# Z-Scores and Percentages from the Mean **URL:** \ **Category:** Factual Questions\ **Created:** [April 23, 2003, 11:05pm UTC](https://boards.straightdope.com/t/z-scores-and-percentages-from-the-mean/170710 "2003-04-23T23:05:05Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Absolute](https://avatars.discourse-cdn.com/v4/letter/a/b2d939/32.png) [@Absolute](https://boards.straightdope.com/u/Absolute)\ **Post date:** [April 23, 2003, 11:05pm UTC](https://boards.straightdope.com/t/z-scores-and-percentages-from-the-mean/170710/1 "2003-04-23T23:05:05Z")
I’m taking a statistics course in school now, and we’ve just done z-scores. My question is simple: given a certain z-score, how do I calculate the percentage of the results (in a normal distribution) between that score and the mean (score 0). We were given a table of these values for successive z-scores, but it’s rather cumbersome to use. I can’t find anything on Google about this, only more tables like the one I got and programs that reference a table internally. ---
**Author:** ![The\_Tim](https://avatars.discourse-cdn.com/v4/letter/t/ec9cab/32.png) [@The\_Tim](https://boards.straightdope.com/u/The_Tim)\ **Post date:** [April 23, 2003, 11:10pm UTC](https://boards.straightdope.com/t/z-scores-and-percentages-from-the-mean/170710/2 "2003-04-23T23:10:35Z")
Try searcing for Gauss and distribution, or “Gaussian distribution”. That’ll probably get you what you want. ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [April 23, 2003, 11:13pm UTC](https://boards.straightdope.com/t/z-scores-and-percentages-from-the-mean/170710/3 "2003-04-23T23:13:55Z")
Do you mean you don’t understand how to use the tables or just that you find doing so rather boring? ---
**Author:** ![Absolute](https://avatars.discourse-cdn.com/v4/letter/a/b2d939/32.png) [@Absolute](https://boards.straightdope.com/u/Absolute)\ **Post date:** [April 23, 2003, 11:26pm UTC](https://boards.straightdope.com/t/z-scores-and-percentages-from-the-mean/170710/4 "2003-04-23T23:26:49Z")
> [@](#): > > \*Originally posted by Jabba \* > \*\*Do you mean you don’t understand how to use the tables or just that you find doing so rather boring? \*\* Boring, in addition to inefficient and innacurate. ---
**Author:** ![Absolute](https://avatars.discourse-cdn.com/v4/letter/a/b2d939/32.png) [@Absolute](https://boards.straightdope.com/u/Absolute)\ **Post date:** [April 23, 2003, 11:29pm UTC](https://boards.straightdope.com/t/z-scores-and-percentages-from-the-mean/170710/5 "2003-04-23T23:29:08Z")
> [@](#): > > \*Originally posted by The Tim \* > \*\*Try searcing for Gauss and distribution, or “Gaussian distribution”. That’ll probably get you what you want. \*\* I’ve tried searching using those terms too, but all I can find are tables or programs that use tables. ---
**Author:** ![The\_Tim](https://avatars.discourse-cdn.com/v4/letter/t/ec9cab/32.png) [@The\_Tim](https://boards.straightdope.com/u/The_Tim)\ **Post date:** [April 23, 2003, 11:33pm UTC](https://boards.straightdope.com/t/z-scores-and-percentages-from-the-mean/170710/6 "2003-04-23T23:33:13Z")
Are you sure? Because when I googled for Gaussian distribution to make sure I wasn’t giving you bad advice the first link had an explanation of the function. [http://mathworld.wolfram.com/GaussianDistribution.html](http://mathworld.wolfram.com/GaussianDistribution.html) Here’s the link I got. I just gave it a glance but it looks to have a nice overview of the probability function and its use. ---
**Author:** ![Absolute](https://avatars.discourse-cdn.com/v4/letter/a/b2d939/32.png) [@Absolute](https://boards.straightdope.com/u/Absolute)\ **Post date:** [April 23, 2003, 11:45pm UTC](https://boards.straightdope.com/t/z-scores-and-percentages-from-the-mean/170710/7 "2003-04-23T23:45:21Z")
I’m still in high school, taking “Honors PreCalculus”, which is delving into statistics for the next month. It’s rather hard for me to understand the link you supplied. Basically, I want a way of getting the following sort of information without using a table: given a normal distribution, 34% of the data points will fall within one standard deviation of the mean. 47.5% of the data points will fall within two standard deviations of the mean. Etc. If this can’t be done without using operations my TI-89 can do (or ones that I can learn within a reasonable amount of time), I guess I’ll stick with my tables until I get to college. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [April 23, 2003, 11:49pm UTC](https://boards.straightdope.com/t/z-scores-and-percentages-from-the-mean/170710/8 "2003-04-23T23:49:47Z")
The problem is, the percentage within n standard deviations of the mean in a normal distribution can’t be calculated using any functions you’d be comfortable with. The tables are your onl realistic option at this point. Sorry. ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [April 23, 2003, 11:53pm UTC](https://boards.straightdope.com/t/z-scores-and-percentages-from-the-mean/170710/9 "2003-04-23T23:53:10Z")
I don’t know of any other way than using the tables, other than using numerical integration, which is going to be vastly more laborious. That’s why there are published tables, because there’s no quicker way. ---
