Denominator in Jensen's inequality

Question

What is the intuition behind having a denominator in Jensen's inequality? For example, why do we need $n$ in a version of Jensen's inequality when all weights are equal: $$ \phi\left(\frac{\sum x_i}{n}\right)\leq\frac{\sum\phi(x_i)}{n}, $$ where $\phi$ is a convex function.

Answer 1

If you add $\frac 1 n $ $n$ times you get 1. This is crucial for Jensen's inequality. For example, $\phi (x)=x^{2}$ is convex but it is not true that $\phi (\sum x_i) \leq \sum \phi (x_i)$ in this case, as you can easily see by expanding $(\sum x_i)^{2}$.

Answer 2

You can apply $\phi$ to the average of the $x_i$'s, or you can average the $\phi(x_i)$'s. Jensen says that the second way gives a larger value than the first, if $\phi$ is convex.