1. Today: Standard Deviations & Z-Scores Any questions from last time?

2. Homework #2 (Due 9/7) Chapters 3, 4, 5 – Central Tendency, Variability, and the Z Transformation CH 3: 4, 5, 13, 14, 24, 27 CH 4: 1, 7, 8, 17, 22, 24, 25, 28 CH 5: 2, 3, 8, 9, 18, 20, 24, 25, 28 SPSS: Find the mean and population standard deviation of the variables Height and ShoeSize using the file 34011data.sav. Create z- transformed versions of both variables & save them to the file. Email your files (data and output) to the instructor (abmeyer@ilstu.edu).abmeyer@ilstu.edu

3. Topics for today Measures of Variability ◦ Standard Deviation & Variance (Population) ◦ Standard Deviation & Variance (Samples) Effects of linear transformations on mean and standard deviation The Z transformation Skip to slide 14

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

5. 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

6. 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. 

7. 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 1 2 3 4 5 6 7 8 9 10 2 - 5 = -3  X - μ = deviation scores -3

8. 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 1 2 3 4 5 6 7 8 9 10 2 - 5 = -3 4 - 5 = -1  X - μ = deviation scores Computing standard deviation (population)

9. 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 1 2 3 4 5 6 7 8 9 10 2 - 5 = -3 4 - 5 = -1 6 - 5 = +1  X - μ = deviation scores 1 Computing standard deviation (population)

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

11. 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 2 - 5 = -3 4 - 5 = -1 6 - 5 = +1 8 - 5 = +3 X -μ= deviation scores = (-3) 2 + (-1) 2 + (+1) 2 + (+3) 2 = 9 + 1 + 1 + 9 = 20 Computing standard deviation (population)

12. Step 3: Compute the Variance (the average of the squared deviations)  Divide by the number of individuals in the population. variance = σ 2 = SS/N Computing standard deviation (population)

13. Step 4: Compute the standard deviation. Take the square root of the population variance. standard deviation = σ = Computing standard deviation (population)

14. 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 Computing standard deviation (population)

15. Self-monitor your understanding We are about to learn how to calculate sample standard deviations. Before we move on, any questions about how to calculate population standard deviations? Any questions about these terms: ◦ deviation scores ◦ squared deviations ◦ sum of squares ◦ Variance ◦ standard deviation Any questions about these symbols : SS

16. 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 Computing standard deviation (sample)

17. Step 1: Compute the deviation scores ◦ subtract the sample mean from every individual in our distribution. Our sample 2, 4, 6, 8 1 2 3 4 5 6 7 8 9 10 2 - 5 = -3 4 - 5 = -1 6 - 5 = +1 8 - 5 = +3 X - M = Deviation Score M

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

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

20. Step 3: Determine the variance Computing standard deviation (sample) Population variance = σ 2 = SS/N Recall: The variability of the samples is typically smaller than the population’s variability Sample variance = s 2 To correct for this we divide by (n-1) instead of just n

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

22. Self-monitor your understanding Next, we’ll find out how changing our scores (adding, subtracting, multiplying, dividing) affects the mean and standard deviation. Before we move on, any questions about the sample standard deviation? About why we divide by (n-1)? About the following symbols: ◦s2◦s2 ◦s◦s

23. Properties of means and standard deviations Change/add/delete a given score MeanStandard deviation changes –Changes the total and the number of scores, this will change the mean and the standard deviation

24. Properties of means and standard deviations –All of the scores change by the same constant. M old Change/add/delete a given score MeanStandard deviation Add/subtract a constant to each score changes

25. Properties of means and standard deviations –All of the scores change by the same constant. Change/add/delete a given score MeanStandard deviation changes Add/subtract a constant to each score M old

26. Properties of means and standard deviations –All of the scores change by the same constant. Change/add/delete a given score MeanStandard deviation changes Add/subtract a constant to each score M old

27. Properties of means and standard deviations –All of the scores change by the same constant. Change/add/delete a given score MeanStandard deviation changes Add/subtract a constant to each score M old

28. Properties of means and standard deviations –All of the scores change by the same constant. –But so does the mean Change/add/delete a given score MeanStandard deviation changes Add/subtract a constant to each score changes M new

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

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

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

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

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

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

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

36. Properties of means and standard deviations –It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Change/add/delete a given score MeanStandard deviation changes No changechangesAdd/subtract a constant to each score M old M new

37. Properties of means and standard deviations Change/add/delete a given score MeanStandard deviation Multiply/divide a constant to each score changes No changechangesAdd/subtract a constant to each score 20 21 22 23 24 21 - 22 = -1 23 - 22 = +1 (-1) 2 (+1) 2 s = M

38. Properties of means and standard deviations –Multiply scores by 2 Change/add/delete a given score MeanStandard deviation Multiply/divide a constant to each score changes No changechanges Add/subtract a constant to each score 42 - 44 = -2 46 - 44 = +2 (-2) 2 (+2) 2 s = 40 42 44 46 48 M S old =1.41

39. Self-monitor your understanding Next, we’ll find out how to convert our scores to z-scores. Before we move on, any questions about how changing our scores (by adding, subtracting, multiplying, or dividing by a constant) changes the mean and standard deviation?

