Working out a percentage of from values

Question

Using the following numbers as an example:

293.21
478.93
485.44
544.78
569.48
593.82

I know how work out the average (mean) value:

50% = ( 293.21 + 478.93 + 485.44 + 544.78 + 569.48 + 593.82 ) / 6 = 494.2766666667

My question is, instead of working out the average (mean) value which I am visualising as 50% or half way. How do I work out 25%, 75% of those values?

I could probably work out 25% by taking the smallest value, and the average value and figuring out the average of those to get 25% for example? But 25% and 75% were just examples. What I really want to understand is how to get any percentage or position from those values, i.e. 1%, 23%, 95% and so on.

Finding it difficult to explain what I am trying to achieve.

So for a more visual example of what I am trying to do is this:

293.21 <- 0% is here
478.93
485.44
<- 50% is about here which is 494.2766666667
544.78
569.48
593.82 <- 100% is here

How do I work out 77% for example?

Answer 1

You are confusing two "averages", the mean and the median. The mean is the value you have calculated, the sum of the list divided by the number of entries. For a symmetric distribution, the mean will be in the middle of the distribution, which is your thought calling it $50\%$. The median is defined as the value that is greater than half the entries. The lower quartile ($25\%$ point) is the value that is greater than $25\%$ of the entries. You can find them by sorting the list and counting. For many asymmetric distributions, the median seems like a better average than the mean. For example, consider $1,2,3,4,5,6,7,8,9,10,110$. The mean is $15$, the median is $6$. Which seems like the best single number to represent the distribution? The lower quartile is about $3$ or $4$ These counting statistics fit what you are looking for.

Answer 2

I believe what you are looking for is not the average, but rather the median. The value of the median is such that half of the values are above it and half are below it. This sometimes corresponds with the mean (or average) but not always (see for instance this graph of household incomes).

Now, if the median is a specific case of the percentile: the median is the $50\%$ percentile. To calculate the $75\%$ percentile, you first sort the values in the list. Then you choose the element that is three quarters of way down the sorted list.

The same goes for any percentile you want to calculate - once you have sorted the list, you just choose the element which is $x\%$ down the list.