Question

Everyone in the class takes a test and receives a score, and the average is calculated. Everyone compares their test score to the average test score, and each person’s distance from the mean test score is their deviation. People who had really low test scores or really high test scores will have large distances, or deviations, from the mean, while people who had test scores that were similar to the mean will have small distances, or deviations. If a person’s test score is equal to the average score, then their deviation would be equal to zero.

For the purposes of this homework assignment, think of standard deviation as roughly the average deviation, or distance, from the mean.

A) A test was recently given in a class and the average test score was 85. The standard deviation was calculated and found to be equal to 5. Thus, a test score that is exactly one standard deviation below the mean would be 80. What test score is exactly one standard deviation above the mean?

B) A test score of 95 is exactly two standard deviations above the mean. What test score would be exactly two standard deviations below the mean?

C) Height was recently measured in a group of male students from Xavier. The average height was found to be 70 inches (5’10”), and the standard deviation was found to be 3 inches. What height is equal to 1 standard deviation below the mean?

D) What height is equal to two standard deviations above the mean?

E) What height is equal to three standard deviations above the mean?

F) Height was also measured in a group of female Xavier students and the mean height was found to be 65 inches (5’5”), and the standard deviation was calculated as 4 inches. What height would be equal to 1 standard deviation below the mean?

G) Using the information from Problem 6, what height would be equal to 1.5 standard deviations below the mean?

Answer 1

A) Here mean = 85 and standard deviation( $\sigma$ ) = 5

therefore, the test score is exactly one standard deviation above the mean = 85 + 1* $\sigma$ = 85 + 5 = 90

B) The test score would be exactly two standard deviations below the mean = 85 - 2* $\sigma$ = 85 - 2*5 = 85 - 10 = 75

C) Here mean = $\mu$ = 70 and standard deviation =   $\sigma$ = 3

The  height is equal to 1 standard deviation below the mean = 70 - 1*3 = 67 inches.

D) The height is equal to two standard deviations above the mean = 70 + 2*3 = 70 + 6 = 76 inches.

E) The height is equal to three standard deviations above the mean = 70 + 3*3 = 70 + 9 = 79 inches.

F) Here mean = $\mu$ = 65 inches and $\sigma$ = 4

The height would be equal to 1 standard deviation below the mean = 65 -1*4 = 61 inches.

G) The height would be equal to 1.5 standard deviations below the mean = 65 - (1.5*4) = 65 - 6 = 59 inches.