Z-score: Measuring by using standard deviations

- Suppose you're given a data set. Perhaps you're even given many of the key statistical tools needed to start forming a statistical opinion, mean, median, range, standard deviation. By now, we should understand that these tools help us start asking important questions about our data set. We also learned that these tools help us in studying not just the entire data set, but also each individual value in the data set.

In particular, the standard deviation allows us to see whether or not individual data points might be considered outliers. Is your data point within one standard deviation of the mean, two standard deviations? How would I know if it is exactly 2.37 standard deviations from the median? Luckily, there's a very simple formula to help us find just how many standard deviations our data points lie from the mean.

This is what we call our Z-Score, and as I said, the formula to find your Z score is quite simple. As you can see, all you need is the data set's mean and standard deviation. Then, all you do is plug in one of the values in the data set. Let's plug in our largest value in the data set, 231. For this data set, we have a mean of 130.1 and a standard deviation of 47.85.

As we can see, we get a Z-Score of 2.11. That means the data point, 231, is 2.11 standard deviations from the mean in the positive direction. Let's do this same calculation for our lowest value. The only thing we change for this calculation is that we switch out 50 for 231. Notice what happens here. Now our Z-Score is negative 1.67.

This means that this data point is 1.67 standard deviations from our mean in the negative direction, which, of course, makes sense since 50 is well below our mean value, 130.1. Next time someone says that a certain outcome is 2.8 standard deviations from the mean, not only will you know what that means, you'll also know how it was calculated. Plus, knowing this simple formula will be helpful in determining whether an individual data point might be considered an outlier.