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.

What is Sum of Squares?

Sum of squares is the method in statistics that helps evaluate the dispersion of the given data set. The sum of squares is found by taking individual squares of the terms and then adding them to find their sum. In algebra, algebraic identity (a+b)² = a² + b² + 2ab gives the sum of squares of two numbers. The general formula to calculate the sum of natural numbers is:

Sum of Square Formula

Now let’s discuss all the formulas used to find the sum of squares in algebra and statistics.

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) = ∑ⁿ_i=0 (x_i – 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 for “n” Natural Numbers

Natural numbers are also known as positive integers and include all the counting numbers, starting from 1 to infinity. If 1, 2, 3, 4,… n are n consecutive natural numbers, then the sum of squares of “n” consecutive natural numbers is represented by 1² + 2² + 3² +… + n² and symbolically represented as Σn².

Sum of Square of n Natural Numbers = Σn² = 1² + 2² + 3² +… + n²

The required sum of squares for ‘n’ natural number formula is,

[Tex]\bold{\sum{n^2}= \frac{n(n+1)(2n+1)}{6}} [/Tex]

This formula is proved using Mathematical Induction Method.

Sum of Squares of First “n” Even Numbers

The formula for the sum of squares of the first “n” even numbers, i.e., 2² + 4² + 6² +… + (2n)² is given as follows:

∑(2n)² = 2² + 4² + 6² +… + (2n)²

[Tex]\bold{\sum{(2n)^2}= \frac{2n(n+1)(2n+1)}{3}} [/Tex]

This formula can be obtained using,

∑(2n)² = ∑4n² = 4∑n²

As [Tex]\sum{n^2}= \frac{n(n+1)(2n+1)}{6} [/Tex]

Thus, ∑(2n)² = 2[n(n+1)(2n+1)]/3

Which is the required formula.

Sum of Squares of First “n” Odd Numbers

Formula for the sum of squares of the first “n” odd numbers, i.e., 1² + 3² + 5² +… + (2n – 1)², can be derived using the formulas for the sum of the squares of the first “2n” natural numbers and the sum of squares of the first “n” even numbers.

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

[Tex]\bold{\sum{(2n-1)^2}= \frac{n(2n+1)(2n-1)}{3}} [/Tex]

This formula can be obtained using,

∑(2n –1)² = [1² + 2² + 3² + … + (2n – 1)² + (2n)²] – [2² + 4² + 6² + … + (2n)²]

Now, applying the formula for sum of squares of “2n” natural numbers and “n” even natural numbers,

∑(2n–1)² = 2n/6 (2n + 1)(4n + 1) – (2n/3) (n+1)(2n+1)

⇒ ∑(2n–1)² = n/3 [(2n+1)(4n+1)] – 2n/3 [(n+1)(2n+1)]

⇒ ∑(2n–1)² = n/3 (2n+1) [4n + 1 – 2n – 2]

⇒ ∑(2n–1)² = [n(2n+1)(2n–1)]/3

Thus, the required formula is verified.

Sum of Squares in Statistics

In statistics, the value of the sum of squares tells the degree of dispersion in a dataset. It evaluates the variance of the data points from the mean and helps for a better understanding of the data. The large value of the sum of squares indicates that there is a high variation of the data points from the mean value, while the small value indicates that there is a low variation of the data from its mean. 

The formula used to calculate the sum of squares in Statistics is,

Sum of Squares of n Data points = ∑ⁿ_i=0 (x_i – x̄)²

where,

• ∑ represents Sum of Series
• x_i represents each value in Set
• x̄ represents Mean of Values
• (x_i – x̄) represents Deviation from Mean Value
• n represents Number of Terms in Series

Steps to Find Sum of Squares

Follow the steps given below to find the Total Sum of Squares in Statistics.

Step 1: Count the number of data points in the given dataset. (say n)

Step 2: Find the mean of the given data set.

Step 3: Find the definition of the data set from the mean value.

Step 4: Find the square of deviation of individual terms.

Step 5: Find the sum of all the square values.

Sum of Squares Error

Sum of Square Error (SSE) is the difference between the observed value and the predicted value of the deviation of the data set. SSE is also called the SSR or sum of square residual. The formula to calculate the sum of square error is,

SSE = ∑ⁿ_i=0 (y_i – f(x_i))²

where, 

• y_i is the i^th Value of Variable to be Predicted
• f(x_i) is the Predicted Value
• x_i is the i^th value of the explanatory variable

Sum of Square Error can also be calculated using the formula,

SSE = SST – SSR

Where,

• SST is Sum of Squares Total
• SSR is Sum of Squares Regression

Sum of Square Table

Sum of the square table is added below,

Sum of Square Table

Read More,

• Arithmetic Progression
• Geometric Progression

Examples on Sum of Square Formulas

Example 1: Find the sum of the given series: 1² + 2² + 3² +…+ 55².

