Sum of Squares Formula: Meaning & Solved Examples

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Sum of Squares Formula is used to calculate the sum of two or more squares of numbers. Basically, the sum of squares is the addition of the squared numbers. The sum of the squares can be calculated with the help of two formulas namely by algebra and by mean.

Sum of squares formula is used to describe how well a model represents the data being modelled. The sum of squares is the measure of deviation from the mean value of the data. Thus, it is calculated as the difference between the total summation of the squares and the mean.

Table of Contents

Sum of Squares
Sum of Squares Formula
Solved Examples
Things to Remember
Previous Years’ Questions
Sample Questions

Key Terms: Sum of Squares, Statistics, Mean, Algebra, Deviation, Variance, Sum of Squares Formula, Natural Numbers

Sum of Squares

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The sum of squares refers to the sum of the squares of the given numbers. Different techniques can be used to find the sum of squares of given numbers.

In statistics, the sum of squares is the sum of the squares of the variation of a dataset. In this, the mean of the data and the variation of each data point from the mean is calculated, squared and added eventually.
In algebra, the sum of the square of two numbers is calculated using the (a + b)² identity.
The sum of squares of the first n natural numbers is also calculated using a formula derived using the principle of mathematical induction.

Sum of Squares Formula in Algebra

Sum of Squares Formula in Algebra

Related Topics
Difference Between Variance and Standard Deviation	Mean Absolute Deviation	Statistics Formula
Arithmetic Mean Formula	Algebra Formula	Algebraic Expressions and Identities

Sum of Squares Formula

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The sum of squares is used to evaluate the overall variance of a data set from its mean value. A large value of variance is denoted by a large sum of squares which means that individual data differs a lot from its mean value.

The formulas to calculate the sum of the squares of two given values are as follows:

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

For the given sum of squares formulas,

∑ = 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 Formula for n Natural Numbers

Sum of Squares of n Natural Numbers Formula

Read More: Sequence and Series

Solved Examples

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Example 1: A given dataset has points 2, 4, 13, 10, 12, and 7. What will be the sum of squares for the given data?

Solution: Given data points are 2, 4, 13, 10, 12, and 7.

Mean of the given data, x̄ = Sum of Data Points / Number of Data Points

Mean = 48 / 6 = 8

Using the sum of squares formula,

Σ(xi + x̄)² = (2 – 8)² + (4 – 8)² + (13 – 8)² + (10 – 8)² + (12 – 8)² + (7 – 8)²

= (–6)² + (–4)² + (5)² + (2)² + (4)² + (–1)²

= 36 + 16 + 25 + 4 + 14 + 1

= 96

Thus, the sum of squares for the given data points is 96.

Example 2: Find the value of 4² + 6² using the sum of squares formula.

Solution: Given that a = 4 and b = 6

Using sum of squares formula,

a² + b² = (a + b)² − 2ab

Substituting the values of a and b in the formula,

4² + 6² = (4 + 6)² − 2(4)(6)

= 100 − 2(24)

= 100 − 48

= 52

Hence, the value of 4² + 6² is 52.

Example 3: What will be the sum of the following series 1² + 2² + 3² ……. 100²?

Solution: Here, we will use the sum of squares formula for n terms,

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

n = 100

Sum = [100(100+1)(2×100+1)] / 6

= (100 × 101 × 201) / 6

= 338350

Thus, the sum of the given series is 338350.

Chapter-Related Topics
Arithmetic Sequence	Infinite Series Formula	Arithmetic Sequence Explicit Formula
Series Formula	Difference between Sequence and Series	Arithmetic Sequence Recursive Formula

Things to Remember

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The sum of squares is the sum of two or more squares in a given expression.
The sum of squares gives us a measure of deviation from the mean value of the data.
There are different formulas to calculate the sum of squares of given numbers.
In algebra, the sum of squares is calculated using the identity a² + b² = (a + b)² − 2ab.
In Statistics, the sum of the square of two numbers is determined using the formula Σ(xi + x̄)².
The sum of squares of the first n natural numbers is calculated through the formula [n(n + 1)(2n + 1)] / 6.

Previous Years’ Questions (PYQs)

The sum of squares of deviations for 10 observations taken from mean 50…
If S_mdenotes the sum of first m terms of a G.P. of n terms
The 100th term of the sequence 1, 2, 2, 3, 3, 3,....
A sequence containing a finite number of terms is called a…
For a positive integer n let a(n)... (JEE Advanced 1999)
If the 2nd, 5th and 9th terms of a non-constant… (JEE Main 2016)
If 4^x= 16^y= 64^z, then… (JKCET 2011)
Sum of n terms of the series 1 + 11…
Sum to 10 terms of the series 1+2…
The 10th common term between the series

Sample Questions

Ques. A given dataset has the following numbers 4, 6, 5, 12, 9, and 10. What will be the sum of squares for the given data? (3 Marks)

Ans. The sum of the given data points is 4 + 6 + 5 + 12 + 9 + 10 = 46.

