http://www.graphpad.com/quickcalcs/PValue1.cfm  stat computation

Chi Square gives probability that a random sample gives no better fit.


chi square= (n-1)* s^2/d^2

n=number of samples     s^2= sample variance  d^2=population variance

Statistics books give tables but not the formula generating it.
xx= Chi Square value    n = degrees of freedom = cases-1

Define gam(n)=Integral e^(-x)*x^(n-1),x,0,30)


          1                                                       
P= ---------------------- * Integral(x2 to 70) e^[(-x/2)) * 
     (n/2)
   2         * gam(n/2)                

                                            x^(n-2)/2)  ) dx



HP program

 2^(-n/2) / ((n/2)-1)! ) * Integral (xx,60, EXP(-x/2)*x^((n-3)/2),x)'

Degrees of freedom= 4-1=3
Example
Observed    Expected
A 70          75
B 80          75
C 85          75
D 84          70

chi square = x2 = Sum  (observed-expected)^2 / expected)

x2=2.33


                 alpha=area x2 to oo = p
n=3
If x2=7.81473    alpha=.05
      9.34830          .025
     11.34830          .01
     12.8382           .005
     16.266            .001  


Example

The probality there is no significant difference in the A B C D cases 
is between .001 and .005 so the assumption there is no significant 
difference bwtween A B C D is accepted.   Normally 0.05 is the rejection
critical value.
=======================================
Graph  Clear F= menu

0.242-      *   
     |
     |   *     *
     |   
     |          |*
     | *        |***              n=3   P=Area under curve from xx to 14
     |          |*****
                |*******
     |          |***********
     |*         |***************
     ------------------------------*------
     x=0       x2 = 2.333                   oo

=========
Go to Stats  F6 Chi2 GOF
Enter
Observed List {70,80,85,70}
Expected List {75,75,75,75}
df=Deg of Freedom  3 = 4-1

Calculate
Chi-2.333
P value 0.506 = probability the results is random and not significant


In Stats select Inverse Chi-square
=
Deg of freedom= 3
Area = 0.95 = 1 -.05
Inverse 7.8147 which is xx for 0.05 probability = area