7. Product and Process Comparisons
7.4. Comparisons based on data from more than two processes
7.4.2. Are the means equal?

## 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, the correction for the mean.

STEP 2 Compute the total SS.

The total SS = sum of suquares of all observations - CM

The 829.390 SS is called the "raw" or "uncorrected " sum of squares.

STEP 3 Compute  SST,  the treatment sum of squares.

First we compute the total (sum) for each treatment.

Then

STEP 4 Compute SSE, the error sum of squares. SSE is also frequently denoted by

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 = 17.54

STEP 5 Compute MST,  MSE and their ratio, F.

MST is the mean square of treatments, MSE is the  mean square of error

MST = SST / (k-1) =   27.897 / 2 = 13.949
MSE = SSE / (N-k) = 17.452/ 12 = 1.453

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 before.