Sum of Squares Formula

Introduction to Sum of Squares Formula

Maths is all about numbers. Numbers are put in two categories, one is an odd number and the other one is an even number. The whole numbers that cannot be divided into pairs are known as odd numbers. When divided by 2 the odd numbers give a reminder ‘1’.

The sum of two odd numbers gives an even number. The product of two or more than two odd numbers gives an odd number. The product of an even number and the product of an odd number is always an even number. In the number line, the first odd number is 1.

The integer which is not an odd number is an even number. When an odd number is divided by two the result always comes out as a fraction.


Sum of Squares of n Natural Numbers Formula

  • Sum of Squares

The Sum of squares is the sum of the squares of numbers. Generally, it is the addition of the squared numbers. The squared terms can be of two terms, three terms, or even of ‘n’ terms. The first n even terms or the odd terms are the set of the natural numbers or the consecutive numbers, etc. This is the basic math used to perform the arithmetic operation of the addition of the squared numbers.

In arithmetic operations, we often come across the sum of ‘n’ numbers. There are many formulas as well as techniques for the calculation of the sums of squares. In the statistics. It is always equal to the sum of the squares of the variation between the individual values and also the mean that is Σ(xi + x̄)2.

  • Sum of Squares Formula

The squares formula is always used to calculate the sum of two or more than two squares in an expression. To describe how well a model can represent the data being modeled the sum of squares formula is always used. The sum of the squares is the measure of the deviation from the mean value of the data. Therefore it is calculated as the total summation of the squares minus of the mean.

The sum of the squares can be calculated using the formulas that are by the algebra and by the mean. The formula to calculate the sum of the squares of the two values are:

What is the sum of squares formula in statistics, algebra, and in 'n' terms?

  • In Statistics: Sum of Squares = Σ(xi + x̄)2

  • In Algebra: Sum of Squares of Two Values = a2 + b2 = (a + b)2 − 2ab

  • For “n” Terms: Sum of Squares Formula for “n” numbers = 12 + 22 + 32 ……. n2 = [n(n + 1)(2n + 1)]/6

Where,

  • ∑ = sum

  • xi = each value in the set 

  • x̄ = mean

  • xi – x̄ = deviation

  • (xi – x̄)2 = square of the deviation

  • a, b = numbers

  • n = number of terms


Sum of Squares Formulas

In Statistics

Sum of Squares:

= Σ(xi + x̄)2

In Algebra

Sum of Squares of Two Values:

= a2 + b2 = (a + b)2 − 2ab

For “n” Terms

Sum of Squares Formula for “n” numbers

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


  • ∑ = sum

  • xi = each value in the set

  • x̄ = mean

  • xi – x̄ = deviation

  • (xi – x̄)2 = square of the deviation

  • a, b = numbers

  • n = number of terms

FAQs (Frequently Asked Questions)

Q1. What is the Difference Between the Sum of Squares of First n Even Numbers and Odd Numbers?

Ans. The addition of squares of the first even natural numbers:

Σ(2n)2 = 22 + 42 + 62 + 82 + …+ (2n)2

While the addition of squares of the first odd natural numbers

Σ(2n-1)2 = 12 + 32 + 52 + … + (2n – 1)2

Q2. Why is it Important to Learn Formulas and Equations?

Ans. Computer chips used in all the machines we use in our daily routine like washers, dryers, backs, etc. All the chips that we use in these machines based on mathematical equations, formulas, and algorithms.

Mathematical equations and formulas are also used in traffic control, aircraft, space program and medicine, etc.

Q3. When was Formula Invented in Mathematics?

Ans: The first math “formula” was invented by the Babylonians and the derivation was of the square root of 2. About 1800 B.C. An Egyptian papyrus contains the first ever quadratic solution and about 1650 B.C. Another papyrus contains solving linear equations and also the first example of cotangent.