A standard type of problem in basic statistics is to calculate the z-score of a value, given that the data is normally distributed and also given the mean and standard deviation. All of the following problems use the z-score formula. For all of them assume that we are dealing with a normal distribution. A version of this worksheet without solutions is also available.
- Scores on a history test have average of 80 with standard deviation of 6. What is the z-score for a student who earned a 75 on the test?
- The weight of chocolate bars from a particular chocolate factory has a mean of 8 ounces with standard deviation of .1 ounce. What is the z-score corresponding to a weight of 8.17 ounces?
- Books in the library are found to have average length of 350 pages with standard deviation of 100 pages. What is the z-score corresponding to a book of length 80 pages?
- The temperature is recorded at 60 airports in a region. The average temperature is 67 degrees Fahrenheit with standard deviation of 5 degrees. What is the z-score for a temperature of 68 degrees?
- A group of friends compares what they received while trick or treating. They find that the average number of pieces of candy received is 43, with standard deviation of 2. What is the z-score corresponding to 20 pieces of candy?
- The mean growth of the thickness of trees in a forest is found to be .5 cm/year with a standard deviation of .1 cm/year. What is the z-score corresponding to 1 cm/year?
- A particular leg bone for dinosaur fossils has a mean length of 5 feet with standard deviation of 3 inches. What is the z-score that corresponds to a length of 62 inches?
Solution: The process for all of these problems is similar: subtract the mean from the given value, then divide by the standard deviation. This gives a z-score of (75 - 80)/6 = -0.833.
Solution: The z-score for this problem is (8.17 - 8)/.1 = 1.7.
Solution: The z-score for this problem is (80 - 350)/100 = -2.7.
Solution: Here the number of airports is information that is not necessary to solve the problem. The z-score for this problem is (68-67)/5 = 0.2.
Solution: The z-score for this problem is (20 - 43)/2 = -11.5.
Solution: The z-score for this problem is (1 - .5)/.1 = 5
Solution: Here we need to be careful that all of the units we are using are the same. There will not be as many conversions if we do our calculations with inches. Since there are 12 inches in a foot, five feet corresponds to 60 inches. The z-score for this problem is (62 - 60)/3 = .667.