Sum of Squares Formula
To calculate the sum of two or more squares in an expression, the sum of squares formula is used. Also, the sum of squares formula is used to describe how well the data being modeled is represented by a model. Let us learn these along with a few solved examples in the upcoming sections.
What Is the Sum of Squares Formula?
The sum of squares formula is used to describe how well the data being modeled is represented by a model. To calculate the sum of two or more squares in an expression, the sum of squares formula is used. Thus, a few sum of squares formulas are,
 In statistics –
Sum of Squares = \(\sum (x_i + \bar{x})^2\)
 In Algebra 
Sum of Squares = \(a^2 + b^2 = (a + b)^2 − 2ab\)
 For “n” Terms.
Sum of Squares = \(1^2 + 2^2 + 3^2 ……. n^2 = \dfrac{[n(n + 1)(2n + 1)]}{6}\)
• ∑ = represents sum
• xi = each value in the set
• x̄ = mean of the values
• xi – x̄ = deviation from the mean value
• (xi – x̄)^{2} = square of deviation
• a, b = arbitrary numbers
• n = number of terms in the series
Let us have a look at a few solved examples on the sum of squares formula to understand the concept better.
Solved Examples Using Sum of Squares Formula

Example 1: Using sum of squares formula, find the value of 4^{2} + 6^{2}?
Solution:
To find : value of 4^{2} + 6^{2}
Given: a = 4, b = 6
Using Sum of Squares Formula,
a^{2} + b^{2} = (a + b)^{2} − 2ab
4^{2} + 6^{2} = (4 + 6)^{2} − 2(4)(6)
= 100 − 2(24)
= 100 − 48
= 52Answer: The value of 4^{2} + 6^{2 }is 52.

Example 2 : Calculate the sum of the following series.
1^{2} + 2^{2} + 3^{2} ……. 100^{2}Solution:
To Find: Sum of the series
Using Sum of Squares Formula for n terms,
Sum of Squares = \(1^2 + 2^2 + 3^2 ……. n^2 = \dfrac{[n(n + 1)(2n + 1)]}{6}\)
Given: n =100
= \(\dfrac{[100(100 + 1)(2 \times 100 + 1)]}{6}\)
= \(\dfrac{[100(101)(201)]}{ 6}\)
= 338350.Answer: The sum of the given series is 338350.