Sum of Squares

The sum of squares means adding the squares of numbers.

Example, for numbers 1, 2, 3, the sum of squares would be 1² + 2² + 3³ = 14

Sum of squares represents various things in various fields of mathematics.

In Statistics (Dispersion)

Measures how much the data values deviate from the mean.

∑ⁿ_i=0 (x_i - x̄)²

• x_i: each data value
• \bar{x}: mean of the dataset
• n: number of observations

In Algebra

Used to rewrite the sum of squares in terms of the square of a sum.

• For 2 Numbers: a² + b² = (a + b)² - 2ab
• For 3 Numbers: a² + b² + c² = (a + b + c)² - 2ab - 2bc - 2ca

First n Natural Numbers

Gives the sum of squares of the first n counting numbers.

\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}

First n Even Numbers

Used to find the sum of squares of even numbers like 2, 4, 6, ..., 2n.

\sum_{i=1}^{n} (2i)^2

First n Odd Numbers

Used to calculate the sum of squares of odd numbers like 1, 3, 5, ..., (2n − 1).

1^2 + 3^2 + 5^2 + \cdots + (2n - 1)^2 = \frac{n(2n+1)(2n-1)}{3}

Sum of Squares Error (SSE)

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 residuals. The formula to calculate the sum of square error is,

SSE = ∑ⁿ_i=0 (y_i - f(x_i))²

• y_i is the i^th value 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

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

Read in Detail

• Arithmetic Progression
• Geometric Progression
• Variance and Standard Deviation

Solved Examples

Example 1: Find the sum of the given series: 1 + 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.