Solution:

To find the value of 1² + 2² + 3² +…+ 55².

Sum of Squares Formula for n terms

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

Given, n = 55

Sum of Squares = [55(55+1)(2×55+1)] / 6

⇒ Sum of Squares = (55 × 56 × 111) / 6

⇒ Sum of Squares = 56,980‬

Thus, the sum of the given series 1² + 2² + 3² +…+ 55² is 56,980‬.

Example 2: Find the value of (3² + 8²), using the sum of squares formula.

Solution:

Find 3² + 8² using sum of square formula,

Given, 

• a = 3
• b = 8

Using sum of square formula,

a² + b² = (a + b)² − 2ab

⇒ 3² + 8² = (3 + 8)² − 2(3)(8)

⇒ 3² + 8² = 121 – 2(24)

⇒ 3² + 8² = 121 − 48

⇒ 3² + 8² = 73.

Thus, the value of (3² + 8²) is 73.

Example 3: Find the sum of squares of the first 25 even natural numbers.

Solution:

Sum of Squares of first 25 Even Natural Numbers(S) = 2² + 4² + 6² +… + 48²+ 50²……(1)

Now simplifying eq(1)

S = 2²( 1² + 2² + 3² +…+25²)

Using Sum Squares Formula for n terms, we have

∑n² = [n(n+1)(2n+1)]/6

Here, n = 25

S= 2²( 1² + 2² + 3² +…+25²) = 4[25(25+1)(2(25)+1)/6]

⇒ S = (2/3) × (25) × (26) × (51)

⇒ S = 22100

Hence, the sum of squares of the first 25 even natural numbers is 22100.

Example 4: A dataset has points 2, 4, 13, 10, 12, and 7. Find the sum of squares for the given data.

Solution: 

Given,

We have 6 data points 2, 4, 13, 10, 12, and 7.

Sum of given data points = 2 + 4 + 13 + 10 + 12 + 7 = 48. 

Mean of the given data,

Mean, x̄ = (Sum of data value) / (Number of data value)

⇒ x̄ = 48 / 6

⇒ x̄ = 8

Now,

∑ⁿ_i=0 (x_i – x̄)² = (2 – 8)² + (4 – 8)² + (13 – 8)² + (10 – 8)² + (12 – 8)² + (7 – 8)²

⇒ ∑ⁿ_i=0 (x_i – x̄)² = (–6)² + (–4)² + (5)² + (2)² + (4)² + (–1)²

⇒ ∑ⁿ_i=0 (x_i – x̄)² = 36 + 16 + 25 + 4 + 14 + 1

⇒ ∑ⁿ_i=0 (x_i – x̄)² = 96

Hence, the sum of squares for the given data is 96.

Example 5: Find the sum of the squares of 4, 9, and 11 using the sum of squares formula for three numbers.

Solution: 

Given,

• a = 4
• b = 9
• c = 11

Using Sum of Squares Formula,

a² + b² + c² = (a + b +c)² − 2ab − 2bc − 2ca

⇒ 4² + 9² + 11² = (4 + 9 + 11)² −(2×4×9) − (2×9×11) − (2×11×4)

⇒ 4² + 9² + 11² = 576 − 72 − 198 − 88

⇒ 4² + 9² + 11² = 218

Hence, the value of (4² + 9² + 11²) is 218.

Example 6: Find the sum of squares of the first 10 odd numbers.

Solution:

Sum of Squares of the first 10 odd numbers (S): 1² + 3² + 5² +… +17² + 19²

Sum of squares of first “n” Odd Numbers ∑(2n–1)² = [n(2n+1)(2n–1)]/3

Here, n is 10.

S = [10×(2×10 + 1)(2×10 – 1)]/3

⇒ S = [10 × 21 × 19]/3

⇒ S = 10 × 7 × 19 = 1330

Hence, the value of the sum of squares of the first 10 odd numbers is 1330.

FAQs on Sum of Squares

What is Sum of Squares Definition?

In statistics sum of squares is a tool that evaluates the dispersion of a dataset. We can easily calculate the sum of squares by first individually finding the square of the terms and then adding them to find their sum.

What is Sum of Squares Error?

Sum of Square Error (SSE) is the difference between the actual value and the predicted value of the data set. It is also called the Residual Sum of the Square.

What is Total Sum of Squares?

The sum of squares of the difference of each data point from the mean value of the data point is called the Total Sum of Squares or simply the Sum of squares. It is calculated using the formula,

TSS = ∑ⁿ_i=0 (x_i – x̄)²

What is the Expansion of Sum of Squares Formula?

The formula to find the value of a² + b² is called the sum of squares formula in algebra and it is calculated as,

a² + b² = (a + b)² – 2ab

What are Sum of Squares Formula used in Algebra?

Various sum of squares formulas used in algebra are,

• a² + b² = (a + b)² − 2ab
• a² + b² + c² = (a +b + c)² − 2ab − 2bc − 2ca