Mean, Median, Mode and Standard Deviation: A Plain Guide

A long list of numbers tells you very little. Two kinds of summary help: where the middle is, and how spread out the numbers are.

The three averages

• Mean: add the numbers and divide by how many there are.
• Median: the middle number when they are in order. With an even count, it is the average of the two in the middle.
• Mode: the number that appears most often. There can be more than one mode, or none.

When the median is better

One huge value drags the mean toward it. If five people earn 30, 32, 35, 38 and 40 thousand, the mean is 35. If a sixth earns 400, the mean jumps to 96 but the median only moves to 36.5. For things like house prices and incomes, the median describes a typical case better. The Mean, Median & Mode Calculator shows all three at once, with the range and quartiles.

Standard deviation

Standard deviation measures how far numbers typically sit from the mean. A small value means they cluster tightly, and a large one means they are spread out. To work it out, find the mean, take each number minus the mean, square it, add the squares, divide, and take the square root.

Sample or population

If your numbers are every value there is, divide by the count n, which is the population standard deviation. If they are a sample from something bigger, divide by n minus 1, which gives the sample figure and corrects for the sample missing some of the spread. For the data 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5, the population standard deviation is exactly 2, and the sample figure is about 2.14. The Standard Deviation Calculator shows both, with the working.

More maths

For anything beyond these, such as powers, roots and logarithms, the Scientific Calculator has you covered.