Download Practice with standard deviations Work through these exercises with

Practice with standard deviations
Work through these exercises with your computer partner and/or instructor. Answers are
on the next page.
1. The numbers 20, 25, 25, and 30 are the highway miles per gallon of an SUV, two
family sedans, and a compact car. Compute the mean and sample standard deviation of
these numbers.
2. Below are three sets of numbers. You can compute their standard deviations simply
knowing the fact that the standard deviation of the numbers 1, 2, 3, 4, and 5 is 1.58.
a. What is the sample standard deviation of the numbers 4, 5, 6, 7, 8?
b. What is the sample standard deviation of the numbers -1, -2, -3, -4, -5?
c. What is the sample standard deviation of the numbers 5, 10, 15, 20, 25?
3. Data on 30 college-age women gave a mean height of 66 inches and a standard
deviation of 3 inches. What interval would include most of the heights of these women?
4. It is sometimes useful to compute how many standard deviations an observation is
from the mean. For instance suppose scores on an exam have a mean of 75 and a standard
deviation of 6. A score of 85 would be 1.67 standard deviations above the mean because
(observation – mean)/s.d. = (85-75)/6 = 1.67.
This is called a z-score and is denoted by z.
a. Assuming that scores have a mean of 75 and a standard deviation of 6, how many
standard deviations from the mean is a score of 90? a score 70?
b. An observation is considered unusually large or small if the observation is more than 2
standard deviations from the mean, that is, if the absolute value of the z-score is more
than 2. Which of the scores in part a is unusually large or small?
Answers next page
Answers
1. The mean is 25. The sample standard deviation is 4.08.
2. a. These numbers were obtained by adding 3 to the original numbers. The mean
changes by 3, but the standard deviation stays the same. So the standard deviation is 1.58
b. The negative numbers are the mirror image of the original numbers so they have the
same variability. The standard deviation is also 1.58. It is not possible for a standard
deviation to be negative, so -1.58 could not be the answer.
b. These numbers were obtained by multiplying the original numbers by 5. This changes
the standard deviation by a factor of 5. So the standard deviation is 5(1.58) = 7.9.
3. Most of the observations will be within two standard deviations of the mean. Because
the mean is 66 and 2s.d. = 2(3) = 6, the two s.d. interval about the mean is 60 inches to 72
inches.
4. a. For a grade of 90, z = (90 - 75)/6 = 2.5, so 90 is 2.5 standard deviations above the
mean. For a grade of 70, z = (70 - 75)/6 = -0.83, so 80 is .83 standard deviations below
the mean.
b. The grade of 90 is unusually large because it is more than 2 standard deviations above
the mean.