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Sum of Squares Calculator

To find algebric & statistical sum of squares, input the comma-separated values and click calculate button using sum of squares calculator

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Sum of squares calculator is used to find the algebraic and statistical sum of squares of given terms, numbers, sequences, or observations. This calculator provides the step-by-step solutions of the algebraic and statistical sum of squares.

What is the sum of squares?

In algebra, the sum of squares is used to find the sum of two or more squared terms. For example, use sum of squares method, if two squared terms are given with an addition sign among them such as a2 + b2.

In statistics, the sum of squares is the variation of the given step of data from its mean. It subtracts the mean from all the observations and find the square of differences to find the statistical sum of squares.

Formulas of algebraic and statistical sum of squares

Formula of the algebraic sum of squares is:

Algebraic sum of squares = a12 + a22 + a32 + a42 + … + an2

This formula can be written in summation form.

Algebraic sum of squares = Σ an2

Formula of the statistical sum of squares is:

Statistical sum of squares = Σ (x - x̄)

How to calculate the algebraic and statistical sum of squares?

Follow the below examples to learn how to calculate the algebraic and statistical sum of squares.

Example

Find the algebraic and statistical sum of squares if the given observations are:
2, 3, 6, 7, 8, 10, 13

Solution

Step 1: First of all, find the algebraic sum of squares by taking the square of each term.

Algebraic sum of squares = (2)2 + (3)2 + (6)2 + (7)2 + (8)2 + (10)2 + (13)2
                                              = 4 + 9 + 36 + 49 + 64 + 100 + 169
                                              = 431

Step 2: Now find the mean of given observations to find the statistical sum of squares.

Mean = x̄ = sum of observations / total numbers

Mean = x̄ = (2 + 3 + 6 + 7 + 8 + 10 + 13) / 7

Mean = x̄ = 49/7 = 7

Step 3: Find the differences and square of differences of each term from the mean.

(x – x̄) (x – x̄)2
2 – 7 = -5 (-5)2 = 25
3 – 7 = -4 (-4)2 = 16
6 – 7 = -1 (-1)2 = 1
7 – 7 = 0 (0)2 = 0
8 – 7 = 1 (1)2 = 1
10 – 7 = 3 (3)2 = 9
13 – 7 = 6 (6)2 = 36

Step 4: Now add the square of differences to get the statistical sum of squares.

Statistical sum of squares = 25 + 16 + 1 + 0 + 1 + 9 + 36

Statistical sum of squares = Σ (x - x̄) = 88

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