Percentages not adding up

Question

I have a series of percentages:

132/220 (60%) and 88/220 (40%).

Now when I break them down into subcategories and then recalculate the percentages them come out 5% different.

81/140 (58%) and 59/140 (42%) (percentages ROUNDED).
21/40 (52.5%) and 19/40 (47.5%).
14/20 (70%) and 6/20 (30%).
16/20 (80%) and 4/20 (20%).

then we I do the averages I get:

65.125% and 34.875%

I have no idea what is happening???

Answer 1

This is "normal." Suppose you have $2$ big exams, each out of $100$, and you get $40\%$ in each. Average: $40\%$.

You also have $2$ little assignments, each out of $25$, and you get a perfect score in each. Average: $100\%$.

The average of the two averages is $70\%$.

However, the total mak out of $250$ was $130$. Average: $52\%$.

Your example is less extreme, but the same phenomenon is involved.

For a correct computation, you should take a weighted average of the averages.

So correct would be $(40\%)\frac{200}{250}+(100\%)\frac{50}{250}$.

Answer 2

You seem to be suggesting that $\frac{132}{220} = \frac{81}{140} + \frac{21}{40} + \frac{14}{20} + \frac{16}{20}$. This is not true. The simplest example, $\frac{4}{6}$ is not equal to $\frac{3}{5}+\frac{1}{1}=1.6$.

Answer 3

If the samples/subcategories have different sizes, the average of the averages is not the same as the average of all numbers.

You have to make a weighted average: multiply each percentage by the size of the group, add them together and divide by the total size.

Answer 4

There might be a typo. Although others have already answered this question adequately.

The type being 19/20 I believe you meant to type 19/40.