In the previous section on standard scores, we introduced you to the concept of standardizing data and using its standardized value to understand how typical it is for a data set with a given mean and standard deviation. In this section, we’ll dive deeper into z-scores and show you how to use a z-table.
Employing the classic example used when teaching z-scores, let’s examine the idea of test taking. If you take a test that is out of 100 points and only score 50 points, you may be tempted to interpret that as a bad score. After all, many point systems are based on an absolute scale. Meaning, no matter how a class performs, 50 points generally considered to be a failing grade.
If this isn’t clear to you, think about what would happen in the opposite scenario, where a grade is based on a relative scale. Instead of being scored out of 100 points, let’s say your teacher decides to base the test on the highest grade in the class. This is a relative scale because it is graded relative to how the class performs. If the highest grade in the class is 55 points, you might have actually performed extremely well.
This example is meant to illustrate the problem with raw scores, which are simply unaltered data points or calculations form your data set. Your raw score of 50 out of 100 or 55 can’t actually tell us how well you did when compared to the entire class. You could have scored the average amount of points, or perhaps have been the worst.
In statistics, raw scores can be transformed in order to be more comparable. One method of transformation is percentiles - which is discussed in other sections of this guide - where your grade can be compared to the percentage of people scoring above and below you.
Another method is called standardization, which is the basis of z-scores. Standardizing data removes whatever units that data set is in and allows each data point to be compared in terms of standard deviations. Recall the formula for standardization below, which happens to also be the same formula for finding the z-score.
|Sample Z-Score||Population Z-Score|
Continuing from our example, let’s say the mean score for the test taken is 74 and the standard deviation is 8. Calculating the z-score, we would get the following.
Recall the rules for interpreting z-scores. Because we are standardizing the data, the rules for a standard normal distribution apply. These interpretations are summarized below.
|3, -3||Scored 3 above or below the mean, where 99.7% of the data lie between|
|2, -2||Scored 2 above or below the mean, where 95% of the data lie between|
|1, -1||Scored 1 above or below the mean, where 68% of the data lie between|
|0||Scored at the mean.|
This means that your score in the example was 3 standard deviations below the mean, where 99.7% of test takers scored. However, z-scores can also be used for an even deeper analysis with the use of z-tables.