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Disclaimer: I understand that this question might be a little strange, since it isn't as much about mathematical properties of ANOVA as about its definition. This question is motivated by recent consultation of semestral paper.

ANOVA stands for ANalysis Of VAriance. Suppose I am using it on experimental data divided in several groups. Would it be correct to say that I'm conducting Analysis of Variance on my data?

Now technically, all I care about is testing the equality of means using ANOVA method. So should I instead just say that I'm testing equality of means using ANOVA?

Another question is about the ANOVA null hypothesis. From my understanding it's pretty straightforward. Unless specified otherwise, the null hypothesis states that all data is from one population (and therefore all means are equal).

And obviously the alternative hypothesis being that the data isn't from one population (and thus the means are not equal).

Do I actually need to restate this, when presenting results of the analysis?

What I'm trying to figure out, is if I just misunderstood some formality requirements or the ANOVA itself.

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    $\begingroup$ "Analysis of variance" is an odd name. Despite its name, its main focus is on comparing means. But how can we compare means (of two or more groups classified in one or more ways)? Only by comparing variation around means within a group and variation between means in different groups. The idea behind ANOVA is to cast that comparison on a scale where it can be made. Many people, like myself, who don't like the name much try to avoid that name, by talking of regression or a general linear model or a generalized linear model. $\endgroup$
    – Nick Cox
    Commented Apr 8, 2018 at 12:47
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    $\begingroup$ At the opposite extreme there are texts and courses that seem to imply it's the central technique in all of statistical science! What you are outlining all seems sound to me. $\endgroup$
    – Nick Cox
    Commented Apr 8, 2018 at 12:48

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You have not misunderstood the ANOVA test, and your description of the running of the test is comprehensive and correct...though you probably don't need to state all of that in a final write-up.

I will echo the first couple of comments that have posted. The ANOVA monicker is confusing as it focuses on variance to test for mean differences, and some authors do indeed suggest that all statistical analyses (at least all ordinary least squares (OLS) analyses) are built on the ANOVA. It is for this reason that I stress the difference between the ANOVA summary table and the ANOVA test when teaching it to my students.

At the core of any application of ANOVA is the attempt to determine if a significant proportion of the variance of the dependent variable has been explained by the model under consideration. Thus, comparing group means results in an ANOVA table, running a regression produces an ANOVA table, running an ANCOVA produces an ANOVA table, even mixed effects analyses produces an ANOVA table. So, I find it helps to stress that the ANOVA summary table is just part of the protocol, and the ANOVA test for group means is a specific test.

Of course, most hypothesis tests can be recast in different notations or formulations for a variety of comparable research questions. In the general ANOVA (that summary table that is used for any of the OLS analyses), the ANOVA is testing the hypothesis: $$H_0 : R^2 = 0$$ where $R = \text{cor}(y,\hat{y})$. When this is applied to just the comparison of group means, we often call this the ANOVA test (for mean differences).

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