In this article, I’ll answer the question “what is a z-score” and show you how we use them in statistics with a real world example.  You can also check out the Z-Score Calculator and be guided through a solution for your problem.  Or, have a look at How to Find a Z-Score for z-score examples and guided help.

Relative Standing

The z-score of a number tell us the number’s “relative standing” in a data set. Relative standing is a measure of how many standard deviations above, or below, a data value is from the mean.

For example, suppose a data set consists of the heights of 10 year old boys.  Suppose the mean of that data set is 58 inches and the standard deviation is 2 inches.  A 10 year old boy whose height is 62 inches has a z-score of 2.0, since 62 is 2 standard deviations above 58.  So, the raw score 62 has a z-score of 2.0.  A 10 year old boy whose height is 56 inches has a z-score of -1.0, since 56 is 1 standard deviation below 58.  The raw score 56 has a z-score -1.0. 

In general, we’d expect the z-score for the height a very tall 10 year old boy to be positive and large.  We would expect the z-score for the height of a very short 10 year old boy to be negative and small.  What is a z-score for a boy with a height of 60?  The z-score would be 1 since 60 is 1 standard deviation above the mean.

Positive, Negative, and 0 Z-Scores

Z-scores can be negative, positive, or 0, and they can have a decimal portion as well. A data value in a data set that is equal in value to the mean of the data set has a z-score that is equal to 0.  What is a z-score for the height of a 10 year old boy who is 58 inches tall?  That would be a z-score of 0.

We calculate positive z-scores from raw scores that are larger than the mean.  The z-score for the height a 10 year old boy who is taller than 58 inches will be positive. Knowing that a z-score is positive immediately tells you that the raw score, height in our example, is greater than the mean.

Similarly, we calculate negative z-scores from raw scores that are smaller than the mean. The z-score for the height of a 10 year boy who is 54 inches tall will be negative. Knowing that a z-score is negative immediately tells you that the raw score is smaller than the mean.

Outliers

Z-scores generally range from -3.0 to +3.0.  For bell shaped distributions, the empirical rule says 99.7% of all the data values have z-scores between -3.0 and +3.0.  We consider any z-score that is either less than -3.0 or greater than +3.0 to be an outlier.

Comparing Similar Types

We use the term “relative” in explaining z-scores because a z-score tells us where a raw score lies when compared to the other raw scores of the data set.  10 year old “John Doe” may have a z-score of 2 in the data set of heights for all 10 year old boys.  The z-score is a measure John’s height relative to the mean of the heights of other 10 year old boys.  It wouldn’t make sense to measure John’s height against the mean for 20 year old boys, it’s a measure relative to the other values in its data set, only.

What’s Next?

Now, head over to How to Find a Z-Score for a guided tour of the z-score formula with specific examples.  Afterward, solve your own specific problems with the Z-Score Calculator. The online z-score calculator gives you both the answer and worked out solution to your problem.