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I want to answer the question: On average how many sticks of butter per dollar.

Given the following

        Dollars     Butter     Butter/Dollars
        1           2           2
        1           2           2
        1           2           2
        2           2           1
        2           2           1
        2           3           1.5
        2           3           1.5
        1           3           3

Sum     12          19          14
Avg     1.5         2.375       1.75

AvgDollars/AvgButter = Sum(Dollars)/sum(butter)= 2.375/1.5= 1.58, is my answer I believe.

However, I have no intuition as to what the Avg(Butter/Dollars) = 1.75 represents or why its incorrect. Can anyone explain?

ARs
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  • It's just $\frac {14}8$. Literally the average of "butter per dollars". As you remark, this is not the same as "average butter per average dollar". Your computation would be unchanged if, say, you permuted the dollar column, but that's clearly not what is intended. – lulu Jun 30 '19 at 23:35

2 Answers2

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What you've noticed is the difference $$ \frac{\langle x\rangle}{\langle y\rangle} = \frac{x_1+x_2+\dots +x_n}{y_1+y_2+\dots +y_n} \neq \frac{\frac{x_1}{y_1}+\frac{x_2}{y_2} + \dots \frac{x_n}{y_n}}{n} = \langle\frac x y\rangle $$ where $\langle A\rangle$ denotes the average value of $A$. It shows that the ratio of the averages is generally not equal to the average of the ratio.

Adam Latosiński
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The table just computes the average of the third column, $\frac {14}8=1.75$

It is simply not the case that the average (butter per dollar) is equal to the average(butter)/average(dollar). At least not in general.

Consider the following

$$ \begin{array}{c|lcr} n & \text{Dollars} & \text{Butter} & \text{Butter Per Dollar} \\ \hline 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 1 \end{array} $$

In this instance the two do in fact coincide! Both are $1$.

But now suppose we permute the Butter Column, to get:

$$ \begin{array}{c|lcr} n & \text{Dollars} & \text{Butter} & \text{Butter Per Dollar} \\ \hline 1 & 1 & 2 & 2 \\ 2 & 2 & 1 & \frac 12 \end{array} $$

The ratio of the two averages is still $1$, of course. We haven't changed those averages. But the average(butter per dollar) is now $\frac {2.5}2=1.25$

lulu
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