# How can I explain the intuition behind ANOVA?

I need to explain the intuition behind what ANOVA is doing to a non-technical person. Is there a visual that explains the idea? A visual that illustrates the key idea in the context of a one-way ANOVA with perhaps 3 factor levels might be helpful?

Let us suppose that the person has taken some statistical courses as a student in the distant past but has forgotten the details of even performing a z-test. However, he/she remembers that hypothesis testing is used to check if the observed effects are due to random chance or due to a real change in the parameter of interest.

• Please add to your question what you expect the listener to have for statistics knowledge. Do they know a t-test? Do they have some familiarity with hypothesis testing?... etc. If they have absolutely none there's not much detail I'd ever want to go into.
– John
Oct 16, 2012 at 20:35
• Of possible interest: How to visualize what ANOVA does?
– chl
Oct 16, 2012 at 21:19
• How about using an example of three firework exploding in the sky? The three explosion heights would be their group means. Given the same three heights, the difference can feel greater if the firework explosion radius is small (low within group variation); and they can also feel indifferent if the explosion radius is huge relative to their explosion height differences. Hence, both explosion heights (between) and explosion radius (within) have to be considered. Oct 17, 2012 at 0:18
• I found David Lane's online book very useful. onlinestatbook.com/2/analysis_of_variance/intro.html Aug 20, 2019 at 1:17
• I wrote a completely non-mathematical explanation of why we use variances to compare means. Here is a link to the blog post. Apr 17, 2020 at 11:35

ANOVA is a statistical technique used to determine whether a particular classification of the data is useful in understanding the variation of an outcome. Think about dividing people into buckets or classes based on some criteria, like suburban and urban residence. The total variation in the dependent variable (the outcome you care about, like responsiveness to an advertising campaign) can be decomposed into the variation between classes and the variation within classes. When the within-class variation is small relative to the between-class variation, your classification scheme is in some sense meaningful or useful for understanding the world. Members of each cluster behave similarly to one another, but people from different clusters behave distinctively. This decomposition is used to create a formal F test of this hypothesis.

I found David Lane's online book very useful.

In a more fundamental way, there's an invited paper in Annals of Statistics by T.P. Speed called "What is Analysis of Variance?". It took me a few attempts, but at the end it was very informative. The essence of the paper is to show that ANOVA is simply a decomposition of variance into a summation of variances belonging to smaller groups. Another important take away is that you can use ANOVA for more general variances (covariances), which I though was interesting.

You could explain that ANOVA is a decomposition of the data as components that correspond to different groups or variables or sources of variation. An example is $${\tiny \begin{pmatrix} 89 & 88 & 97 & 94 \\ 84 & 77 & 92 & 79 \\ 81 & 87 & 87 & 85 \\ 87 & 92 & 89 & 84 \\ 79 & 81 & 80 & 88 \end{pmatrix} = \begin{pmatrix} 86 & 86 & 86 & 86 \\ 86 & 86 & 86 & 86 \\ 86 & 86 & 86 & 86 \\ 86 & 86 & 86 & 86 \\ 86 & 86 & 86 & 86 \end{pmatrix} + \begin{pmatrix} \phantom{-}6 & \phantom{-}6 & \phantom{-}6 & \phantom{-}6 \\ -3& -3 & -3 & -3 \\ -1 & -1 & -1 & -1 \\ \phantom{-}2 & \phantom{-}2 & \phantom{-}2 & \phantom{-}2 \\ -4 & -4 & -4 & -4 \end{pmatrix} + \begin{pmatrix} -2 & -1 & 3 & 0 \\ -2 & -1 & 3 & 0 \\ -2 & -1 & 3 & 0 \\ -2 & -1 & 3 & 0 \\ -2 & -1 & 3 & 0 \end{pmatrix} + \begin{pmatrix} -1 & -3 & \phantom{-}2 & \phantom{-}2 \\ \phantom{-}3 & -5 & \phantom{-}6 & -4 \\ -2 & \phantom{-}3 & -1 & \phantom{-}0 \\ \phantom{-}1 & \phantom{-}5 & -2 & -4 \\ -1 & \phantom{-}0 & -5 & \phantom{-}6 \end{pmatrix} }$$
which represents observations from a two-way ANOVA design (without repliation), with rows and columns as the two groups. The algebraic model is $$y_{ti} = \mu + \beta_i + \tau_t + \epsilon _{ti}$$ and the corresponding data decomposition is calculated as $$y_{ti} = \bar{y} + \left\{\bar{y}_i-\bar{y}\right\} + \left\{ \bar{y}_t-\bar{y}\right\} + \left\{y_{ti}-\bar{y}_i - \bar{y}_t +\bar{y}\right\}.$$ For one-way ANOVA an example is $${\tiny \begin{pmatrix} 62 & 63 & 68 & 56 \\ 60 & 67 & 66 & 62 \\ 63 & 71 & 71 & 60 \\ 59 & 64 & 67 & 61 \\ & 65 & 68 & 63 \\ & 66 & 68 & 64 \\ & & & 63 \\ & & & 59 \end{pmatrix} = \begin{pmatrix} 64 & 64 & 64 & 64 \\ 64 & 64 & 64 & 64 \\ 64 & 64 & 64 & 64 \\ 64 & 64 & 64 & 64 \\ & 64 & 64 & 64 \\ & 64 & 64 & 64 \\ & & & 64 \\ & & & 64 \end{pmatrix} + \begin{pmatrix} -3 & 2 & 4 & -3 \\ -3 & 2 & 4 & -3 \\ -3 & 2 & 4 & -3 \\ -3 & 2 & 4 & -3 \\ & 2 & 4 & -3 \\ & 2 & 4 & -3 \\ & & & -3 \\ & & & -3 \end{pmatrix} + \begin{pmatrix} \phantom{-}1 & -3 & \phantom{-}0 & -5 \\ -1 & \phantom{-}1 & -2 & \phantom{-}1 \\ \phantom{-}2 & \phantom{-}5 & \phantom{-}3 & -1 \\ -2 & -2 & -1 & \phantom{-}0 \\ & -1 & \phantom{-}0 & \phantom{-}2 \\ & \phantom{-}0 & \phantom{-}0 & \phantom{-}3 \\ & & & \phantom{-}2 \\ & & & -2 \end{pmatrix} }%end tiny$$ and the algebra can be written in the same way.

This is mostly a comment, since it is not a full explanation, but could be a helpful component of any explanation, and could be suited to the necessary level. Such tables is used a lot in this famous book.

• Why the downvote? May 5, 2020 at 10:56

I've noticed that there are few sources that actually explain why variation between is close to variation within under null hypothesis. For simple ANOVA, the intuitive explanation is:

Variation within is simply an estimation of the population variance by averaging the variances of the individual groups (under the assumption that the variances of all groups are equal), by averaging the variances of individual groups we take the best shot at estimating the population variance.

As to variation between, it’s an interesting one. Under null hypothesis we assume that all groups come from the same population. Each group is a separate sample with its own mean. Under null hypothesis, the variance of the sample mean is a good estimator of the population variance (var(sample_mean) * sample_size), and if all the groups really come from the same population, this estimator of pop variance drawn from the sample means will tend to be close to population variance, otherwise, it will OVERESTIMATE the population variance, thus leading to a higher mean square between (which is an estimator of the population variance) and thus to higher F-statistic.

More on that here:

https://onlinestatbook.com/2/analysis_of_variance/one-way.html