing the value of a single score may not affect the mode or median, but it will affect the mean.

14. __ 5 X=2.4 In order for the mean to be meaningful, the variable of interest must be measures on an interval scale. 4 3 Frequency 2 1 0 Jewish Muslim Catholic Buddhist Protestant Score Mean __ In many cases the mean is the preferred measure of central tendency, both as a description of the data and as an estimate of the parameter. X=7.07

15. __ __ X=36.8 X=36.5 __ X=93.2 Mean The mean is sensitive to extreme scores and is appropriate for more symmetrical distributions.

16. Symmetry • a symmetrical distribution exhibits no skewness • in a symmetrical distribution the Mean = Median = Mode

17. median mode mean Skewed distributions • Skewness refers to the asymmetry of the distribution • A positively skewed distribution is asymmetrical and points in the positive direction. Mode = 70,000$ Median = 88,700$ Mean = 93,600$ • mode < median < mean

18. median mode mean Skewed distributions • A negatively skewed distribution • mode > median > mean

19. Measures of central tendency

20. Distributions • Center: mode, median, mean • Shape: symmetrical, skewed • Spread

21. Measures of Spread • the dispersion of scores from the center • a distribution of scores is highly variable if the scores differ wildly from one another • Three statistics to measure variability • range • interquartile range • variance

22. Range • largest score minus the smallest score • these two have same range (80) but spreads look different • says nothing about how scores vary around the center • greatly affected by extreme scores (defined by them)

23. Interquartile range • the distance between the 25th percentile and the 75th percentile • Q3-Q1 = 70 - 30 = 40 • Q3-Q1 = 52.5 - 47.5 = 5 • effectively ignores the top and bottom quarters, so extreme scores are not influential • dismisses 50% of the distribution

24. Deviation measures • Might be better to see how much scores differ from the center of the distribution -- using distance • Scores further from the mean have higher deviation scores

25. Deviation measures • To see how ‘deviant’ the distribution is relative to another, we could sum these scores • But this would leave us with a big fat zero

26. Deviation measures So we use squared deviations from the mean This is the sum of squares (SS) __ SS= ∑(X-X)2

27. For a population: For a sample: Variance We take the “average” squared deviation from the mean and call it VARIANCE (to correct for the fact that sample variance tends to underestimate pop variance)

28. Variance • Find the mean. • Subtract the mean from every score. • Square the deviations. • Sum the squared deviations. • Divide the SS by N or N-1.

29. Standard deviation The standard deviation is the square root of the variance The standard deviation measures spread in the original units of measurement, while the variance does so in units squared. Variance is good for inferential stats. Standard deviation is nice for descriptive stats.

30. Example N = 28 X = 50 s2 = 140.74 s = 11.86 N = 28 X = 50 s2 = 555.55 s = 23.57

31. Measures of Center Measures of Spread Range Inter-quartile Range VarianceStandard deviation Mode Median Mean * * * * * * Measures of Symmetry Skewness Descriptive Statistics: Quick Review

32. Mean Variance Standard Deviation Descriptive Statistics: Quick Review For a population: For a sample:

33. Exercise • Treat this little distribution as a sample and calculate: • Mode, median, mean • Range, variance, standard deviation

34. Measures of Spread Measures of Center Range Inter-quartile Range VarianceStandard deviation Mode Median Mean * * * Measures of Symmetry Measures of Position Skewness Percentile Deviation ScoreZ-score * * Descriptive Statistics: Overview

35. How to describe a data point in relation to its distribution Measures of Position

36. Measures of Position Quantile Deviation Score Z-score

37. Divides ranked scores into four equal parts 25% 25% 25% 25% (median) (minimum) (maximum) Quantiles Quartile

38. Divides ranked scores into ten equal parts 10% 10% 10% 10% 10% 10% 10% 10% 10% 10% Quantiles Decile

39. Quantiles Percentile rank Divides ranked scores into 100 equal parts number of scores less than x Percentile rank of score x= • 100 total number of scores

40. Deviation Scores For a population: For a sample:

41. What if we want to compare scores from distributions that have different means and standard deviations? • Example • Nine students scores on two different tests • Tests scored on different scales

42. Nine Students on Two Tests

43. Nine Students on Two Tests

44. Z-Scores • Z-scores modify a distribution so that it is centered on 0 with a standard deviation of 1 • Subtract the mean from a score, then divide by the standard deviation For a population: For a sample:

45. Z-Scores

46. Z-Scores A distribution of Z-scores… • Always has a mean of zero • Always has a standard deviation of 1 • Converting to standard or z scores does not change the shape of the distribution: z scores cannot normalize a non-normal distribution A Z-score is interpreted as “number of standard deviations above/below the mean”

47. Exercise On their third test, the class average was 45 and the standard deviation was 6. Fill in the rest.

48. Standard Deviation Descriptive Statistics: Quick Review For a population: For a sample: Mean Variance Z-score

49. Messing with Units • If you add or subtract a constant from each value in a distribution, then • the mean is increased/decreased by that amount • the standard deviation is unchanged • the z-scores are unchanged • If you multiply or divide each value in a distribution by a constant, then • the mean is multiplied/divided by that amount • the standard deviation is multiplied/divided by that amount • the z-scores are unchanged

50. Example