1. Transformations of Data
Section 9.3
Transformations of Data

2. Objectives:
1. To find z-scores and other
transformed scores for data.
2. To determine the effect of
transformations on the mean and
standard deviation.

3. Original
Transformed 4 6 7 9 10 9 11 12 14 15
Mean 7.5
St Dev 2.26
12.5
2.26

4. Theorem 9.1: Translated Data
Let x1, x2, . . ., xn represent the original data. If y1, y2, . . ., yn are obtained by adding a constant k to the original data values, then y = x + k and
sy = sx.

5. Original
Transformed 4 6 7 9 10 12 18 21 27 30
Mean 7.5
St Dev 2.26
22.5
6.77

6. Theorem 9.2: Scaled Data
If y1, y2, . . ., yn are found from x1, x2, . . ., xn by multiplying each by the same constant k, then y =k x and sy = ksx.

7. Definition
z-score The transformed score found by subtracting the mean from the individual score, and dividing by the standard
deviation: x s z - =

8. The statistic z is a measure of the deviation of an individual score from the mean in units of standard deviation.

9. Practice:
If you scored a 135 on an IQ test that has a mean of 100 and standard deviation of 16, how many standard deviations are you away from the mean? x s z - = 16 =
≈ 2.2

10. A z-score of 2.2 on the IQ test means that the score earned (135) was 2.2 standard deviations (16) above the mean (100). If the individual score x is the same as the mean, the z-score is 0. A score below the mean will result in a negative z-score.

11. Practice: Use the given set of scores to find the following values.
63, 74, 67, 76, 77, 71, 68, 66
Find the mean.

12. Practice: Use the given set of scores to find the following values.
63, 74, 67, 76, 77, 71, 68, 66
Find the standard deviation.

13. Practice: Use the given set of scores to find the following values.
63, 74, 67, 76, 77, 71, 68, 66
Find the z-score of the lowest score.

14. Practice: Use the given set of scores to find the following values.
63, 74, 67, 76, 77, 71, 68, 66
Find the z-score of the highest score.

15. Practice: Use the given set of scores to find the following values.
63, 74, 67, 76, 77, 71, 68, 66
Transform the rest of the scores to z-scores.

16. Homework: pp

17. If the mean is 83 and the standard deviation is 7, find the z-score for each test score below.
1. 88

18. If the mean is 83 and the standard deviation is 7, find the z-score for each test score below.
5. 73

19. If the mean of a set of data values is 75 and the standard deviation is 10, find the mean and standard deviation for the data transformed as follows.
9. y = 1/5x - 10

20. SAT scores are calculated from z-scores using the transformation SAT = 100z+500.
16. Give the mean score on the SAT.

21. SAT scores are calculated from z-scores using the transformation SAT = 100z+500.
17. Give the standard deviation on the SAT.

22. SAT scores are calculated from z-scores using the transformation SAT = 100z+500.
18. Give the SAT score of someone who scored 2.5 standard deviations above the mean.

23. SAT scores are calculated from z-scores using the transformation SAT = 100z+500.
19. What does an SAT score of 563 mean?

24. ■ Cumulative Review
26. A line passes through (3, 4) with an angle of inclination of 20°. Write its equation in slope- intercept form.

25. ■ Cumulative Review
27. State the three Pythagorean identities.

26. ■ Cumulative Review
28. In class, three quiz scores range from 10 to 20 with a median of Find the mean, midrange, and mode.

27. ■ Cumulative Review
29. Graph r = cos .

28. ■ Cumulative Review
30. If sin  = , give the other five trig functions in terms of a and b. a b