yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

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


2m read
·Nov 11, 2024

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.

More Articles

View All
The Venus Project: mistakes that advocates make
So there’s been an exchange between Stefan Molyneux and Peter Joseph on YouTube lately, and I’ve been commenting on both videos and communicating with advocates of the Venus Project. In this video, I’ll try to correct some of the most important misconcept…
Confidence interval for hypothesis test for difference in proportions | AP Statistics | Khan Academy
A university offers a certain course that students can take in person or in an online setting. Teachers of the course were curious if there was a difference in the passing rate between the two settings. Data from a recent semester showed that 80 percent o…
Introduction to verb tense | The parts of speech | Grammar | Khan Academy
Hello grammarians! Today, I want to introduce the idea of the verb tense. The way I want to do that is to express the following: if you can master grammatical tenses, you will become a time wizard—a literal, actual time wizard. Because tense is nothing mo…
Period of a Pendulum | Simple harmonic motion and rotational motion | AP Physics 1 | Khan Academy
So a simple pendulum is just a mass hanging from a string, and if you were to pull this mass—sometimes it’s called a pendulum bob—if you were to pull it back and then let go, gravity would act as a restoring force, and this mass would swing back and forth…
Why your passwords suck..
Passwords are a string of nonsensical characters that separate us from our finances, our medical records, our school information, our entire digital life. It’s amazing how much power these random characters hold over us, how much they can do. How a simple…
How to find a good deal / off market properties in Real Estate
What’s up, you guys? It’s Graham here. So one of the questions I get asked a lot is how do you find a good deal or how do you find something off-market in real estate. So I’m going to be sharing my thoughts about this. Now, I’ve been buying properties si…