Sum of Squares Calculator

How to use the Sum of Squares Calculator

1. 1Enter n, or your own list of numbers.
2. 2Pick the units or options from the menus if the sum of squares tool offers them.
3. 3Read the sum of squares in the large result box.
4. 4Look at the extra rows for related figures.
5. 5Press Copy result to keep the answer.

When you need the sum of squares

Students meet the sum of squares in algebra, series and induction proofs. Statisticians use it as the first step of variance, regression and analysis of variance, where the sum of squared deviations measures how spread out data is.

The closed formula saves you from adding thousands of terms by hand, and big values of n stay exact here because the arithmetic uses whole-number precision. For a list, the sum of squares about the mean is always smaller than or equal to the sum of squares about any other point.

How the sum of squares is worked out

The sum of the first n squares is n(n + 1)(2n + 1) ÷ 6. The sum of the first n cubes is (n(n + 1) ÷ 2)². The sum of the first n odd numbers is n². For n = 10 the squares add to 10 × 11 × 21 ÷ 6 = 385, the cubes to 55² = 3,025, and the first ten odd numbers to 100. For the list 1, 2, 3 the sum of squares is 14 and the mean is 2.

Every answer on this page comes from that one rule, so you can check any result by hand with a calculator. The figures shown underneath, such as other units, are conversions of the same answer.

• Both formulas and your own list
• Gives cubes and odd numbers too
• Exact for very large n

Mistakes to avoid with sum of squares

Do not mix up Σx² (square each number, then add) with (Σx)² (add, then square). They give different answers. Also check that n is a whole number: the formulas only hold for whole values.

On phone and computer

On a phone, type into the boxes one after another and read the sum of squares underneath; Copy result puts it on the clipboard for a message or a note. On a computer the inputs sit on the left and the answer on the right, so you can change one figure and watch the sum of squares move. The sum of squares calculator works the same on both.

Checking your sum of squares by hand

A good habit is to test the sum of squares calculator with a case you already know. For n = 10 the squares add to 10 × 11 × 21 ÷ 6 = 385, the cubes to 55² = 3,025, and the first ten odd numbers to 100. For the list 1, 2, 3 the sum of squares is 14 and the mean is 2.

If your own figures give a surprising answer, repeat the working yourself: The sum of the first n squares is n(n + 1)(2n + 1) ÷ 6. The sum of the first n cubes is (n(n + 1) ÷ 2)². The sum of the first n odd numbers is n². Small differences usually come from rounding, from mixing units, or from a typing slip in one of the inputs. A sum of squares leads to spread. Use the Standard Deviation Calculator for the usual measure, the Variance Calculator for the squared version, and the Factorial Calculator for large products.

Features

• Σx² is not (Σx)²
• Sum of squares about the mean measures spread
• The closed formula needs whole n
• Clear messages when the sum of squares inputs make no sense
• Copy the result with one tap
• Private: nothing is uploaded

Frequently asked questions