# 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 '12 at 20:35
• Of possible interest: How to visualize what ANOVA does? – chl Oct 16 '12 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. – Penguin_Knight Oct 17 '12 at 0:18
• I found David Lane's online book very useful. onlinestatbook.com/2/analysis_of_variance/intro.html – idnavid Aug 20 '19 at 1:17

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.