Mean of the given data, x̄ = Sum of Data Points / Number of Data Points

Mean = 46 / 6 = 7.67

Using the sum of squares formula,

Σ(xi + x̄)² = (4 - 7.67)² + (6 - 7.67)² + (5 - 7.67)² + (12 - 7.67)² + (9 - 7.67)² + (10 - 7.67)²

= (-3.67)² + (-1.67)²+ (-2.67)² + (4.33)² + (1.33)² + (2.33)²

= 13.4689 + 2.7889 + 7.1289 + 18.7489 + 1.7689 + 5.4289

= 49.3334

Thus, the sum of squares for the given data is 49.3334.

Ques. What will be the sum of the squares of 10 and 22 if calculated directly? Verify the answer obtained using the sum of squares formula. (3 Marks)

Ans. Given numbers are 10 and 22.

On direct calculation, 10² + 22² = 100 + 484 = 584

Using the sum of square formula a² + b²= (a + b)² - 2ab, we get

10² + 22² = (10 + 22)² - 2 × 10 × 22

= 322 - 440

= 1024 - 440 = 584

Hence Verified.

Ques. Find out three consecutive natural numbers if the sum of their squares is 77. (3 Marks)

Ans. Consider n to be the number. For three consecutive numbers, we need to find n, n+1 and n+2.

According to the question, n² + (n+1)² +(n+2)² = 77

On expansion, n² + n²+ 2n + 1+ n² + 4n + 4 = 77

3n² + 6n + 5 = 77

3n² + 6n = 72

Dividing by equation by 3 we get,

n² + 2n - 24 = 0

On solving the quadratic equation through middle-term splitting, we get

(n+6)(n-4) = 0

As n cannot be negative, we take n = 4.

Putting the value of n in n, n+1 and n+2, we get 4, 5 and 6.

Thus, the required numbers are 4, 5 and 6.

Ques. Calculate the sum of the squares of 19 and 22 directly and using the sum of squares formula. Verify both answers. (3 Marks)

Ans. Given numbers are 19 and 22.

On direct calculation, 19² + 22² = 361 + 484 = 845

Using the sum of squares formula,

a² + b²= (a + b)² - 2ab

192 + 222 = (19 + 22)2 - 2 × 19 × 22

= 412 - 836 = 1681- 836

= 845

Both the answers are same, hence verified.

Ques. Calculate what will be the sum of squares of the first 20 natural numbers. (3 Marks)

Ans. The formula for the sum of squares of first n natural numbers is

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

Here, n is 20.

[n(n+1)(2n+1)] / 6 = [20(20+1)(40+1)] / 6

= [20(21)(41)] / 6

= 2870

Thus, the sum of squares of first 20 natural numbers is 2870.

Ques. What will be the sum of the squares of the first 30 odd numbers? (3 Marks)

Ans. The formula for the sum of the squares of n odd numbers is

[n(2n+1) (2n-1)]/3

Here, n is given as 30. Substituting the value of n, we get,

30(61)(59)/ 3 = 35990

Thus, the sum of the squares of the first 30 odd numbers is 35990.

Ques. Calculate the value of (3² + 8²) using the sum of squares formula. (3 Marks)

Ans. Given that a = 3 and b = 8.

Using the sum of squares formula, we get

a² + b²= (a + b)² - 2ab

3² + 8² = (3 + 8)² − 2(3)(8)

= 121 – 2(24)

= 121 − 48 = 73

Thus, the value of (3² + 8²) is 73.

Ques. What will be the sum of the given series 1² + 2² + 3² +…+ 55²? (3 Marks)

Ans. Using the sum of squares formula for n terms, we get

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

Here, n = 55

= [55(55+1)(2×55+1)] / 6 = (55 × 56 × 111) / 6 = 56,980‬

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

Ques. What is sum of squares? (3 Marks)

Ans. The sum of squares is defined as the sum of the squares of a given set of numbers. In the case of statistics, it is the sum of the squares of the variation of a given dataset. In algebra, the sum of squares is calculated using the (a + b)2 identity. We can also calculate the sum of squares of the first n natural numbers using the formula derived using the principle of mathematical induction.

Ques. Find out the value of the sum of squares of the first 40 natural numbers. (3 Marks)

Ans. Using the sum of squares formula for natural numbers,

Σn² = [n(n+1)(2n+1)]/6

n = 40

Σ40² = (40/6) (40 + 1)(2 x 40 + 1) = (20/3) (41)(81) = (20)(41)(27) = 22140

Thus, the sum of squares of the first 40 natural numbers is 22140.

Ques. What will be the sum of 4² + 5²? Use the sum of squares formula to calculate the same. (3 Marks)

Ans. Using the sum of squares formula,

a² + b² = (a+b)² - 2ab

4² + 5² = (4 + 5)² – 2 x 4 x 5 = 9² – 40 = 81 – 40 = 41

Thus, the sum of 4² + 5² is 41.

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Sum of Squares Formula: Meaning & Solved Examples

Sum of Squares

Sum of Squares Formula

Solved Examples

Things to Remember

Previous Years’ Questions (PYQs)

Sample Questions

CBSE CLASS XII Related Questions

1.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

2.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

3.
Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

4.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

6.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

Similar Mathematics Concepts

Comments

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Sum of Squares Formula: Meaning & Solved Examples

Sum of Squares

Sum of Squares Formula

Solved Examples

Things to Remember

Previous Years’ Questions (PYQs)

Sample Questions

CBSE CLASS XII Related Questions

1. Find: If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

2. Find: If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

3.Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

4.Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

6. Find: The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

2.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

3.
Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

4.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

6.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]