average percentage

[Paulpacheco]

I have figured the average percentage of one group of numbers to another group on numbers two different ways and ended up with two different results.
I do not understand why I get two different results and which is correct.
The first way I figured the sum of the first two columns and divided the second onto the first to get the percentage.
The second way I figured the percentage of each line first (third column) then added the percentages and divided by the number of said percentages to get the average.
See below.

$0.71 $0.71 100.00%
$0.99 $0.99 100.00%
$3.71 $3.71 100.00%
$82.07 $82.07 100.00%
$0.17 $0.20 85.00%
$1.75 $0.54 324.07%
$0.10 $0.18 55.56%
$4.51 $6.24 72.28%
$4.83 $5.80 83.28%
$1.00 $1.64 60.98%
$80.70 $80.70 100.00%
$20.00 $37.49 53.35%
$2.65 $3.91 67.77%
$0.10 $0.08 125.00%
$0.49 $0.33 148.48%
$0.68 $0.70 97.14%
$0.68 $0.84 80.95%
$205.14 $226.13 1753.86%
/$226.13 /17[
90.72% 103.17%

 

[lookagain]

I have figured the average percentage of one group of numbers to another group on numbers two different ways and ended up with two different results.
I do not understand why I get two different results and which is correct.
The first way I figured the sum of the first two columns and divided the second onto the first to get the percentage.
The second way I figured the percentage of each line first (third column) then added the percentages and divided
by the number of said percentages to get the average.

Each exam, for instance, must be based on the same number of maximum
points for an average percentage to work.

Your two methods should not be expected to give results that are the same.

Suppose for any pair:

\(\displaystyle a \ \ \ \ \ b \ \ \ \ \ \frac{a}{b}\)

\(\displaystyle c \ \ \ \ \ d \ \ \ \ \ \frac{c}{d}\)
-------------------------------------------


In general,


\(\displaystyle \frac{a + b}{c + d} \ \ne \ \frac{1}{2}\bigg(\frac{a}{b} + \frac{c}{d}\bigg) \ = \frac{ad +bc}{2bd}\)

 

[soroban]

Hello, Paulpacheco!

Your title says it all: average percentge . . . a very bad idea.


Suppose you took a two-part exam.

In part 1, you got 5 out of 10 points.
. . Your average is 50%.

In part 2, you got 90 out of 90 points.
. . Your average is 100%.
\(\displaystyle \text{So your teacher awards you: }\:\frac{50\% + 100\%}{2} \:=\:75\%\)
Is that a fair grade?

Certainly not!

You got 95 out of 100 points.
Your overall average is 95%.