40. The Z transformation If you know the mean and standard deviation of a distribution, you can convert a given score into a Z score or standard score. This score is informative because it tells you where that score falls relative to other scores in the distribution.

41. Locating a score Where is our raw score within the distribution? ◦ The natural choice of reference is the mean (since it is usually easy to find).  So we’ll subtract the mean from the score (find the deviation score). –The direction will be given to us by the negative or positive sign on the deviation score –The distance is the value of the deviation score

42. Locating a score X 1 = 162 X 2 = 57 X 1 - 100 = +62 X 2 - 100 = -43 Reference point Direction

43. Locating a score X 1 = 162 X 2 = 57 X 1 - 100 = +62 X 2 - 100 = -43 Reference point Below Above

44. Transforming a score –The distance is the value of the deviation score However, this distance is measured with the units of measurement of the score (such as inches, ounces, likert rating, etc). Convert the score to a standard (neutral) score. In this case a z-score. Raw score Population mean Population standard deviation

45. Transforming scores X 1 = 162 X 2 = 57 X 1 - 100 = +1.20 50 X 2 - 100 = -0.86 50 A z-score specifies the precise location of each X value within a distribution. Direction: The sign of the z-score (+ or -) signifies whether the score is above the mean or below the mean. Distance: The numerical value of the z-score specifies the distance from the mean by counting the number of standard deviations between X and μ.

46. Transforming a distribution We can transform all of the scores in a distribution –We can transform any & all observations to z-scores if we know the distribution mean and standard deviation. –We call this transformed distribution a standardized distribution. Standardized distributions are used to make dissimilar distributions comparable. –e.g., your height and weight One of the most common standardized distributions is the Z- distribution.

47. Properties of the z-score distribution µ µ transformation 15050 = 0 X mean = 100

48. Properties of the z-score distribution  transformation 15050 X mean = 100 = 0 = +1 X +1std = 150 +1

49. Properties of the z-score distribution  transformation 15050 X mean = 100 X +1std = 150 = 0 = +1 = -1 X -1std = 50 +1

50. Properties of the z-score distribution Shape - the shape of the z-score distribution will be exactly the same as the original distribution of raw scores. Every score stays in the exact same position relative to every other score in the distribution. Mean - when raw scores are transformed into z-scores, the mean will always = 0. The standard deviation - when any distribution of raw scores is transformed into z-scores the standard deviation will always = 1.

51. Self-monitor your understanding Next, we’ll find out how to convert z-scores back into raw scores. Before we move on, any questions about z- scores (what they are, how to compute them from raw scores, properties of the z distribution)?

52.  15050  +1 From z to raw score We can also transform a z-score back into a raw score if we know the mean and standard deviation information of the original distribution. Z = (X - μ ) --> (Z)( σ ) = (X - μ ) --> X = (Z)( σ ) + μ σ transformation Z = -0.60 X = (-0.60)( 50) + 100 X = 70

53. Let’s try it with our data To transform data on mothers’ height into standard scores, use the formula bar in excel to subtract the mean and divide by the standard deviation. Can also choose standardize (x,mean,sd) Show with fathers’ height Observe how height and shoe size can be more easily compared with standard (z) scores

54. Z-transformations with SPSS You can also do this in SPSS. Use Analyze …. Descriptive Statistics…. Descriptives …. Check the box that says “save standardized values as variables.”

55. Homework #2 (Due 9/7) Chapters 3, 4, 5 – Central Tendency, Variability, and the Z Transformation CH 3: 4, 5, 13, 14, 24, 27 CH 4: 1, 7, 8, 17, 22, 24, 25, 28 CH 5: 2, 3, 8, 9, 18, 20, 24, 25, 28 SPSS: Find the mean and population standard deviation of the variables Height and ShoeSize using the file 34011data.sav. Create z- transformed versions of both variables & save them to the file. Email your files (data and output) to the instructor (abmeyer@ilstu.edu).abmeyer@ilstu.edu