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## AP®︎/College Statistics

# Z-score introduction

A z-score is an example of a standardized score. A z-score measures how many standard deviations a data point is from the mean in a distribution.

## Want to join the conversation?

- When I paused to calculate the standard deviation myself, I came up with 1.83, not 1.69. It looks like Sal got 1.69 by taking the sqrt of the biased sample variance instead of the unbiased sample variance, which we were taught to do in the previous videos. Why is this?(10 votes)
- Nevermind I answered my own question: it's not sample variance because it's the entire population of turtles, so you calculate the variance by dividing by N instead of n-1(44 votes)

- How did you get the number 1.69?(10 votes)
- 1.69 is the standard deviation. Here is the article on how to calculate standard deviation:

https://www.khanacademy.org/math/probability/data-distributions-a1/summarizing-spread-distributions/a/calculating-standard-deviation-step-by-step(17 votes)

- When calculating the z score with my Ti-84 calculator for 6, I roughly got 1.7751.. which, when rounded, would be 1.78 am I wrong about rounding it up?(7 votes)
- When I did it, I got 1.7748... It's conventional, at least for the AP, to round to the thousandths place. But as long as your answer makes sense based on the math you got, which it appears it does due to the proximity of your answer to his and mine, you should be good.(4 votes)

- I have calculated the standard deviation for this video on my own. However, in your answer, you calculated it as if it was the population mean instead of the sample mean. To calculate the sample mean, we have to divide by (n-1); however, here, I see instead of diving by 6, it was divided by n, which was 7.

I am confused right now.(3 votes)- At around0:40, Sal says that "The entire population of winged turtles is 7." Since we are dealing with all the individuals in the population (a very small population), we don't have to divide by n-1.(8 votes)

- Can z-scores be negative if the value is less than the mean? Or would it be the absolute value of the z-score since it's measuring just how many standard deviations the value is away from the mean?(4 votes)
- From what I could tell, z-scores can be negative.(4 votes)

- Why does Sal say the z score of -0.59 is "a little bit more than half a standard deviation" below the mean when a standard deviation is 1.69? wouldn't half be approx. 0.7?(3 votes)
- A 1 in a z-score means 1 standard deviation, not 1 unit. So if the standard deviation of the data set is 1.69, a z-score of 1 would mean that the data point is 1.69 units above the mean. In Sal's example, the z-score of the data point is -0.59, meaning the point is approximately 0.59 standard deviations, or 1 unit, below the mean, which we can easily see since the data point is 2 and the mean is 3.(4 votes)

- How did he get a standard deviation of 1.69? When I did it, I got about 1.8257419... So, what did I do wrong?(3 votes)
- You took the sample standard deviation instead of the population standard deviation.(3 votes)

- Why does he do the data point minus the mean? are we not supposed to do the Mean minus the data point?(3 votes)
- TL;DR... It gives us the correct sign.

Long answer:

We want the absolute difference between the numbers but also the direction the point is from the mean. When finding the standard deviation this doesn't matter, since we're only interested in the absolute value of the discrepancy between each point and the mean, as standard deviation is an absolute value. If we didn't look at the absolute values, any dataset with both positive and negative data points would be messed up when we find the sum of each difference before dividing by (n) or (n-1) and then finding the square root.

Here we have a mean of (3), and a data point with a value of (2). When we subtract (3) from (2) to find the difference, that gives us a negative answer, (-1), which we then divide by the standard deviation to see how far the difference between the mean and the data point are, in terms of standard deviations (the definition of a z-score). If we were to subtract the data point from the mean, (which would be (2) from (3), or (3) - (2)), we would get the same absolute difference between the two values but we might come away thinking our z-score is positive, since we'd get a positive difference of (1) before dividing by the standard deviation, which is always positive.

I will say that-- unless there's a reason that becomes apparent later-- it would probably be better practice to subtract the data point minus the mean when finding standard deviation too, just to be consistent. You'd still get the correct absolute value for each difference as long as you use the absolute value bars.

But here are some other examples with various negative and positive signs to prove that subtracting the data point minus the mean always works, but that the reverse (mean minus data point) doesn't work, with decimal places just to prove that's not a factor either in case you were curious, as I was):

1. suppose:

mean = (-5.9), x = (-2.2)

Say we try to find the difference between the two by doing mean minus point:

(-5.9) - (-2.2) = (-3.7)

This would be the correct absolute difference of (3.7), but the negative symbol also implies that our data point, (-2.2), was below our mean, (-5.9), which of course is not true. If we were taking the absolute value of the difference, this wouldn't matter, but here we want the difference and the direction. If we subtract the mean from the point however:

(-2.2) - (-5.9) = (3.7)

We get the correct difference and the correct positive sign.

Let's do the same thing with different values, one positive and one negative:

2. suppose:

mean = (-4.25) , x = (10.75)

and first we try mean minus point:

(-4.25) - (10.75) = (-15)

then point minus mean:

(10.75) - (-4.25) = (15)

Here's another, with the positive and negative signs on the opposite side:

3. suppose:

mean = (1.5) , x = (-9.5)

(1.5) - (-9.5) = (11)

(-9.5) - (1.5) = (-11)

Lastly, if both values are positive:

4. suppose:

mean = (12.46) , x = (7.27)

(12.46) - (7.27) = (5.19)

(7.27) - (12.46) = (-5.19)(2 votes)

- I don't understand the reasoning behind the z-score equation. Why subtract the mean? Why divide by the standard deviation?(3 votes)
- you subtract the mean from the equation because the equation is looking for the number of standard deviations from the mean, which should not include the mean.(2 votes)

- i dont have the sample size numbers. it just gives me an average. so what would i plug in as X for # of variables(2 votes)
- As long as you have the population mean (𝜇) and the population standard deviation (𝜎), then you can calculate the z-score for the sample mean (𝑥̅) as

𝑧 = (𝑥̅ − 𝜇)∕𝜎(4 votes)

## Video transcript

- [Instructor] One of the
most commonly used tools in all of statistics is
the notion of a Z-score. And 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, 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 and there's a total of
seven winged turtles, the entire population of
these winged turtles is seven. And so you go and you're actually able to measure all the winged turtles and you care about their
length and you also wanna care about, how are
those lengths 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. So 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 and from this, and I encourage you at
any point if you want. Pause this video and see
if you wanna 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. And then using these
data points and the mean you can calculate the
population standard deviation. And once again, as 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, I'll round
to the hundredth 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. So here I'm gonna put our Z-score. And 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 right over
here's gonna tell you how far you are above or below the mean, but you wanna know how
many standard deviations you are from the mean,
so then you'll divide by the population standard deviation. So for example, this first
data point right over here if I wanna calculate the
Z-score I will take two. From that I will subtract three and then I will divide by 1.69. I will divide by 1.69. And if you've got a calculator out this is going to be -1 divided by 1.69 and if you use a calculator you would get, this is going to be approximately -0.59. And the Z-score for this data
point is going to be the same. That is also going to be -0.59. One way to interpret this is, this is a little bit more
than half a standard deviation below the mean, and 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 six
minus our mean, so minus three. All of that over the standard deviation. All of that over 1.69 and
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 gonna be a really useful way to think about how usual or how unusual a certain data point is. And 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.