Statistics for Social Sciences: Describing Distributions

1. Statistics for the Social Sciences Describing Distributions Psychology 340 Spring 2010

2. Announcements • Homework #1: will accept these on Th (Jan 21) without penalty • Quiz problems • Quiz 1 is now posted, due date extended to Tu, Jan 26th (by 11:00) • Don’t forget Homework 2 is due Tu (Jan 26)

3. Outline (for week) • Characteristics of Distributions • Finishing up using graphs • Using numbers (center and variability) • Descriptive statistics decision tree • Locating scores: z-scores and other transformations

4. Distributions • Three basic characteristics are used to describe distributions • Shape • Many different ways to display distribution • Frequency distribution table • Graphs • Center • Variability

5. Shapes of Frequency Distributions • Unimodal, bimodal, and rectangular

6. Shapes of Frequency Distributions • Symmetrical and skewed distributions Positively Negatively • Normal and kurtotic distributions

7. Frequency Graphs • Histogram • Plot the different values against the frequency of each value

8. Frequency Graphs • Histogram by hand • Step 1: make a frequency distribution table (may use grouped frequency tables) • Step 2: put the values along the bottom, left to right, lowest to highest • Step 3: make a scale of frequencies along left edge • Step 4: make a bar above each value with a height for the frequency of that value

9. Frequency Graphs • Histogram using SPSS (create one for class height) • Graphs -> Legacy -> histogram • Enter your variable into ‘variable’ • To change interval width, double click the graph to get into the chart editor, and then double click the bottom axis. Click on ‘scale’ and change the intervals to desired widths • Note: you can also get one from the descriptive statistics frequency menu under the ‘charts’ option

10. Frequency Graphs • Frequency polygon - essentially the same, put uses lines instead of bars

11. Displaying two variables • Bar graphs • Can be used in a number of ways (including displaying one or more variables) • Best used for categorical variables • Scatterplots • Best used for continuous variables

12. Bar graphs • Plot a bar graph of men and women in the class • Graphs -> bar • Simple, click define • N-cases (the default) • Enter Gender into Category axis, click ‘okay’

13. Bar graphs • Plot a bar graph of shoes in closet crossed with men and women • What should we plot? (and why?) • Average number of shoes for each group? • Graphs -> bar • Simple, click define • Other statistic (default is ‘mean’) – enter pairs of shoes • Enter Gender into Category axis, click ‘okay’

14. Scatterplot • Useful for seeing the relationship between the variables • Graphs -> Legacy Dialogs • Scatter/Dot • Simple Scatter, click ‘define’ • Enter your X & Y variables, click ‘okay’ • Can add a ‘fit line’ in the chart editor • Plot a scatterplot of soda and bottled water drinking

15. Describing distributions • Distributions are typically described with three properties: • Shape: unimodal, symmetric, skewed, etc. • Center: mean, median, mode • Spread (variability): standard deviation, variance

16. Describing distributions • Distributions are typically described with three properties: • Shape: unimodal, symmetric, skewed, etc. • Center: mean, median, mode • Spread (variability): standard deviation, variance

17. Which center when? • Depends on a number of factors, like scale ofmeasurement and shape. • The mean is the most preferred measure and it is closely related to measures of variability • However, there are times when the mean isn’t the appropriate measure.

18. Which center when? • Use the median if: • The distribution is skewed • The distribution is ‘open-ended’ • (e.g. your top answer on your questionnaire is ‘5 or more’) • Data are on an ordinal scale (rankings) • Use the mode if: • The data are on a nominal scale • If the distribution is multi-modal

19. Divide by the total number in the population Add up all of the X’s Divide by the total number in the sample The Mean • The most commonly used measure of center • The arithmetic average • Computing the mean • The formula for the population mean is (a parameter): • The formula for the sample mean is (a statistic): • Note: your book uses ‘M’ to denote the mean in formulas

20. The Mean • Number of shoes: • 5, 7, 5, 5, 5 • 30, 11, 12, 20, 14, 12, 15, 8, 6, 8, 10, 15, 25, 6, 35, 20, 20, 20,25, 15 • Suppose we want the mean of the entire group? • Can we simply add the two means together and divide by 2? • NO. Why not?

21. The Weighted Mean • Number of shoes: • 5, 7, 5, 5, 5,30, 11, 12, 20, 14, 12, 15, 8, 6, 8, 10, 15, 25, 6, 35, 20, 20, 20,25, 15 • Suppose we want the mean of the entire group? Can we simply add the two means together and divide by 2? • NO. Why not? Need to take into account the number of scores in each mean

22. Both ways give the same answer The Weighted Mean • Number of shoes: • 5, 7, 5, 5, 5, 30, 11, 12, 20, 14, 12, 15, 8, 6, 8, 10, 15, 25, 6, 35, 20, 20, 20, 25, 15 Let’s check:

23. The median • The median is the score that divides a distribution exactly in half. Exactly 50% of the individuals in a distribution have scores at or below the median. • Case1: Odd number of scores in the distribution Step1: put the scores in order Step2: find the middle score • Case2: Even number of scores in the distribution Step1: put the scores in order Step2: find the middle two scores Step3: find the arithmetic average of the two middle scores

24. major mode minor mode The mode • The mode is the score or category that has the greatest frequency. • So look at your frequency table or graph and pick the variable that has the highest frequency. so the mode is 5 so the modes are 2 and 8 Note: if one were bigger than the other it would be called the major mode and the other would be the minor mode

25. Describing distributions • Distributions are typically described with three properties: • Shape: unimodal, symmetric, skewed, etc. • Center: mean, median, mode • Spread (variability): standard deviation, variance

26. Variability of a distribution • Variability provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together. • In other words variabilility refers to the degree of “differentness” of the scores in the distribution. • High variability means that the scores differ by a lot • Low variability means that the scores are all similar

27. m Standard deviation • The standard deviation is the most commonly used measure of variability. • The standard deviation measures how far off all of the scores in the distribution are from the mean of the distribution. • Essentially, the average of the deviations.

