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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

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

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
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 bysigmahat(e)**2).

    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
    F = MST / MSE = 9.59
That is it. These numbers are the quantities that are assembled in the ANOVA table that was shown previously. 
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