7.
Product and Process Comparisons
7.4.
Comparisons based on data from more than two processes
7.4.3.
Are the means equal?
7.4.3.4.
|
1-Way ANOVA calculations
|
|
| Formulas
for 1-way ANOVA hand calculations |
Although computer
programs that do ANOVA calculations now are common, for reference purposes
this page describes how to calculate the various entries in an ANOVA table.
Remember, the goal is to produce two variances (of treatments and error)
and their ratio. The various computational formulas will be shown and applied
to the data from the previous example. |
| Step 1:
compute CM |
STEP 1 Compute
CM, the correction for the mean.
![CM = {SUM[i=1 to 3][j=1 to 5]y(ij)}**2/N(total) = (Total of all observations squared)**2/N(total) = (108.1)**2/15 = 779.041](eqns/anova10.gif)
|
| Step 2:
compute total SS |
STEP 2 Compute
the total SS.
The total SS = sum of squares of all observations - CM
![SS(total) = SUM[i=1 to 3][j=1 to 5][y(ij)**2] - CM = 6.9**2 + 5.4**2 + ... + 6.9**2 - CM = 829.390 - 779.041 = 45.349](eqns/anova11.gif)
The 829.390 SS is called the "raw" or "uncorrected " sum of squares. |
| Step 3:
compute SST |
STEP 3 Compute
SST, the treatment sum of squares.
First we compute the total (sum) for each treatment.
T1 = (6.9) + (5.4) + ... + (4.0) = 26.7
T2 = (8.3) + (6.8) + ... + (6.5) = 38.6
T1 = (8.0) + (10.5) + ... + (9.3) = 42.8
Then
![SST = SUM[i=1 to 3][T(i)**2/n(i)] - CM = 26.7**2/5 + 38.6**2/5 + 42.8**2/5 - 770.041 = 27.897](eqns/anova13.gif)
|
| Step 4:
compute SSE |
STEP 4 Compute
SSE, the error sum of squares.
Here we utilize the property that the treatment sum of squares plus
the error sum of squares equals the total sum of squares.
Hence, SSE = SS Total - SST = 45.349 - 27.897 = 17.45. |
| Step 5:
Compute MST, MSE, and F |
STEP 5 Compute
MST, MSE and their ratio, F.
MST is the mean square of treatments, MSE is the mean square of error
(MSE is also frequently denoted by ).
MST = SST / (k-1) = 27.897 / 2 = 13.949
MSE = SSE / (N-k) = 17.452/ 12 = 1.454
where N is the total number of observations and k is the
number of treatments. Finally, compute F as
That is it. These numbers are the quantities that are assembled in the
ANOVA table that was shown previously. |