28. -3 1 2 3 4 5 6 7 8 9 10 μ Computing standard deviation (population) • Step 1: To get a measure of the deviation we need to subtract the population mean from every individual in our distribution. Our population 2, 4, 6, 8 X - μ = deviation scores 2 - 5 = -3

29. -1 1 2 3 4 5 6 7 8 9 10 μ Computing standard deviation (population) • Step 1: To get a measure of the deviation we need to subtract the population mean from every individual in our distribution. Our population 2, 4, 6, 8 X - μ = deviation scores 2 - 5 = -3 4 - 5 = -1

30. 1 1 2 3 4 5 6 7 8 9 10 μ Computing standard deviation (population) • Step 1: To get a measure of the deviation we need to subtract the population mean from every individual in our distribution. Our population 2, 4, 6, 8 X - μ = deviation scores 2 - 5 = -3 6 - 5 = +1 4 - 5 = -1

31. 3 1 2 3 4 5 6 7 8 9 10 μ Computing standard deviation (population) • Step 1: Compute the deviation scores: Subtract the population mean from every score in the distribution. Our population 2, 4, 6, 8 X - μ = deviation scores 2 - 5 = -3 6 - 5 = +1 Notice that if you add up all of the deviations they must equal 0. 4 - 5 = -1 8 - 5 = +3

32. X - σ = deviation scores 2 - 5 = -3 6 - 5 = +1 4 - 5 = -1 8 - 5 = +3 Computing standard deviation (population) • Step 2: Get rid of the negative signs. Square the deviations and add them together to compute the sum of the squared deviations (SS). SS = Σ (X - μ)2 = (-3)2 + (-1)2 + (+1)2 + (+3)2 = 9 + 1 + 1 + 9 = 20

33. Computing standard deviation (population) • Step 3: Compute the Variance (the average of the squared deviations) • Divide by the number of individuals in the population. variance = σ2 = SS/N • Note: your book uses ‘SD2’ to denote the variance in formulas

34. standard deviation = σ = Computing standard deviation (population) • Step 4: Compute the standard deviation. Take the square root of the population variance. • Note: your book uses ‘SD’ to denote the standard deviation in formulas

35. Computing standard deviation (population) • To review: • Step 1: compute deviation scores • Step 2: compute the SS • SS = Σ (X - μ)2 • Step 3: determine the variance • take the average of the squared deviations • divide the SS by the N • Step 4: determine the standard deviation • take the square root of the variance

36. Computing standard deviation (sample) • The basic procedure is the same. • Step 1: compute deviation scores • Step 2: compute the SS • Step 3: determine the variance • This step is different • Step 4: determine the standard deviation

37. Our sample 2, 4, 6, 8 1 2 3 4 5 6 7 8 9 10 X - X = deviation scores X Computing standard deviation (sample) • Step 1: Compute the deviation scores • subtract the sample mean from every individual in our distribution. 2 - 5 = -3 6 - 5 = +1 4 - 5 = -1 8 - 5 = +3

38. SS = Σ (X - X)2 2 - 5 = -3 6 - 5 = +1 = (-3)2 + (-1)2 + (+1)2 + (+3)2 4 - 5 = -1 8 - 5 = +3 = 9 + 1 + 1 + 9 = 20 X - X = deviation scores Apart from notational differences the procedure is the same as before Computing standard deviation (sample) • Step 2: Determine the sum of the squared deviations (SS).

39. 3 X X X X 2 1 4 μ Computing standard deviation (sample) • Step 3: Determine the variance Recall: Population variance = σ2 = SS/N The variability of the samples is typically smaller than the population’s variability

40. Sample variance = s2 Computing standard deviation (sample) • Step 3: Determine the variance Recall: Population variance = σ2 = SS/N The variability of the samples is typically smaller than the population’s variability To correct for this we divide by (n-1) instead of just n

41. standard deviation = s = Computing standard deviation (sample) • Step 4: Determine the standard deviation

42. Changes the total and the number of scores, this will change the mean and the standard deviation Properties of means and standard deviations Mean Standard deviation • Change/add/delete a given score changes changes

43. X old • All of the scores change by the same constant. Properties of means and standard deviations Mean Standard deviation • Change/add/delete a given score changes changes • Add/subtract a constant to each score

44. X old • All of the scores change by the same constant. Properties of means and standard deviations Mean Standard deviation • Change/add/delete a given score changes changes • Add/subtract a constant to each score

45. X old • All of the scores change by the same constant. Properties of means and standard deviations Mean Standard deviation • Change/add/delete a given score changes changes • Add/subtract a constant to each score

46. X old • All of the scores change by the same constant. Properties of means and standard deviations Mean Standard deviation • Change/add/delete a given score changes changes • Add/subtract a constant to each score

47. X new • All of the scores change by the same constant. • But so does the mean Properties of means and standard deviations Mean Standard deviation • Change/add/delete a given score changes changes • Add/subtract a constant to each score changes

48. X old • It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Properties of means and standard deviations Mean Standard deviation • Change/add/delete a given score changes changes • Add/subtract a constant to each score changes

49. X old • It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Properties of means and standard deviations Mean Standard deviation • Change/add/delete a given score changes changes • Add/subtract a constant to each score changes

50. X old • It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Properties of means and standard deviations Mean Standard deviation • Change/add/delete a given score changes changes • Add/subtract a constant to each score changes