Sum of Squares Formula
Sum of squares represents various things in various fields of Mathematics, in Statistics it represents the dispersion of the data set, which tells us how the data in a given set varies to the mean of the data set. The sum of the square formula in various fields of Mathematics is,
In Statistics: Sum of Squares (of n values) = ∑ni=0 (xi – x̄)2 where x̄ is the mean of n-values.
In Algebra: Sum of Squares = a2 + b2 = (a + b)2 – 2ab
Sum of Squares of n Natural Numbers: 12 + 22 + 32 + … + n2 = [n(n+1)(2n+1)] / 6
We can easily find the sum of squares for two numbers, three numbers, and n numbers. Also, we can find the sum of squares of n natural numbers, etc.
Sum of Squares for Two Numbers
Let a and b be two real numbers, then the sum of squares for two numbers formula is,
a2 + b2 = (a + b)2 − 2ab
This formula can be obtained using the algebraic identity of (a+b)2
We know that,
(a + b)2 = a2 + 2ab + b2
Subtracting 2ab on both sides
(a + b)2 − 2ab = a2 + 2ab + b2 − 2ab
⇒a2 + b2 = (a + b)2 − 2ab
Thus, the required formula is obtained.
Sum of Squares for Three Numbers
Let a, b, and c be three real numbers, then the sum of squares for three numbers formula is,
a2 + b2 + c2 = (a +b + c)2 – 2ab – 2bc – 2ca
This formula can be obtained using the algebraic identity of (a+b+c)2
We know that,
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
Subtracting 2ab, 2bc, and 2ca on both sides,
a2 + b2 + c2 = (a + b + c)2 − 2ab − 2bc − 2ca
Thus, the required formula is obtained.
Sum of Squares
Sum of squares in the addition of the square of the numbers i.e. we find the sum of squares by first finding the individual squares and then adding them to find the sum of the squares. We define the sum of squares in statistics as the variation of the data set. In algebra, we can find the sum of squares for two terms, three terms, or “n” number of terms, etc. We can find the sum of squares of two numbers using the algebraic identity,
- (a + b)2 = a2 + b2 + 2ab
We can also find the sum of squares of more than two terms using the concept of Algebra and Mathematical Induction. In this article, we will learn about the different sum of squares formulas, their examples, proofs, and others in detail.
Table of Content
- What is Sum of Squares?
- Sum of Squares Formula
- Sum of Squares for “n” Natural Numbers
- Sum of Squares in Statistics
- Steps to Find Sum of Squares
- Sum of Squares Error
- Sum of Square Table