Recently I came across a methodology called design of experiments, which can be used for making processes more robust.
I was looking through some examples and noticed that for the ANOVA used in this methodology only the means are used. My question is, wheres does the standard deviation go? It is not relevant to take into account??
Best Regards
One-Way ANOVA: The simplest ANOVA design is called a 'one-way' or 'completely randomized' design. That would be an appropriate design to test whether there are differences among the means $\mu_2, \mu_2, \dots \mu_g$ of $g > 2$ populations.
Suppose you want to know whether $g=3$ different (natural or chemical) insecticides are equally effective against a particular pest on a particular crop. You have 24 plots of the crop (of equal size) and you administer each pesticide to $r = 8$ plots $(r = 24/g),$ where the plots receiving each pesticide are randomly chosen. Later you measure the pests on each plot (perhaps the number of pests, the number of plants ruined by pests, the crop yield, etc.)
Then you can find $g$ sample means $\bar X_1, \dots \bar X_g$ and $g$ sample standard deviations $s_1, \dots, s_g$ (each over $r$ plots).
The model for a one-way ANOVA is $$Y_{ij} = \mu_i + e_{ij} = \mu + \alpha_i + e_{ij},$$ where $e_{ij} \stackrel{iid}{\sim} \mathsf{Norm}(0, \sigma).$ The terms $\alpha_i$ are sometimes called the 'effects' of the $g$ kinds of insecticide, respectively.
Now, finally, for the answer to your question: The data are used to obtain two estimates of $\sigma^2.$ One estimate, called MSA, 'the mean square for factor' or 'the mean square among groups' is based on the $\bar X_i$'s. The other estimate is based on the $S_i$'s. It is sometimes called MSE, 'the mean square for error' or the 'the mean square within groups'.
If all all insecticides are equally effective, MSA and MSE are both unbiased estimates of $\sigma^2.$ But if insecticides have differing effects, then MSE is an unbiased estimate of $\sigma^2$ and MSA tends to be too large (by an amount that depends on $\sum_i \alpha_i^2).$
Under the null hypothesis that all $\alpha_i$ are the same (or that $\sum_i \alpha_i^2 = 0),$ the ratio $F = \text{MSA/MSE}$ has Snececor's F-distribution with numerator degrees of freedom $g-1$ and denominator degrees of freedom $g(r-1).$ Accordingly MSA/MSE $\approx 1.$ If MSA/MSE is sufficiently greater than 1, we reject the null hypothesis, and conclude there are likely differences among the insecticides.
I will leave it to you to look at a basic statistics text (or online) for the formulas for MSA and MSE. It may not be immediately obvious that MSA depends only on the $\bar X_i$'s. However MSA is essentially $r$ times the sample SD of the $\bar X_i$'s. Also, MSE is essentially a weighted average of the $S_i^2$'s, with weights determined by sample sizes within the $g$ groups. In the example above, all sample sizes are $r,$ so MSE is a simple average of the $g$ sample variances. So the sample variances go into MSE.
Notes: (a) For normal data, the sample mean and sample SD are independent random variables. It follows that MSA and MSE are independent. The distribution theory of the $F$ ratio requires independence.
(b) If we reject the null hypothesis and conclude there are statistically significant differences among the group mean, then we need to do 'multiple comparison' procedures to determine the pattern of the differences among groups. Call the groups A, B, and C. All three groups may be different: for example, C >> A >> B. Alternatively, two groups may be indistinguishable and the third different: for example, C $\approx$ B >> A.
(c) In an application where MSA/MSE is substantially smaller than unity, one should question whether assumptions are met. Data may not be normal. Groups may have different population standard deviations. There might somehow be correlation among the data from neighboring plots. And so on.
(d) Particularly if the number of replications $r$ is large, it may be possible to detect statistically significant differences that are not of practical importance. Judging how large a difference is of practical importance is essential in applications, but it is not entirely a statistical issue.
Example; Here is an example with fake data, in which group C has a larger population mean than group A or B.
In the corresponding Minitab ANOVA, MSA = 71.2, MSE = 2.89, F = 24.65, and the P-value < 0.0005 indicates a highly significant result. It seems obvious from the plot that Insecticide C has a higher population mean than the other two, but a formal multiple comparison procedure should be done in order validate that impression. [The sample variance for C is a little larger than the sample variances for A or B, but that difference in variances is not statistically significant.]