I am recently learning ANOVA technique, its different mathematical proofs models etc, but somehow I cannot visualise properly its practical use in real life.
ANOVA has immensely useful practical applications in business, particularly Lean-Six Sigma/operational efficiency.
As you might expect, it is useful whenever you have multiple different discrete x variables and continuous y data. Usually, but not always, this takes the form of categorical labels and associated data.
So, for example, I have in the past used ANOVA to measure whether the waiting times of process operators were longer for any of four different processes in which those operators worked. So I measured the length of time that each operator was waiting for each of the four processes each day, and then ANOVA analysis showed very clearly that the waiting time for one of the processes was longer than the other three (based on a 95% confidence interval).
That's just one application - I could really give scores of examples of situations in which I personally have used it. It could also be used outside operations - in marketing, you might have five advertising campaigns and want to see whether any of them generated higher (or lower) sales figures than the others, etc. You might find that the confidence intervals for campaigns A and B were higher than those for C, D and E, but overlapped with each other, in which case you would take A and B for further testing. Alternatively, all five campaigns might have overlapping confidence intervals, in which case you can maybe question whether that expensive celebrity endorsement for campaign C was really worth it, since it hasn't generated returns which appear to be better than the other campaigns by a statistically significant amount.
Of course, you can also use an ANOVA with two samples, in which case it reduces to a t-test (although some software programmes will give you slightly different results (albeit very, very slightly - to ten decimal places or similar) for a two-sample ANOVA and a t-test, but this is only because the computational methods used are different.
Finally, of course, all of the usual provisos about statistical analysis apply, and you should make sure that anyone (especially a non-technical manager) to whom you show the output of an ANOVA analysis fully understands the provisos and qualifications about confidence intervals, type 1 and 2 errors, sampling, etc.