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Math Help - Applying antiderivatives to normally distributed sets of data (bell curve/z scores)

  1. #1
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    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:
    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
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  2. #2
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    Applying antiderivatives to normally distributed sets of data (bell curve/z scores)-untitled.gif

    That is a z-score at least 5.
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  3. #3
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    Quote Originally Posted by Coukapecker View Post
    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.
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  4. #4
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    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?
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  5. #5
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    Quote Originally Posted by Coukapecker View Post
    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?
    If you Google

    normal distribution

    you will get the function. It cannot be integrated exactly (except in a couple of very special cases) - a numerical procedure for finding the required integral is necessary (most scientific calculators can do this. All graphics and CAS calculators can do it).
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    \displaystyle \Phi (z) = P(Z \leqslant z) = \int_{ - \infty }^z {\frac{{e^{\frac{{ - x^2 }}{2}} }}{{\sqrt {2\pi } }}dx}
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