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Standard normal table for proportion between values

Finding the proportion of a normal distribution that is between two values by calculating z-scores and using a z-table.

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• The z-table I get to use in class doesn't go below 0, how do I deal with this?
• Here is a link to Z-Tables for Z-Scores that are negative and positive.
http://www.math.odu.edu/stat130/normal-tables.pdf

Or you can take the value on the z-Table of the positive Z-Score and subtract it from 1.
Example: Z-Score = -1.20, Z-Table for +1.20 = 0.8849
1-0.8849 = 0.1151. That is the same as if you look at the Z-Table for values with negative Z-Scores.
• Hi Sal,

I am curious about how to find the area under the Normal curve between two values without using a table or a calculator. If you think it would be useful, you could make a video with an example of a definite integral calculation for a Normal curve and put it in the "Applications of integration" unit in AP Calculus BC.

Thank you for reading this! It is very much appreciated. :)
• At -, how do we already know it will be 2 point something?
• Z-scores are measuring the number of standard deviations a certain number is above or below the mean. At , we learn that 750(the mean) - 60(the standard deviation) is 690. If you take another standard deviation away from that, it will equal 630. 624 is more than 120(2 standard deviations) below 750, and, therefore, its z-score will -2 point something.
• Shouldn't this have been calculated "higher than 624" and "less than 768"; Eqn being {Zscrore768 - (1 - Zscore624)}?
(1 vote)
• So you would need to use the cdf: culmative density function of the standard normal distribution to calculate the probability.

Observe P(624<X<768) = P(X<768) - P(X<624) .

After you calculate the z-score you can apply the cdf given by the z-table.
• how do i find the range of values if i have the percentage?
(1 vote)
• you can't, because RANGE is determined by 2 variables, upper boundary and lower boundary
but if you know one of the boundaries, you can use the integral and the anti-derivative(the second fundamental theorem of calculus, if my memory doesn't fool me) to get the other bound
• Is there another solution to solve the problem than having to use a z-table?
(1 vote)
• yes, we have, with the help of integral.
1. find the general PDF for normal distribution from https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm
2. fill in the parameters, μ and σ with 750 and 60 respectively, we got the PDF for this distribution
3. integrate the result from 624 to 768, we got the same result(to evaluate it, you need a calculator), 0.600047001626
• My answer became 0.3642, as I subtracted the z-score of 624, which is 0.0179, from 1 and obtained 0.9821. After that, I further subtracted 0.9821 by 0.6179, resulting in 0.3642.
(1 vote)
• Looking at your reasoning, you got:
- The % of laptops above 624
- The % of laptops below 768

Subtracting the z-score of 0.6179 from your score of 0.9821, you get the % of laptops that are above 624, but not below 768; you essentially only get the % of laptops above 768.

What you would want to do is, instead, subtract your 624 z-score (which is 0.01786) from your 768 z-score (which is 0.61791) to get the % of laptops that are below 768, but not below 624, which is what we want. You would end up with a score of 0.60005, or ~60%.
(1 vote)
• Is there any way to solve for proportions without a z-table?
(1 vote)
• If you know calculus this should be possible (although to be fair, the procedure to integrate this is a bit more involved and non-intuitive)
• Is there a way to get the Z-table values mathematically?

If so, I would love to know!
(1 vote)
• z
∫ (1 / √(2π)) * e⁻⁽ˣ^²⁾ᐟ²
-∞
(1 vote)
• Several people have asked how to solve this without a lookup table; if you have a computer with Python installed on it, the SciPy library has everything you need for stats calc. Here's the code to produce the area under curve and its inverse (for the "what proportion is higher than X" questions):

``import scipy.stats as stimport numpy as npmean = 19.7stdev = 2datapoint = 21.4zscore = np.round((datapoint-mean)/stdev, 4)# SciPy calculates left-tail probabilities by defaultauc = np.round(st.norm.cdf(zscore), 4)print(f"Z-score: {zscore}\tArea under curve: {auc}\tRemainder: {1-auc}")``
(1 vote)

