Transformations of Data Section 9.3 Transformations of Data
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.
Original Transformed 4 6 7 9 10 9 11 12 14 15 Mean 7.5 St Dev 2.26 12.5 2.26
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.
Original Transformed 4 6 7 9 10 12 18 21 27 30 Mean 7.5 St Dev 2.26 22.5 6.77
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.
Definition z-score The transformed score found by subtracting the mean from the individual score, and dividing by the standard deviation: x s z - =
The statistic z is a measure of the deviation of an individual score from the mean in units of standard deviation.
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 - = 16 135 - 100 = ≈ 2.2
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.
Practice: Use the given set of scores to find the following values. 63, 74, 67, 76, 77, 71, 68, 66 Find the mean.
Practice: Use the given set of scores to find the following values. 63, 74, 67, 76, 77, 71, 68, 66 Find the standard deviation.
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.
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.
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.
Homework: pp. 460-461
If the mean is 83 and the standard deviation is 7, find the z-score for each test score below. 1. 88
If the mean is 83 and the standard deviation is 7, find the z-score for each test score below. 5. 73
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
SAT scores are calculated from z-scores using the transformation SAT = 100z+500. 16. Give the mean score on the SAT.
SAT scores are calculated from z-scores using the transformation SAT = 100z+500. 17. Give the standard deviation on the SAT.
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.
SAT scores are calculated from z-scores using the transformation SAT = 100z+500. 19. What does an SAT score of 563 mean?
■ Cumulative Review 26. A line passes through (3, 4) with an angle of inclination of 20°. Write its equation in slope- intercept form.
■ Cumulative Review 27. State the three Pythagorean identities.
■ Cumulative Review 28. In class, three quiz scores range from 10 to 20 with a median of 18. Find the mean, midrange, and mode.
■ Cumulative Review 29. Graph r = 2 + 3 cos .
■ Cumulative Review 30. If sin = , give the other five trig functions in terms of a and b. a b