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Calculus to find the z-score?

  1. Mar 1, 2013 #1
    Maybe this should be in calculus instead...

    I'm getting reaquainted with stats and I ran across the z-table again. I always wondered how the value in the table are populated.

    As I understand it, the values in the table are just the value along the X axis and the corresponding area for a normal distribution. Wouldn't this be the same as find the area under the curve using an integral in calculus?

    I don't know the equation for the curve used in creating the z-table.

    Anyway, if what I said is true, then I should be able to make an equation for whatever curve is generated by the data. I made up some data below with a strong left curve.

    X value Y value
    1 1
    2 2
    3 3.3
    4 5
    5 7
    6 9
    7 11
    8 11
    9 9
    10 2

    1. How do I make an equation using these data points?
    2. Once I have that equation, how do I integrate?

    Note: I will answer Qs 1 and 2 soon. I am wondering if my original thoughts on how a z-table are created are true. When I asked in class some 2 years ago I remember my prof saying "you don't want to go into that, it's complicated"! :P

    Thanks PF crew!
  2. jcsd
  3. Mar 1, 2013 #2


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  4. Mar 2, 2013 #3


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    We generally use suitable quadrature formula for the integral values. Simpson's 2/3 or Weddle formula are very good with short intervals like 0.1 to 0.5 etc. We can take smaller intervals for higher desired degree of accuracy.
    PS. What is Y in your data and how the question of z score arises is not clear.
    Last edited: Mar 2, 2013
  5. Apr 22, 2013 #4
    Thanks for sending me in the right direction.
    The Y values are the # of observations.

    What's the point of using the Z table at all? I'm trying to predict the likelihood of demand going above the average demand (and by how much). Why can't I just look at the past 10 orders and use a more precise function than a Z table?
    I sell t-shirts. Here’s the data for t-shirts sold in the past 10 weeks. (Fictional product and units)
    Code (Text):

    What I want to know is how much inventory(X) of t-shirts I need to not run out 95% of the time. Basically, what is the x value when the probability (shirts sold < X) = .95.

    T-shirt demand doesn’t follow a nice neat normal distribution. There are booms and busts but seldom is the average number of shirts sold. Therefore, I would not want to use a normal distribution.
    Some t-shirts have multiple, but predictable, humps. That is, if you plotted the distribution it would have 3 or 4 crests and valleys.
    Can I just use the data to make the “z-table”.
    I have actually forgotten how to turn those values into a distribution graph. Help on that would be appreciated.

    Thank you,
  6. Apr 26, 2013 #5
    Does anyone know how to find the f(x) function for the probability density function? I did not understand the wikipedia article.

    The problem is to find the breakeven point between sales and spoilage. I have the mean sales and the standard of deviation for the sales. I know that if the product is sold, I will get $100. If the product spoils I will lose $50. I sell 1000 units on average with a standard deviation of 250 units. How many units should I order above the mean? Basically I am looking for X. Problem is I can't use a z table in a breakeven calculation. I need to have the actual function.

    So if the function was x^2 *250x - .5x^2 *250x I could solve for X. As it stands I'm not sure what to do to find said function. It's got to be a simple one given that I only need half the function.


    Ideas? The normal distribution curve has to be a polynomial
  7. Apr 27, 2013 #6
    No. The normal distribution curve is not a polynomial. If N is the "normal function" with parameters μ and σ, then ##\displaystyle N(x;\mu,\sigma)=\frac{e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}}{\sigma\sqrt{2\pi}}##. This is not a polynomial.

    If you want a polynomial, consider ##\displaystyle \sum_{n=0}^{\infty}\frac{d^n}{dx^n}\left.\left[\frac{e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}}{\sigma\sqrt{2\pi}}\right]\right|_a\frac{(x-a)^n}{n!}##. :rofl:
  8. Apr 30, 2013 #7
    I put the equation into excel and got a line that looks more like an exponential than a normal curve. Mean= 0, Stdev = 2, X=1-99.
  9. May 3, 2013 #8
    Out of curiosity, how on earth does a line look like an exponential? :tongue:

    You should be careful to put in the equation correctly. The definition of the normal curve is the formula given above.
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