Standard normal table for proportion below | AP Statistics | Khan Academy

A set of middle school students' heights are normally distributed with a mean of 150 cm and a standard deviation of 20 cm. Darnell is a middle school student with a height of 161.405, so it would have a shape that looks something like that. That's my hand-drawn version of it. There's a mean of 150 cm, so right over here, that would be 150 cm.

They tell us that there's a standard deviation of 20 cm, and Darnell has a height of 161.405. Drawing it exactly, but you get the idea, that is 161.405 because they tell us what the standard deviation is. We know the difference between Darnell's height and the mean height, and then once we know how many standard deviations he is above the mean, that's our z-score. We can look at a z-table that tells us what proportion is less than that amount in a normal distribution.

So let's do that. I have my TI-84 emulator right over here, and let's see. Darnell is 161.405. Now the mean is 150 centimeters. 150 is equal to—we could have done that in our head—11.405 cm. Now, how many standard deviations is that above the mean? Well, they tell us that a standard deviation in this case for this distribution is 20 cm.

So we'll take 11.405 divided by 20, so we will just take our previous answer. This just means our previous answer divided by 20 cm, and that gets us 0.57025. So we can say that this is approximately 0.57 standard deviations above the mean.

Now, why is that useful? Well, you could take this z-score right over here and look at a z-table to figure out what proportion is less than 0.57 standard deviations above the mean. So let's get a z-table over here.

What we're going to do is we're going to look up this z-score on this table and the way that you do it is this: The first column, each row tells us our z-score up until the 10th place, and then each of these columns after that tells us which hundreds we're in. So, for 0.57, the 10's place is right over here, so we're going to be in this row, and then our hundred's place is this seven. So we'll look right over here.

So 0.57 tells us the proportion that is lower than 0.57 standard deviations above the mean, and so it is 0.7157. Another way to think about it is, if the heights are truly normally distributed, 71.57% of the students would have a height less than Darnell's.

But the answer to this question, "What proportion of students' heights are lower than Darnell's height?" Well, that would be 0.7157, and they want our answer to four decimal places, which is exactly what we have done.