ANOVA test real-life application What are the real-life application of ANOVA test ?I used to solve question during my college day and never understood the real life application.
Thanks,
 A: Just as a two-sample t test may allow you to judge whether two
groups come from populations with the same mean, the simplest
ANOVA allows you to judge whether $g > 2$ groups come from populations
with the same mean. 
Suppose an engineer is trying to improve the efficiency of a
particular device and has two possibilities for improvement in
mind. Let $X$ be a measure of efficiency, and suppose there are
10 devices made according to each of the proposed new specifications.
Then we can measure efficiency of 10 of the current standard devices
(Group 1) and of 10 devices with each proposed modification
(Groups 2 and 3). 
The model for this experimental design would be
$$X_{ij} = \mu_i + e_{ij},$$
for $i = 1,2,3$ and $j=1,2,\dots 10,$ where $\mu_i$ is the mean efficiency in the population
from which the $i$th group is sampled, and 
$e_{ij} \stackrel{iid}{\sim} \mathsf{Norm}(0, \sigma)$
represent random variability among the sampled devices.
Key assumptions are that the efficiencies $X_{ij}$ are
normal and that they have the same standard deviation $\sigma.$
Roughly speaking, variability among the three group averages $\bar X_i$ is compared against an estimate of the underlying variance $\sigma^2.$
If the relative variability among the $\bar X_i$'s is markedly larger than would be
anticipated for groups from the same population mean, then we have
evidence that the three designs differ. If so, the next step would be
to make pairwise comparisons among the three groups to try to discover what the
pattern of different efficiencies might be.
You can find the details of the relevant statistical tests in many elementary
level statistics books. [Perhaps you will see an ANOVA table with
MS(Group), MS(Error), and an F-statistic which is their ratio.]
Note: There are alternative, slightly messier, procedures in case the three
designs have different variabilities or the efficiency measurements are
not normally distributed.