Video transcript

- [Instructor] A set of laptop prices are normally distributed with a mean of \$750 and a standard deviation of \$60. What proportion of laptop prices are between \$624 and \$768? So let's think about what they are asking. So we have a normal distribution for the prices. So it would look something like this. This is just my hand-drawn sketch of a normal distribution. So it would look something like this. It should be symmetric, so I'm making it as symmetric as I can hand draw it. And we have the mean right in the center. So the mean would be right there. And that is \$750. They also tell us that we have a standard deviation of \$60. So that means one standard deviation above the mean would be roughly right over here, and that would be 750 plus 60. So that would be \$810. One standard deviation below the mean would put us right about there, and that would be 750 minus \$60, which would be \$690. And then they tell us, what proportion of laptop prices are between \$624 and \$768. So the lower bound, \$624, that's going to actually be more than another standard deviation less, so that's going to be right around here. So that is \$624. And 768 would put us right at about right at about there, and once again this is just a hand-drawn sketch. But that is 768. And so what proportion are between those two values? So we wanna find essentially the area under this distribution between these two values. The way we are going to approach it, we're going to figure out the z-score for 768, it's going to be positive because it's above the mean, and then we're going to use a z-table to figure out, what proportion is below 768. So essentially we're going to figure out this entire area. We're even going to figure out the stuff that's below 624. That's what that z-table will give us. Then we'll figure out the z-score for 624. That will be negative 2 point something, and we will use the z-table again to figure out the proportion that is less than that. And so then we can subtract this red area from the proportion that is less than 768 to get this area in between. So let's do that. Let's figure out first the z-score for 768. And then we'll do it for 624. The z-score for 768, I'll write it like that, is going to be 768 minus 750 over the standard deviation, over 60. So this is going to be equal to 18 over 60, which is the same thing as six over, let's see if we divide the numerator and the denominator by three, 6/20, and this is the same thing as 0.30. So that is the z-score for this upper bound. Let's figure out what proportion is less than that. For that, we take out a z-table, get our z-table. And let's see we wanna get 0.30. And so this is 0.3, this first column, and we've done this in other videos, this goes up until the tenths place for our z-score, and then if we wanna go to our hundredths place, that's what these other columns give us. But we're at 0.3, so we're going to be in this row, and our hundredths place is right over here, it's a zero, so this is the proportion that is less than \$768. So .6179. So 0.61, 0.6179. So now let's do the same exercise but do it for the proportion that's below \$624. The z-score for 624 is going to be equal to 624 minus the mean of 750, all of that over 60. And so what is that going to be? I'll get my calculator out for this one, don't wanna make a careless error. 624 minus 750 is equal to, and then divide by 60, is equal to negative 2.1. So that lower bound is 2.1 standard deviations below the mean, or you could say it has a z-score of negative 2.1. Is equal to negative 2.1. And so to figure out the proportion that is less than that, this red area right over here, we go back to our z-table. And we actually go to the first part of the z-table. So same idea but this starts at negative, a z-score of negative 3.4, 3.4 standard deviations below the mean. And just like we saw before, this is our zero hundredths, one hundredth, two hundredths, so on and so forth. And we wanna go to negative 2.1. We could say negative 2.10 just to be precise. So this is going to get us, let's see negative 2.1, there we go. And so we are negative 2.1, and it's negative 2.10, so we have zero hundredths, so we're gonna be right here on our table. So we see the proportion that is less than 624 is .0179 or 0.0179. So 0.0179. And so if we wanna figure out the proportion that's in between the two, we just subtract this red area from this entire area, the entire proportion that's less than 768 to get what's in between. 0.6179, once again I know I keep repeating it, that's this entire area right over here, and we're gonna subtract out what we have in red, minus 0.0179, so we're gonna subtract this out, to get 0.6. So if we wanna give our answer to four decimal places, it would be 0.6000, or another way to think about it is exactly 60% is between 624 and 768.