# Why the log of the mean is so much different than the mean of logs

I analyse some data, where I have two vectors of values, vector A and vector B. I compute the following two things:

for x, y in A, B:
ratios.append(x / y)

for x, y in A, B:
log_ratios.append(ln(x/y))


Next I compute

log_mean = ln(mean(ratios))
mean_logs = mean(log_ratios)


So in simply words, given my data I compute two mean values - a mean of logs, and a log of means. I noted that those two values are very far away, i.e.,

log_means = 20.148329613107876
mean_logs = 1.6568702569456684


I did a small computation, comparing how the equations look like, so for the log of mean I would get $$log\_mean = log\left(\frac{\sum_{i=0}^n\frac{x_i}{y_i}}{n}\right) = log\left(\sum_{i=0}^n\frac{x_i}{y_i}\right) - log(n)$$ whereas $$mean\_log = \frac{\sum_{i=0}^n \log\left(\frac{x_i}{y_i}\right)}{n} = \frac{\sum_{i=0}^n\left(log(x_i) - log(y_i)\right)}{n}$$

So I see, that mathematically, I obtain two different values, however I'm having hard time to intuitively understand why this difference is so big. Can someone help me a bit with it?

• As $\log(a+b)\neq \log(a)+\log(b)=\log(ab)$, I'm not surprised. Jul 2 '18 at 20:13
• If you have one ratio close to $0$ in the sample, that hardly has an influence on the mean of the ratios, and hence also little impact for the logarithm of that mean. But the logarithm of that ratio is a negative number that can have a huge modulus, and then it makes a big bonking difference for the mean of the logarithms. Jul 2 '18 at 20:22

## 3 Answers

This is the AM-GM inequality in disguise.

You have, setting $a_i = \frac{x_i}{y_i}$, $$\left( a_1 a_2\dots a_n \right)^{1/n} \leq \frac{a_1+\dots+ a_n}{n} \tag{AM-GM}$$ Taking the log on each side, $$\frac{1}{n}\log \left( a_1 a_2\dots a_n \right) \leq \log \frac{a_1+\dots+ a_n}{n}$$ or, equivalently, $$\frac{\log a_1 + \dots + \log a_n}{n} \leq \log \frac{a_1+\dots+ a_n}{n} \,.$$ (Note that you can also prove your inequality with Jensen's inequality, since $\log$ is concave).

In both cases (AM-GM or Jensen's inequalities), you get that the inequality holds if, and only if, $a_1=\dots = a_n$. That tells you why you have that inequality, and why it's not an equality.

As to see why it's that "much" of a strict inequality in your case, intuitively you can think of it as a robust version of the above statement: "the more the $a_i$'s are different (the least balanced), the more this inequality will be far from an equality." (This is very handwavy, but true here.)

• Great, that helps a lot! Jul 2 '18 at 20:43
• Regarding that last point, there exist references on how to quantity the "gap" in the inequality (typically, that's called "defect"). It's not necessarily that easy to parse, but if you really are interested, it exists (e.g., The Variance Drain and Jensen's Inequality (Becker) Jul 2 '18 at 20:45

This is because $\ln$ is a concave function. In contrast, if you performed the same calculations with a convex function e.g. $\exp$, you would see that taking the mean first produces a smaller quantity.

You are finding two different quantities. One is logarithm of the arithmetic mean and the other is logarithm of the geometric mean.

$$\frac {\log a + \log a }{2} = \frac {\log ab}{2} = \log \sqrt {ab} \ne \log \frac {a+b}{2}$$

For example for $a=20$ and $b=80$ we get

$$\log \sqrt {ab}= \log 40$$ while $$\log \frac {a+b}{2}=\log 50$$