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.
Try searcing for Gauss and distribution, or “Gaussian distribution”. That’ll probably get you what you want.
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.
I’ve tried searching using those terms too, but all I can find are tables or programs that use tables.
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
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.
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.
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.
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.
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.
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)
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?
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.
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.
Clearly you just copied it from Equation 4. 
Unless you are sitting next to a compute,r using tables are quite fast and gives you sufficient accuracy.
OK… anal point here, but double those numbers. There are two sides to every tail 
Thanks to everyone who gave me advice.
Achernar, your formula worked perfectly.
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.