Sum of squares in statistics can be defined as the variation of the data set. Sum of squares can be applied in regression analysis, optimization problems, or error measurement for quantifying how individual elements in a set deviate from the central tendency. 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,
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 FormulaNow let's discuss all the formulas used to find the sum of squares in algebra and statistics.
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 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,
\bold{\sum{n^2}= \frac{n(n+1)(2n+1)}{6}}
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
\bold{\sum{(2n)^2}= \frac{2n(n+1)(2n+1)}{3}}
This formula can be obtained using,
∑(2n)2 = ∑4n2 = 4∑n2
As \sum{n^2}= \frac{n(n+1)(2n+1)}{6}
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
\bold{\sum{(2n-1)^2}= \frac{n(2n+1)(2n-1)}{3}}
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 = ∑ni=0 (xi - x̄)2
where,
- ∑ represents Sum of Series
- xi represents each value in Set
- x̄ represents Mean of Values
- (xi – 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 = ∑ni=0 (yi - f(xi))2
where,
- yi is the ith Value of Variable to be Predicted
- f(xi) is the Predicted Value
- xi is the ith 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 TableRead More,
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,
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,
∑ni=0 (xi – x̄)2 = (2 – 8)2 + (4 – 8)2 + (13 – 8)2 + (10 – 8)2 + (12 – 8)2 + (7 – 8)2
⇒ ∑ni=0 (xi – x̄)2 = (–6)2 + (–4)2 + (5)2 + (2)2 + (4)2 + (–1)2
⇒ ∑ni=0 (xi – x̄)2 = 36 + 16 + 25 + 4 + 14 + 1
⇒ ∑ni=0 (xi – 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,
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.
Conclusion
Sum of squares is a fundamental concept in mathematics, statistics, and data analysis, representing the sum of the squared differences between data points and a reference value, typically the mean. It is a critical measure used to assess the variability or dispersion within a data set, forming the basis for many statistical methods, including variance and standard deviation.
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