Friday, April 14, 2023

Аverage percentage

Average percentage. Mathematics For Blondes.
Average percentage

On Reddit, a person asked an interesting question about percentages:

Help with explaining negative compounding

Hey guys, not a maths teacher and I need to teach a concept related to work where something is negatively compounded.

For example:

Of a 1000 items 20% are rejected, leaving 800. At the next stage of those 800 30% are rejected, leaving 560. At next stage of the 560 all are rejected at 100% rejection.

The "average" rejection rate is NOT (20+30+100)/3 equalling 50%, it's 42% because "maths". I do actually have the formula and I personally understand why it's 42% instead of 50 but can anyone help me try to explain in a manner that makes it simple to understand WHY it's 42 and not 50?

I wrote my comment:

This is an incorrect application of mathematics. The average failure rate can only be calculated within one process. The average percentage between processes is absurd. Mathematically it looks like this:


I showed that the proposed concept of "average percentage" does not allow you to get zero at the end of the calculation, which is not true. The man explained how he sees his decision:

I'm not sure I understand.

I understand the formula, as you have written, as it is what I use, and what I demonstrated above in a reply to an answer. But I don't understand what you mean about the average percentage between processes being absurd.

If I use my previous example:

First use = 1 (Used once, no rejection)
Second use = 1*(1-0.2) = 0.8 (20% rejected)
Third use = 0.8*(1-0.3) = 0.56 (30% rejected)
Fourth use 0.56*(1-1) = 0 (End of life product, 100% rejected)

Total average = 1/(1+0.8+0.56) = 0.42

In the end, using the average of 0.42, gives us an approximate product rejection, or if we want to call it something else, it's average "life", before it needs a brand new replacement.

Now I will show you what is wrong in this decision.

Let's go back to 1000 items. Absurdity is the summation of numbers with different units of measure. Percentages are taken from different numbers:

I use     100%=1000   1%=10
II use    100%=1000   1%=10
III use   100%=800    1%=8
IV use   100%=560    1%=5.6

To determine the average percentage, we are actually adding numbers with different units of measure. We received 42%, but it is not known from what number these percentages are to be calculated. It's like adding one dress to one gallon. We can get two. But what does the number two mean? Two gallons, two dresses, or two wet dresses?


To get wet dresses, you need to use multiplication:


This math matches reality much better. By the way. In mathematics, the absence of a number is denoted by the number "zero". The absence of a letter in the grammar is indicated by a "space". Mathematicians say that zero is a number. Philologists have never said that a space is a letter. In units of measurement, it is more logical to use a space.


Now let's look at percentages. To be correct, we must count the number of items rejected at each stage as a percentage of the total number of items.

I use      0 items      0%
II use    200 items   20%
III use   240 items   24%
IV use   560 items   56%

It's even better to count as a percentage the number of items that fail after each stage with a cumulative total. This will be a more realistic picture of what is happening.

I use      0 items      0%
II use    200 items   20%
III use   440 items   44%
IV use   1000 items  100%

Personally, I argue like this. In this case, it is not appropriate to calculate the average percentage.

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