Sum Of Squares Calculator

What the sum of squares calculator does

The sum of squares is the total amount your data varies around its own average. It adds up how far each value sits from the mean, after squaring those distances. This calculator works it out from your numbers and shows the count and the mean it used.

It is a quiet workhorse of statistics. On its own it is a measure of total variation, and it is also the raw material that variance, standard deviation, and analysis of variance are all built from. Below is how it works and where it leads.

How to use it

1. Enter your numbers in the box, separated by commas, spaces, or new lines.
2. Press Calculate for the sum of squares, count, and mean, or Reset to clear it.

How the sum of squares is worked out

Three steps. Find the mean of your numbers. For each number, measure how far it is from the mean and square that distance. Then add all those squared distances together.

Sum of squares = the sum of (each value minus the mean), squared

Worth being clear on one point, because the name is used loosely elsewhere: this is the sum of the squared distances from the mean, not the sum of the numbers squared. The distances are squared so that values above and below the mean both add to the total rather than cancelling out to zero.

Why the sum of squares matters

The sum of squares is the first step of almost every measure of spread. Divide it by the count and you get the population variance. Divide it by one less than the count and you get the sample variance. Take the square root of either and you have a standard deviation. So the number this calculator gives you is the quantity sitting inside all of those.

It also stands at the heart of analysis of variance, or ANOVA, the method for comparing groups. There, the total sum of squares is split into the part explained by the differences between groups and the part left over within them, and that split is what the whole test rests on. Whenever you see variation being measured or divided up, the sum of squares is usually underneath it.

A worked example: eight numbers

Take the eight numbers 2, 4, 4, 4, 5, 5, 7, 9. Their mean is 40 ÷ 8 = 5. The distances from 5 are minus 3, minus 1, minus 1, minus 1, 0, 0, 2, and 4, and squaring each gives 9, 1, 1, 1, 0, 0, 4, and 16.

Adding those up gives a sum of squares of 32. From here the rest follows: 32 divided by 8 is the population variance of 4, and 32 divided by 7 is the sample variance of about 4.57. The single figure of 32 is the seed both of those grow from.

Entering your data, and the rounding

You can separate your numbers with commas, spaces, or new lines, in any mix, and the order does not matter, since the sum of squares depends only on each value's distance from the mean. The sum of squares and the mean are shown to two decimal places with trailing zeros trimmed.

Questions people ask

What is the sum of squares?

The total of the squared distances between each value and the mean of the data. It measures how much the data varies around its average, all in one number.

Is this the sum of squared numbers?

No. This is the sum of the squared distances from the mean, the version used in statistics. It is not the sum of each number squared, which is a different quantity.

How does it relate to variance?

Variance is the sum of squares divided by the count for a population, or by one less than the count for a sample. So the sum of squares is the numerator of variance.

Where is the sum of squares used?

In variance and standard deviation, and at the centre of analysis of variance, where the total sum of squares is split into the variation between groups and the variation within them.

References

A quick note on where the methods here come from. The sum of squares, as the sum of squared deviations from the mean and the foundation of variance and analysis of variance, is set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook covering variance and its building blocks.

1. NIST/SEMATECH e-Handbook of Statistical Methods (measures of spread and analysis of variance). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (measures of the spread of the data). https://openstax.org/details/books/introductory-statistics-2e