# Calculus to find the z-score?

1. Mar 1, 2013

### GiTS

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. Mar 1, 2013

3. Mar 2, 2013

### ssd

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
4. Apr 22, 2013

### GiTS

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):

10724
13119
13576
13711
13849
19285
19734
19978
20760
21077

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,
-GITS

5. Apr 26, 2013

### GiTS

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.

Thanks!

Ideas? The normal distribution curve has to be a polynomial

6. Apr 27, 2013

### Mandelbroth

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:

7. Apr 30, 2013

### GiTS

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.

8. May 3, 2013

### Mandelbroth

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.