Z-score introduction | Modeling data distributions | AP Statistics | Khan Academy

One of the most commonly used tools in all of statistics is the notion of a z-score. One way to think about a z-score is it's just the number of standard deviations away from the mean that a certain data point is. So let me write that down: number of standard deviations. I'll write it like this: number of standard deviations from our population mean for a particular data point.

Now let's make that a little bit concrete. Let's say that you're some type of marine biologist, and you've discovered a new species of winged turtles. There's a total of seven winged turtles; the entire population of these winged turtles is seven. So you go, and you're actually able to measure all the winged turtles. You care about their length, and you also want to care about how those lengths are distributed. Lengths of winged turtles.

All right, and let's say—and this is all in centimeters—these are very small turtles. So you discover, and these are all adults: there's a two centimeter one, there's another two centimeter one, there's a three centimeter one, there's another two centimeter one, there's a five centimeter one, a one centimeter one, and a six centimeter one. So we have seven data points. From this, I encourage you, at any point, if you want, pause this video and see if you want to calculate what is the population mean. Here we're assuming that this is the population of all the winged turtles.

Well, the mean in this situation is going to be equal to—you could add up all these numbers and divide by seven, and you would then get three. Then using these data points and the mean, you can calculate the population standard deviation. Once again, as a review, I always encourage you to pause this video and see if you can do it on your own. But I've calculated that ahead of time: the population standard deviation in this situation is approximately around to the hundredths place, 1.69.

So with this information, you should be able to calculate the z-score for each of these data points. Pause this video and see if you can do that. So let me make a new column here. Here I'm going to put the z-score. If you just look at the definition, what you're going to do for each of these data points—let's say each data point is x—you’re going to subtract from that the mean, and then you're going to divide that by the standard deviation.

The numerator idea over here is going to tell you how far you are above or below the mean, but you want to know how many standard deviations you are from the mean. So then you'll divide by the population standard deviation. For example, this first data point right over here, if I want to calculate its z-score, I will take 2 from that, I will subtract 3, and then I will divide by 1.69. I will divide by 1.69, and if you got a calculator out, this is going to be negative 1 divided by 1.69. If you use a calculator, you would get this is going to be approximately negative 0.59, and the z-score for this data point is going to be the same; that is also going to be negative 0.59.

One way to interpret this is this is a little bit more than half a standard deviation below the mean. We could do a similar calculation for data points that are above the mean. Let's say this data point right over here—what is its z-score? Pause this video and see if you can figure that out. Well, it's going to be 6 minus our mean, so minus 3, all of that over the standard deviation—all of that over 1.69. This, if you have a calculator—and I calculated it ahead of time—this is going to be approximately 1.77. So more than one but less than two standard deviations above the mean.

I encourage you to pause this video and now try to figure out the z-scores for these other data points. Now an obvious question that some of you might be asking is why. Why do we care how many standard deviations above or below the mean a data point is? In your future statistical life, z-scores are going to be a really useful way to think about how usual or how unusual a certain data point is. That's going to be really valuable once we start making inferences based on our data.

So I will leave you there. Just keep in mind it's a very useful idea, but at the heart of it, a fairly simple one. If you know the mean, you know the standard deviation. Take your data point, subtract the mean from the data point, and then divide by your standard deviation. That gives you your z-score.