Calculating and Interpreting Expected and Chi-Square Tables

Date: 04/10/98 at 22:31:30
From: Lola Kelly
Subject: Stats- chi squared

How do you get the frequency expected when doing chi squared? I have 
a 3-by-3 table. Help. Please.

Date: 04/11/98 at 08:06:14
From: Doctor Anthony
Subject: Re: Stats- chi squared

I will work through a 3x3 contingency table. The method will be 
directly applicable to any similar problem. The following table refers 
to the performance of a baseball team on various pitches classified as 
good, medium and bad. The null hypothesis is that the state of the 
pitch does not affect the performance of the team.

            Wins     Draws     Losses     Total
  ---------|--------------------------|----------
   Good    | 11        6          4   |     21
  Medium   | 12        7          7   |     26
    Bad    |  7        7         14   |     28
  ---------|--------------------------|-----------
  Total    | 30       20         25   |     75


To find the expected frequencies, we assume independence of the rows 
and columns. To get the expected frequency corresponding to the 11 at 
top left, we look at row total (21) and column total (30), multiply 
them, and then divide by the overall total (75). So the expected 
frequency is: 

     21*30
     -------  = 8.4
       75

So to complete the expected table, draw up another table similar to 
that above and having the same row and column totals. For each entry 
in this table, we simply calculate (row total*column total)/75. The 
completed table is:

               Wins     Draws     Losses     Total
   -----------|--------------------------|-----------
      Good    | 8.4      5.6        7.0  |    21
     Medium   |10.4      6.9        8.7  |    26
       Bad    |11.2      7.5        9.3  |    28
   -----------|--------------------------|------------
     Total    | 30       20         25   |    75

The number of degrees of freedom is calculated for an m-by-n table as 
(m-1)(n-1), so in this case (3-1)(3-1) = 2*2 = 4.

To calculate X^2, we then have a further table:

       O       E       |O-E|       |O-E|^2/E
     ----------------------------------------
      11      8.4       2.6         0.805         
       6      5.6       0.4         0.0286
       4       7         3          1.2857
      12     10.4       1.6         0.246
       7      6.9       0.1         0.0014
       7      8.7       1.7         0.332
       7     11.2       4.2         1.575
       7      7.5       0.5         0.033 
      14      9.3       4.7         2.375
     -------------------------------------------      
                            Total = 6.70  = X^2

The tabular 95% value of X^2 (degrees of freedom = 4) is 9.49, so the 
value of X^2 that we obtained (6.70) is not significant at the 5% 
level. We conclude that the state of the pitch does not affect the 
performance of the team.
   
-Doctor Anthony, The Math Forum
Check out our web site! http://mathforum.org/dr.math/   

Date: 04/11/98 at 13:00:10
From: Coolbabe29
Subject: Re: Stats- chi squared

I cannot believe what a big help you were. I appreciate your speedy 
reply. My Stats prof. had me so flustered, a simple straightforward 
answer helped so much! THANKS!