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.