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. This z-score, or standard score, is the signed number of standard deviations by which the data points' value is above the mean value of that which is being measured.

Calculating z-scores for normal distribution in statistical analysis allows one to simplify observations of normal distributions, starting with an infinite number of distributions and working down to a standard normal deviation instead of working with each application that is encountered.

All of the following problems use the z-score formula, and for all of them assume that we are dealing with a normal distribution.

### The Z-Score Formula

The formula for calculating the z-score of any particular data set is z = (x - *μ) / σ *where *μ* is the mean of a population and *σ *is the standard deviation of a population. The absolute value of z represents the z-score of the population, the distance between the raw score and population mean in units of standard deviation.

It's important to remember that this formula relies not on the sample mean or deviation but on the population mean and the population standard deviation, meaning that a statistical sampling of data cannot be drawn from the population parameters, rather it must be calculated based on the entire data set.

However, it is rare that every individual in a population can be examined, so in cases where it is impossible to calculate this measurement of every population member, a statistical sampling may be used in order to help calculate the z-score.

### Sample Questions

Practice using the z-score formula with these seven questions:

- Scores on a history test have an average of 80 with a 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 a 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 an average length of 350 pages with a 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 a 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 a 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 a standard deviation of 3 inches. What is the
*z*-score that corresponds to a length of 62 inches?

### Answers for Sample Questions

Check your calculations with the following solutions. Remember that the process for all of these problems is similar in that you must subtract the mean from the given value then divide by the standard deviation:

- The
*z*-score of (75 - 80)/6 and is equal to -0.833. - The
*z*-score for this problem is (8.17 - 8)/.1 and is equal to 1.7. - The
*z*-score for this problem is (80 - 350)/100 and is equal to -2.7. - 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 and is equal to 0.2. - The
*z*-score for this problem is (20 - 43)/2 and equal to -11.5. - The
*z*-score for this problem is (1 - .5)/.1 and equal to 5. - 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 and is equal to .667.

If you have answered all of these questions correctly, congratulations! You've fully grasped the concept of calculating z-score to find the value of standard deviation in a given data set!