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## High school statistics

### Course: High school statistics>Unit 2

Lesson 5: Normal distribution calculations

# Standard normal table for proportion below

AP.STATS:
VAR‑2 (EU)
,
VAR‑2.B (LO)
,
VAR‑2.B.3 (EK)
CCSS.Math:
Finding the proportion of a normal distribution that is below a value by calculating a z-score and using a z-table.

## Want to join the conversation?

• There is no proper presentation of the z-table anywhere on the curriculum of your Statistics and Probability. I guess it's a standard thing which we just have to find somewhere on our own?
• I agree that this was an oversight on KA's part not to provide a table. But I did find this resource from The University of Arizona to be useful:

https://www.math.arizona.edu/~rsims/ma464/standardnormaltable.pdf

It's just a PDF that has the z-table values for a standard normal distribution curve so it avoids all the pesky ads of a lot of the other sites that you might have found while trying to Google for a proper z-table.
• I paused the video before it got to using z-tables, and came up with a different answer. Darnell is +0.57 SD above the mean. 1 whole SD is 34%. 34% * 0.57 = 19.38 (%). Plus the 50% below the mean = 69.38%. I'm guessing I've made a dumb error somewhere along the way, but be great to know why this approach isn't the right one. Thanks.
• The area under the distribution curve is not proportional to the distance away from the mean, so multiplying the number of standard deviations by 0.34 gives us the wrong result.

0.57 ∙ 0.34 = 19.38%, but the correct percentage is 21.57%
• On the AP exam are you provided with a z-Table?
• I don't know about the AP exam, but in Cambridge AS and A2 they do provide you with a table.
• Where does the Z-table come from?
I know how to use it. but I don't know what it is!
• If you need help finding a Z-Table, use this resource from the University of Arizona:

https://www.math.arizona.edu/~rsims/ma464/standardnormaltable.pdf

It's just a PDF that has the z-table values for a standard normal distribution curve. What's nice about it is that it avoids all the pesky advertisements found on a lot of the other sites that you might have come across while trying to find a proper z-table.
• At , Sal says to divide our answer by the standard deviation. If we have a negative difference when subtracting the mean, do we still keep it negative?
• I'm still confused. How do you know if you need to subtract 1-the z score? And why do you need to subtract 1 from the z score?
• Is there some way to get z-table values on a calculator? It reminds me of the days when there were trigonometric and logarithmic tables, but now calculators take care of those, so I'm wondering if these z-table values are also taken care of, or couldn't you just analytically integrate the normal distribution curve for whatever interval you care about?
• `I know this may sound dumb but why are bell curves` `called` `bell curves`
(1 vote)
• Is there any way to calculate this using the NormCDF/NormDIST function in Ti-Inspire/Desmos after standardizing values or do you have to use the z-table? Is the key difference in when you can and cannot do this lying in the wording to the question (Is proportion inherently different than probability?)
(1 vote)
• A z-table uses a standard normal distribution, which has 𝜇 = 0 and 𝜎 = 1.
In this case the z-score is 0.57.

So, on Desmos, type in normaldist(0,1).cdf(0.57)

– – –

If there are 𝑛 students in total, and 𝑚 of them are shorter than Darnell, then the proportion of students that are shorter than Darnell is 𝑚∕𝑛,
which is also the probability that a randomly chosen student is shorter than Darnell.
(1 vote)

## Video transcript

- [Instructor] A set of middle school students' heights are normally distributed with a mean of 150 centimesters and a standard deviation of 20 centimeters. Darnell is a middle school school student with a height of 161.4 centimeters. What proportion of student heights are lower than Darnell's height? So, let's think about what they are asking. So, they're saying that heights are normally distributed. 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 centimeters, so right over here. That would be 150 centimeters. They tell us that there's a standard deviation of 20 centimeters and Darnell has a height of 161.4 centimeters. So, Darnell is above the mean, so let's say he is right over here, and I'm not drawing it exactly, but you get the idea. That is 161.4 centimeters. And we wanna figure out what proportion of students' heights are lower than Darnell's height. So, we wanna figure out what is the area under the normal curve right over here. That will give us the proportion that are lower than Darnell's height. And I'll give you a hint on how to do this. We need to think about how many standard deviations above the mean is Darnell, and we can do that because they tell us what the standard deviation is and 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 to tell us what proportion is less than that amount in a normal distribution. So let's do that. So, I have my TI-84 Emulator right over here and let's see. Darnell is 161.4 centimeters, 161.4. Now, the mean is 150, minus 150 is equal to, we coulda done that in our head, 11.4 centimeters. Now, how many standard deviations is that above the mean? Well, they tell us that the standard deviation in this case for this distribution is 20 centimeters, so we'll take 11.4 divided by 20. So, we will just take our previous answer, so this just means our previous answer divided by 20 centimeters and that gets us 0.57. So we can say that this is 0.57 standard deviations, 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. So, what we're going to do is we're gonna look up this z score on this table, and the way that you do it, this first column, each rows tells us our z score up until the tenths place and then each of these columns after that tell us which hundredths we're in. So, 0.57, the tenths place is right over here, so we're going to be in this row. And then our hundredths place is this seven, so we'll look right over here. So, 0.57, this tells us the proportion that is lower than 0.57 standard deviations above the mean, and so it is 0.7157 or another way to think about 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.