1. Anthony J Greene1 Dispersion Outline What is Dispersion? I Ordinal Variables 1.Range 2.Interquartile Range 3.Semi-Interquartile Range II Ratio/Interval Variables 1.Variance 2.Standard Deviation

2. Anthony J Greene2 Significant Differences? μ 1 = 40 μ 2 =60

3. Anthony J Greene3 Significant Differences? μ 1 = 40 μ 2 =60

4. Anthony J Greene4 Dispersion is the Measure of Spread

5. Anthony J Greene5 Measures of Dispersion

6. Anthony J Greene6 Range

7. Anthony J Greene7 Range The range of a data set is the difference between its maximum and minimum observations: Range = Max – Min. –Use Lower Real Limits: The Min is not merely the lowest score its any score that could be rounded up to the lowest score. –Use Upper Real Limits: Likewise the Max is any score that could be rounded down to the lowest score. –For integer values this generally amounts to adding 0.5 to the highest to get the max, and subtracting 0.5 from the lowest score to get the min.

8. Anthony J Greene8 Quartiles Let n denote the number of observations. Arrange the data in increasing order. The first quartile is at position (n + 1)/4. The second quartile is the median, which is at position (n + 1)/2. The third quartile is at position 3(n + 1)/4. If a position is not a whole number, linear interpolation is used to find the fraction representing the quartile.

9. Anthony J Greene9 Interquartile Range The interquartile range, denoted IQR, is the difference between the first and third quartiles; that is, IQR = Q 3 – Q 1 Roughly speaking, the IQR gives the range of the middle 50% of the observations.

10. Anthony J Greene10 The Interquartile Range

11. Anthony J Greene11 Five Number Summary The five-number summary of a data set consists of the minimum, maximum, and quartiles written in increasing order: Min, Q 1, Q 2, Q 3, Max.

12. Anthony J Greene12 Quartiles

13. Anthony J Greene13 Box & Whiskers Plots

14. Anthony J Greene14 Box & Whiskers Plots

15. Anthony J Greene15 Box & Whiskers Plots

16. Anthony J Greene16 Standard Deviation 68% 95%

17. Anthony J Greene17 Standard Deviation 68% 95%

18. Anthony J Greene18 Standard Deviation 68% 95%

19. Anthony J Greene19 Standard Deviation of a Discrete Random Variable The population standard deviation of a discrete random variable X is denoted by  and is defined by Or the computational formula The variance, V, is the square of the standard deviation V=  2

20. Anthony J Greene20 Variance is the Average Squared Deviation Average Deviation is Zero Average Squared Deviation: V = Σ(x-μ) 2 /N -6 -15 -17 x 2 -20 -22 -23 -27 +1 +2 +4 +6 +9 x 3 +11 +14 x 2 +15 +16 +18 +20 μ = 33

21. Anthony J Greene21 Samples and Populations

22. Anthony J Greene22 Population and Sample Variability

23. Anthony J Greene23 Sample Standard Deviation For a variable x, the standard deviation of the observations for a sample is called a sample standard deviation. It is denoted by s x or, when no confusion will arise, simply by s. We have where n is the sample size: n-1 is referred to as the degrees of freedom

24. Anthony J Greene24 Deviation from the Sample Mean M

25. Anthony J Greene25 Deviation From the Sample Mean M

26. Anthony J Greene26 Sample Variance and Standard Deviation Using Conceptual Formula MM

27. Anthony J Greene27 Computational Columns Using Conceptual Formula M M

28. Anthony J Greene28 Computational Columns Using Computational Formula

29. Anthony J Greene29 APA Format For Mean and St.Dev

30. Anthony J Greene30 Sample Standard Deviation Almost all of the observations in any data set lie within three standard deviations to either side of the mean 95% of the observations lie within two standard deviations to either side of the mean 68% of the observations lie within one standard deviation to either side of the mean

31. Anthony J Greene31 Sample Standard Deviation 68% 95%

32. Anthony J Greene32 Summary of Descriptives Central Tendency 1.Mode 2.Median 3.Mean Dispersion 1. -- 2.Interquartile range or Semi-interquartile range 3.Variance or Standard deviation*

33. Anthony J Greene33 Again, The Basic Idea of Experiments 1.Are there differences between means? 2.Is that difference large enough so that it is not likely to be due to chance factors? Answer: It depends on how far apart the means are and how much dispersion you have in your variables

34. Anthony J Greene34 Effect Size Compared to Random Variation The variability within samples is small and it is easy to see the 5-point mean difference between the two samples.

35. Anthony J Greene35 Effect Size Compared to Random Variation The 5- point mean difference between samples is obscured by the large variability within samples.

36. Anthony J Greene36 Significant Differences? μ 1 = 40 μ 2 =60

37. Anthony J Greene37 Significant Differences? μ 1 = 40 μ 2 =60