How to Calculate Z-Scores in Statistics

How to Calculate Z-Scores in Statistics
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If you scored 80 percent on a test and the class average was 50 percent, your score is above average, but if you really want to know where you are on the "curve," you should calculate your Z-score. This important statistics tool not only takes into account the average of all the test scores but also the variation in the results. To find the Z-score, you subtract class mean (50 percent) from the individual score (80 percent) and divide the result by the standard deviation. If you want, you can convert the resulting Z-score to a percentage to get a clearer idea of where you stand relative to the other people who took the test.

Why Are Z-Scores Useful?

The Z-score, also known as a standard score, provides a way to compare a test score or some other piece of data with a normal population. For example, if you know your score is 80 and that the mean score is 50, you know you scored above average, but you don't know how many other students did as well as you. It's possible that many students scored higher than you, but the mean is low because an equal number of students did abysmally, On the other hand, you may be in an elite group of a few students who truly excelled. Your Z-score can provide this information.

The Z-score provides useful information for other types of tests as well. For example, your weight may be above average for people of your age and height, but many other people may weigh more or you may be in a class by yourself. The Z-score can tell you which it is, and may help you make up your mind whether or not to go on a diet.

Calculating the Z-Score

In a test, poll or experiment with a mean M and a standard deviation SD, the Z-score for a particular piece of data (D) is:

(D - M)/SD = Z-score

This is a simple formula, but before you can use it, you must first calculate the mean and the standard deviation. To calculate the mean, use this formula:

Mean = Sum of all scores/number of respondents

It's easier to explain how to calculate the standard deviation than it is to express it mathematically. You subtract the mean from each score and square the result, then sum up those squared values and divide by the number of respondents. Finally, you take the square root of the result.

Example Calculation of a Z-Score

Tom and nine other people took a test with a maximum score of 100. Tom got 75 and the other people got 67, 42, 82, 55, 72, 68, 75, 53 and 78.

Start by calculating the mean score by adding all the scores, including Tom's, to get 667 and dividing by the number of people who took the test (10) to get 66.7.

Next, find the standard deviation by first subtracting the mean from each score, squaring each result and adding those numbers. Note that all number in the series are positive, which is the reason for squaring them: 53.3 + 0.5 + 660.5 + 234.1 + 161.3 + 28.1 + 1.7 + 53.3 + 216.1 + 127.7 = 1,536.6. Divide that by the number of people who took the test (10) to get 153.7 and take the square root, which equals 12.4.

It's now possible to calculate Tom's Z-score.

Z-score = (Tom's Score - Mean Score)/Standard Deviation = (75 - 66.7)/12.4 = 0.669

If Tom looked up his Z-score on a table of standard normal probabilities, he would find it associated with the number 0.7486. This tells him that he did better than 75 percent of the people who took the test and that 25 percent of the students outperformed him.

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