Sum of Squares Formula

Sum of Squares Formula

The Sum of squares formula is used for computing the addition of two or more square expressions. In mathematics, the Sum of two or more numbers is computed using two different methods. One is the algebra method, and the second is the mean method. 

About the Topic 

The Sum of square formulas is the statistical study used for computing the dispersion of a data sheet. For computing the square through the Sum of squares formula, one takes the Sum of squares of every data or unit given in the expression and adds them. 

The Sum of squares is calculated by two methods, while one more method is used for computing the Sum of squares of natural numbers, but it is not considered a method because it is a GP. In the statistics method, one has to count the number of units in the given data and find the mean of the data. After that, every unit gets subtracted from the mean value, and the square of difference is computed. After that, add the fair value of all the acquired units. While in the algebraic method, one has to put the values in the formula and then Compute the answer by following basic calculation.

Formula

In statistics: Sum of squares of n numbers or units = ∑ni=0 (xi – x̄)²

In algebra expression: Sum of squares of two algebraic expressions = a²+ b² = (a + b)² – 2ab

Sum of squares formula for n natural numbers: 1² + 2² + 3² + … + n² = [n(n+1)(2n+1)] / 6

∑ = demonstrating the sum

xi = It is describing every value in the given set

x̄ = mean value 

xi – x̄ = difference or deviation occurs after subtracting the mean value from each unit 

(xi – x̄)² = square of difference occurs after subtracting the mean value from each unit

a, b = algebraic numbers

n = number of terms in the given series 

Solved Examples

1.Find the value of 3² + 2². 

For finding the value of  3² + 2², we use the algebraic formula of the sum of squares. 

a² + b² = (a + b)² – 2ab

a = 3

b = 2

3² + 2² = (3 + 2)² – 2×3×2

          =  5² – 12

        = 25 – 12

         = 13

2.Find the sum of squares is 10 natural numbers. 

Using the formula of natural numbers, 

1² + 2² + 3² + … + n² = [n(n+1)(2n+1)] / 6

n = 10

So, putting the value of n in the formula, 

= [n(n+1)(2n+1)] / 6

= [10(10+1)(2×10+1]/6

= [10×11×21]/6

=2310/6

=385