One standard type of problem from an introductory statistics course is to calculate the z-score of a particular value. This is a very basic calculation, but is one that is quite important. The reason for this is that it allows us to wade through the infinite number of normal distributions. These normal distributions can have any mean or any positive standard deviation.
The z-score formula starts with this infinite number of distributions and lets us only work with the standard normal distribution. Instead of working with a different normal distribution for each application that we encounter, we only need to work with one special normal distribution. The standard normal distribution is this well-studied distribution.
Explanation of the Process
We assume that we are working in a setting in which our data are normally distributed. We also assume that we are given the mean and standard deviation of the normal distribution that we are working with. By using the z-score formula: z = (x - μ) / σ we can convert any distribution to the standard normal distribution. Here the Greek letter μ the mean and σ is the standard deviation.
The standard normal distribution is a special normal distribution. It has a mean of 0 and its standard deviation is equal to 1.
All of the following problems use the z-score formula. All of these practice problems involve finding a z-score from the information provided. See if you can figure out how to use this formula.
- 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 .1cm/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?
Once you have worked out these problems, be sure to check your work. Or maybe if you are stuck on what to do. Solutions with some explanations are located here.