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## AP®︎/College Statistics

### Course: AP®︎/College Statistics>Unit 4

Lesson 2: Z-scores

# 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? •  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
• How did you get the number 1.69? • 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? • 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. • 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? • 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? • 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.
• 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? • Why does he do the data point minus the mean? are we not supposed to do the Mean minus the data point? • TL;DR... It gives us the correct sign.

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)  