# Calculating mean from 5 number summary

It seems like it should be possible to calculate the mean (usual average) from a 5-number summary of a set of numbers (min, first quartile or Q1, median, third quartile or Q3, and max). You should be able to calculate roughly what a percentile is, then by taking each discrete percentile and then taking the average of those hundred numbers... or better yet by using calculus and taking every percentile point, and the average of every point, you should be able to come really close to the mean, if not compute it directly. I, however, don't know math well enough to do that, nor do I remember any calculus.

In one data set that I'm looking at, the min, Q1, median, Q3, and max are: 0, 3900, 18882, 50145.5, 1250000
And the mean is: 46172.04545 or just under Q3.
How can the mean be calculated from those 5 numbers?

## Answers and Replies

mathman
Science Advisor
You can't exactly. A rough estimate (analogous to trapezoid rule for integral approximation) is:
(min + max + 2(Q1+median+Q3))/8.

phinds
Science Advisor
Gold Member
2021 Award
It seems like it should be possible to calculate the mean (usual average) from a 5-number summary of a set of numbers ...
Did you mean estimate? Surely you can see (as mathman pointed out) that you can't get an exact calculation based on just summary numbers.

You can't exactly. A rough estimate (analogous to trapezoid rule for integral approximation) is:
(min + max + 2(Q1+median+Q3))/8.
Hmm, ok, thanks. I have one data set where the 5-number summary is: 0, 29496, 68552, 124280, 780575. The mean is 80041.24331 and that approximation comes up with 153153.875, so I guess it's a really rough estimate.
Did you mean estimate?
Sure, why not. If I clearly don't know what I'm talking about, feel free to attempt to fill in the gaps. :)

Svein
Science Advisor
It seems like it should be possible to calculate the mean (usual average) from a 5-number summary of a set of numbers (min, first quartile or Q1, median, third quartile or Q3, and max). You should be able to calculate roughly what a percentile is, then by taking each discrete percentile and then taking the average of those hundred numbers... or better yet by using calculus and taking every percentile point, and the average of every point, you should be able to come really close to the mean, if not compute it directly. I, however, don't know math well enough to do that, nor do I remember any calculus.

In one data set that I'm looking at, the min, Q1, median, Q3, and max are: 0, 3900, 18882, 50145.5, 1250000
And the mean is: 46172.04545 or just under Q3.
How can the mean be calculated from those 5 numbers?
You cannot. But if you the distribution is fairly symmetrical (like the familiar bell-curve), the mean and the median are approximately equal.

mathman
Science Advisor
Hmm, ok, thanks. I have one data set where the 5-number summary is: 0, 29496, 68552, 124280, 780575. The mean is 80041.24331 and that approximation comes up with 153153.875, so I guess it's a really rough estimate.

Sure, why not. If I clearly don't know what I'm talking about, feel free to attempt to fill in the gaps. :)
The last number (780575) swamps the other four.