**Author:** ![Absolute](https://avatars.discourse-cdn.com/v4/letter/a/b2d939/32.png) [@Absolute](https://boards.straightdope.com/u/Absolute)\ **Post date:** [April 24, 2003, 12:11am UTC](https://boards.straightdope.com/t/z-scores-and-percentages-from-the-mean/170710/10 "2003-04-24T00:11:55Z")
My calculator can do integration - it is something that I could just tell it to do and not need to know the workings, or does it require more than finding the integral of one function? And if I obviously don’t know what I’m talking about, sorry :). I don’t. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [April 24, 2003, 12:53am UTC](https://boards.straightdope.com/t/z-scores-and-percentages-from-the-mean/170710/11 "2003-04-24T00:53:02Z")
If you’re clever, you can do it. You may run into trouble as the necessary integrals may be improper, but if you’re clever, you can get around that. Here are some facts that’ll help you be clever. Let C(x) denote the probability that a N(0, 1) (that’s a normal distribution with mean 0 and variance 1) variable is less than or equal to x. C(0) = .5 C(-x) = 1 - C(x) ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [April 24, 2003, 3:18am UTC](https://boards.straightdope.com/t/z-scores-and-percentages-from-the-mean/170710/12 "2003-04-24T03:18:41Z")
> [@](#): > > \*Originally posted by Mister V \* > My calculator can do integration - it is something that I could just tell it to do and not need to know the workings, or does it require more than finding the integral of one function? Whatcha got, a TI83? If so, try this out: Y[sub]1/sub=0.5+fnInt(e^(-Z[sup]2[/sup]/2),Z,0,X)/Sqrt(2pi) where Sqrt( is your square root button there, and pi is your constant pi. I think this will work, but it’s been a couple of years since I’ve worked with that calculator. If I got the notation right, you should be able to input stuff like Y[sub]1/sub and get stuff like 0.8413447. Is that even right? ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [April 24, 2003, 3:27am UTC](https://boards.straightdope.com/t/z-scores-and-percentages-from-the-mean/170710/13 "2003-04-24T03:27:47Z")
Oh oops. I see that you wanted the percentage between the given z-score and the mean, not negative infinity. The only necessary change to the function I gave is remove the 0.5+ from the beginning. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [April 24, 2003, 3:54am UTC](https://boards.straightdope.com/t/z-scores-and-percentages-from-the-mean/170710/14 "2003-04-24T03:54:21Z")
You can also do it using power series. Multiply 1 over the square root of 2 \* pi by the sum of (-1)^n \* [1/(2n+1)] \* [1/2^n] \* [1/n!] \* z^(2n+1) starting with n = 0 and going to infinity—or until you get two successive running totals (“partial sums”) that agree to within the degree of accuracy you want. Come back after you’ve had a couple of semesters of calculus and I can explain how I got this formula. ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [April 24, 2003, 4:00am UTC](https://boards.straightdope.com/t/z-scores-and-percentages-from-the-mean/170710/15 "2003-04-24T04:00:10Z")
Clearly you just copied it from [Equation 4](http://mathworld.wolfram.com/Erf.html). 😉 ---
**Author:** ![Urban\_Ranger](https://avatars.discourse-cdn.com/v4/letter/u/e9c0ed/32.png) [@Urban\_Ranger](https://boards.straightdope.com/u/Urban_Ranger)\ **Post date:** [April 24, 2003, 4:10am UTC](https://boards.straightdope.com/t/z-scores-and-percentages-from-the-mean/170710/16 "2003-04-24T04:10:43Z")
> [@](#): > > \*Originally posted by Mister V \* > Boring, in addition to inefficient and innacurate. Unless you are sitting next to a compute,r using tables are quite fast and gives you sufficient accuracy. ---
**Author:** ![CuriousCanuck](https://avatars.discourse-cdn.com/v4/letter/c/d2c977/32.png) [@CuriousCanuck](https://boards.straightdope.com/u/CuriousCanuck)\ **Post date:** [April 24, 2003, 4:43am UTC](https://boards.straightdope.com/t/z-scores-and-percentages-from-the-mean/170710/17 "2003-04-24T04:43:11Z")
> [@](#): > > \*Originally posted by Mister V \* > **34% of the data points will fall within one standard deviation of the mean. 47.5% of the data points will fall within two standard deviations of the mean. Etc.** OK… anal point here, but double those numbers. There are two sides to every tail 🙂 ---
**Author:** ![Absolute](https://avatars.discourse-cdn.com/v4/letter/a/b2d939/32.png) [@Absolute](https://boards.straightdope.com/u/Absolute)\ **Post date:** [April 24, 2003, 11:59am UTC](https://boards.straightdope.com/t/z-scores-and-percentages-from-the-mean/170710/18 "2003-04-24T11:59:11Z")
Thanks to everyone who gave me advice. **Achernar** , your formula worked perfectly. ---
**Author:** ![drhess](https://avatars.discourse-cdn.com/v4/letter/d/94ad74/32.png) [@drhess](https://boards.straightdope.com/u/drhess)\ **Post date:** [April 24, 2003, 2:19pm UTC](https://boards.straightdope.com/t/z-scores-and-percentages-from-the-mean/170710/19 "2003-04-24T14:19:31Z")
Off topic, but how did you get those sub and superscripts to show up? : ) ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [April 24, 2003, 2:42pm UTC](https://boards.straightdope.com/t/z-scores-and-percentages-from-the-mean/170710/20 "2003-04-24T14:42:07Z")
> [@](#): > > \*Originally posted by drhess \* > \*\*Off topic, but how did you get those sub and superscripts to show up? : ) \*\* Whenever you want to see how someone did something, press the ‘quote’ button on the bottom right of their post. It’ll show you their code.