How to Calculate the Average of a List of Numbers

What the Word Average Actually Means

In mathematics the word average has a precise meaning: it is the arithmetic mean, the total of all the values divided by how many there are. In conversation people use the word loosely for a typical value, which can be the median or the mode instead. When a teacher or a spreadsheet says average, they almost always mean the arithmetic mean, so that is the figure this guide is about.

The Formula in One Line

Mean equals the sum of the values divided by the count of values. Written out: Mean = Sum / Count. There is no hidden weighting and no trick to it. The two things that go wrong in practice are taking the sum incorrectly and counting the values incorrectly.

A Worked Example With Five Numbers

Take the numbers 12, 15, 18, 21 and 24. Add them: 12 plus 15 is 27, plus 18 is 45, plus 21 is 66, plus 24 is 90. Count the values: there are 5 of them. Divide: 90 divided by 5 is 18. The average is 18. Note that in this case the median is also 18, because the numbers are evenly spaced.

Why the Order of the Numbers Does Not Matter

Addition is commutative, so you can add the numbers in any order you like and reach the same total. This matters when you are adding a long list by hand: work through the numbers in an order that reduces your chances of a slip, such as largest first, or in the order they appear in a column of a spreadsheet.

Common Mistake One: Forgetting to Divide

The most frequent error is stopping at the sum. If your five numbers total 90, the average is not 90, it is 18. A sum is only ever equal to the average when you have entered exactly one number.

Common Mistake Two: Counting the Wrong Number of Values

If one cell is left blank, or you accidentally include the header row, the count is wrong and so is the average. Always state the count alongside the average. On the calculator above, the count is shown as its own tile precisely so this cannot go unnoticed.

Common Mistake Three: Averaging Values From Different Sized Groups

You cannot average a 90 percent score out of 10 questions and a 90 percent score out of 100 questions as if they were equal, because the first is worth far less. The correct approach is to add the points earned and divide by the points available, which is a weighted average rather than a plain one.

Common Mistake Four: Rounding at Every Step

Rounding the sum before dividing introduces an error that carries through to the answer. Add everything at full precision, divide once, and round only the final result. The calculator rounds at four decimal places and only at the end for exactly this reason.