Question

What's the string of 4 numbers that results in a median of 20?

Answer 1

Any set of numbers `a, b, c, d`, where `a < b < c < d, and b+c=40`

Explanation:

The median is the middle value in a set of numbers. Where there are an even number of numbers, you take the 2 middle values and take their mean (add them and divide by 2).

In this case, with the 4 numbers, say:

`a, b, c, d`, where `a < b < c < d`

Let's talk about `b and c` first:

We need `b and c` to have a mean of 20, which means:

`(b+c)/2=20=>b+c=40`

Therefore, there is an infinite set of numbers that meet this condition (we can have 19 and 21, 18 and 22, `-1` and 41, etc...)

We can have any value for `a` so long as it is less than `b` - which is another infinite list.

We can have any value for `d` so long as it is greater than `c` - which is yet another infinite list.

Answer 2

The four numbers `18, 19, 21, 22` have a median of `20`.
But let's not stop there, because there are hundreds of four number combinations that have a median of `20`.

Explanation:

Your search is for a string of four numbers that will essentially bracket `20` as the middle number even though it will not appear in the series.

The `20` cannot appear because you have an even quantity of numbers `(4)`. If you had five numbers, the series could be `18, 19, 20, 21, 22.`

To make more combinations we can stretch the series:
`17, 18, 22, 23`; or `0, 10, 30, 40`;

Or lop-side it:
`1, 15, 25, 100`; or `1/2, 7 1/2, 32 1/2, 10,000 2/3`

To bracket the `20` in each case, we chose two center numbers in the series whose values would average out to `20`.

In the examples above: `(19+21)/2= 40/2 = 20`

`(18+22)/2 = 40/2 = 20 = (10+30)/2 = (15+25)/2 = (7 1/2+ 32 1/2)/2`

Summary:
Given a median M of an even quantity of numbers;
Double the given median value `=2M`;
Subtract any value m `(cancelM)`from the doubled value; `2M-m = to (m cancel= M).`
The answer obtained above and `m` are your two center numbers.
Arrange an even quantity of numbers in an ascending series.