# Applying antiderivatives to normally distributed sets of data (bell curve/z scores)

• November 2nd 2010, 02:22 PM
Coukapecker
Applying antiderivatives to normally distributed sets of data (bell curve/z scores)
Hi,

Me and my G12 teacher are having a debate about a question he marked wrong on one of my tests.

The question was true or false:
Quote:

Z scores > 4 are undefined. T/F?
I put false, reason being it truly isn't undefined. Z scores can continue on to positive/negative infinity, all of which are defined. Z scores > 4 perhaps are negligable, but most definitely not undefined.

The reason my teacher says it is undefined is because of the way we go about calculating the probability using the z scores. Instead of using the antiderivative (the accurate way to determine percentages from a z score, i would think), we use a table that has z scores and their corrosponding percentages listed. We simply round to the nearest percent.

Since the table we use only goes to a min of -4 and a max of +4, he says that all scores above and below that range are undefined.

The class im taking is a financial math class, which is why we use a simple table rather than figuring out the antiderivative of the normal distribution curve. I hope to show my teacher that the way the table he uses was generated was using calculus, and more importantly z scores above and below -4/+4 are most definitely not undefined.

The problem I've encountered, is I'm a bit rusty since my calculus classes a few years ago, and I can't find the equation of normally distributed data :\.

Through a bit of searching, I found that the normal distribution is a form of the gaussian function.

I got the equation $f(x)=e^(-x^2)$, but the problem is the total area under the curve doesn't come to 1, how can i mold it so it does come to 1, so that when I find the area under a certain section of the curve it will be a percent in decimal form?

any help would be appriciated,

thanks,

Coukapecker
• November 2nd 2010, 02:37 PM
Plato
• November 2nd 2010, 02:58 PM
mr fantastic
Quote:

Originally Posted by Coukapecker
Hi,

Me and my G12 teacher are having a debate about a question he marked wrong on one of my tests.

The question was true or false:

I put false, reason being it truly isn't undefined. Z scores can continue on to positive/negative infinity, all of which are defined. Z scores > 4 perhaps are negligable, but most definitely not undefined.

The reason my teacher says it is undefined is because of the way we go about calculating the probability using the z scores. Instead of using the antiderivative (the accurate way to determine percentages from a z score, i would think), we use a table that has z scores and their corrosponding percentages listed. We simply round to the nearest percent.

Since the table we use only goes to a min of -4 and a max of +4, he says that all scores above and below that range are undefined.

The class im taking is a financial math class, which is why we use a simple table rather than figuring out the antiderivative of the normal distribution curve. I hope to show my teacher that the way the table he uses was generated was using calculus, and more importantly z scores above and below -4/+4 are most definitely not undefined.

The problem I've encountered, is I'm a bit rusty since my calculus classes a few years ago, and I can't find the equation of normally distributed data :\.

Through a bit of searching, I found that the normal distribution is a form of the gaussian function.

I got the equation $f(x)=e^(-x^2)$, but the problem is the total area under the curve doesn't come to 1, how can i mold it so it does come to 1, so that when I find the area under a certain section of the curve it will be a percent in decimal form?

any help would be appriciated,

thanks,

Coukapecker

You are correct, your teacher is wrong. Feel free to refer him/her to this thread to get instruction on the subject.
• November 2nd 2010, 06:36 PM
Coukapecker
Hmm, yes I thought he was lol.

What I'm asking though is what is the function to create a normal distribution graph? And what would be the antiderivative function to get the corrosponding percentage to the z score?

My graphing calculator has a plethora of functions in it that I can use to get a percentage, so I know its possible to do without a chart, but how exactly do you do it?
• November 2nd 2010, 06:44 PM
mr fantastic
Quote:

Originally Posted by Coukapecker
Hmm, yes I thought he was lol.

What I'm asking though is what is the function to create a normal distribution graph? And what would be the antiderivative function to get the corrosponding percentage to the z score?

My graphing calculator has a plethora of functions in it that I can use to get a percentage, so I know its possible to do without a chart, but how exactly do you do it?

$\displaystyle \Phi (z) = P(Z \leqslant z) = \int_{ - \infty }^z {\frac{{e^{\frac{{ - x^2 }}{2}} }}{{\sqrt {2\pi } }}dx}$