Sum of squares is a term used to calculate the addition of the squared sum of the given numbers. The sum of the square is frequently used in algebra & statistics. In algebra, the sum of squares takes an identity (u + v)2 two find the squared sum of the numbers.
In statistics, it is used to find the variation in the dataset with the help of the mean of the given data values. It plays a vital role in the calculations of covariance, standard deviation, variance, etc. It is also used to find the squares of n times number.
In this article, we’ll study the definition and formulas of the algebraic and statistical sum of squares along with examples.
What is the Sum of Squares Total?
The sum of squares is basically the addition of the squared terms of the sequence or data set values. It is used in mathematics as well as statistics. The working and calculation of the sum of squares in mathematics and statistics are different.
It is used to find the sequence, summation, variance, standard deviation, correlation coefficients, and many other kinds of mathematics and statistics. Let us study the algebraic and statistical sum of squares briefly.
Also Read: How to find Square root of 3
Algebraic Sum of Squares & Formula
In algebra, the sum of squares is used to find the summation and addition of squared values. It is widely used to find the identity of mathematics such as u2 + v2 which comes by opening the formula of (u + v)2 i.e., u2 + v2 = (u + v)2 + 2uv.
The formula for the algebraic sum of squares is:
- u2 + v2 + w2
- (u1)2 +(u2)2 + (u3)2 + … + (un)2
- (u + v)2 = u2 + 2uv + v2
How to Calculate the Algebraic Sum of Squares? Examples
The algebraic sum of squares can be calculated easily either by using the formula or a sum of squares calculator. Follow the below steps to find the algebraic sum of squares.
- Take the data values.
- Find the squares of each term of data values.
- Determine the addition of squared terms.
Algebraic Sum of Square Example 1
Determine the algebraic sum of squares, if the given numbers are 3, 8, 12, 15, 19, 45, 51, and 64?
Solution
Step I: First of all, write the given numbers.
3, 8, 12, 15, 19, 45, 51, 64
Step II: Take the formula of the algebraic sum of squares and find the squares of each number.
Algebraic sum of squares = (u1)2 +(u2)2 + (u3)2 + … + (un)2
Algebraic sum of squares = (3)2 +(8)2 + (12)2 + (15)2 + (19)2 + (45)2 + (51)2 + (64)2
Algebraic sum of squares = 9 + 64 + 144 + 225 + 361 + 2025 + 2601 + 4096
Step III: Now find the sum of squared terms.
Algebraic sum of squares = 9 + 64 + 144 + 225 + 361 + 2025 + 2601 + 4096
= 73 + 144 + 225 + 361 + 2025 + 2601 + 4096
= 217 + 225 + 361 + 2025 + 2601 + 4096
= 442 + 361 + 2025 + 2601 + 4096
= 803 + 2025 + 2601 + 4096
= 2828 + 2601 + 4096
= 5429 + 4096
= 9525
Algebraic Sum of Square Example 2
Determine the algebraic sum of squares, if the given numbers are 2, 4, 5, 6, 9, 11, 12, and 14?
Solution
Step I: First of all, write the given numbers.
2, 4, 5, 6, 9, 11, 12, 14
Step II: Take the formula of the algebraic sum of squares and find the squares of each number.
Algebraic sum of squares = (u1)2 +(u2)2 + (u3)2 + … + (un)2
Algebraic sum of squares = (2)2 +(4)2 + (5)2 + (6)2 + (9)2 + (11)2 + (12)2 + (14)2
Algebraic sum of squares = 4 + 16 + 25 + 36 + 81 + 121 + 144 + 196
Step III: Now find the sum of squared terms.
Algebraic sum of squares = 4 + 16 + 25 + 36 + 81 + 121 + 144 + 196
= 20 + 25 + 36 + 81 + 121 + 144 + 196
= 45 + 36 + 81 + 121 + 144 + 196
= 81 + 81 + 121 + 144 + 196
= 162 + 121 + 144 + 196
= 283 + 144 + 196
= 427 + 196
= 623
Statistical Sum of Squares
In statistics, the sum of squares is the dispersion of the data values from the mean of the given terms. It is used in various kinds of statistics to calculate them. A Sum of squares helps the students to find the variance, covariance, sample standard deviation, correlation, etc.
The formula of the statistical sum of squares is:
Statistical sum of squares = Σ (x – x̄)
How to Calculate the Statistical Sum of Squares? Examples
The statistical sum of squares can be found easily by using its formula. Follow the below steps to find it manually.
- Take the given data values.
- Find the mean of given data values.
- Subtract the mean from each data value.
- Take the sum of each subtracted value.
Statistical Sum of Square Example 1
Determine the statistical sum of squares for the given data, 12, 19, 29, 33, 45, 61, 72, and 89?
Solution
Step I: First of all, write the given data values from the problem.
12, 19, 29, 33, 45, 61, 72, 89
Step II: Find the total number of terms and the sum of given values to calculate the mean.
Total number of observations = 8
Sum of the given data values = 12 + 19 + 29 + 33 + 45 + 61 + 72 + 89
Sum of the given data values = 360
The mean of the given data values = (Sum of the given data values) / (Total number of observations) = x̄/n
The mean of the given data values 360/8 = 180/4 = 90/2
The mean of the given data values = 45
Step III: Subtract each term of the data set by its mean, find the squares of each difference term, and add them to find the statistical sum of squares.
| Given data values (x) | (x – x̄) | (x – x̄)2 |
| 12 | 12 – 45 = -33 | (-33)2 = 1089 |
| 19 | 19 – 45 = -26 | (-26)2 = 676 |
| 29 | 29 – 45 = -16 | (-16)2 = 256 |
| 33 | 33 – 45 = -12 | (-12)2 = 144 |
| 45 | 45 – 45 = 0 | (0)2 = 0 |
| 61 | 61 – 45 = 16 | (16)2 = 256 |
| 72 | 72 – 45 = 27 | (27)2 = 729 |
| 89 | 89 – 45 = 44 | (44)2 = 1936 |
| Σ x = 360 | Σ (x – x̄)2 = 4668 |
Hence,
Statistical sum of squares = Σ (x – x̄)2 =4668
Statistical Sum of Square Example 2
Determine the statistical sum of squares for the given data, 2, 6, 10, 16, 20, 24, 30, and 36?
Solution
Step I: First of all, write the given data values from the problem.
2, 6, 10, 16, 20, 24, 30, 36
Step II: Find the total number of terms and the sum of given values to calculate the mean.
Total number of observations = 8
Sum of the given data values = 2 + 6 + 10 + 16 + 20 + 24 + 30 + 36
Sum of the given data values = 144
The mean of the given data values = (Sum of the given data values) / (Total number of observations) = x̄/n
The mean of the given data values 144/8 = 72/4 = 36/2
The mean of the given data values = 18
Step III: Subtract each term of the data set by its mean, find the squares of each difference term, and add them to find the statistical sum of squares.
| Given data values (x) | (x – x̄) | (x – x̄)2 |
| 2 | 2 – 18 = -16 | (-16)2 = 256 |
| 6 | 6 – 18 = = -12 | (-12)2 = 144 |
| 10 | 10 – 18 = -8 | (-8)2 = 64 |
| 16 | 16 – 18 = -2 | (-2)2 = 4 |
| 20 | 20 – 18 = 2 | (2)2 = 4 |
| 24 | 24 – 18 = 6 | (6)2 = 36 |
| 30 | 30 – 18 = 12 | (12)2 = 144 |
| 36 | 36 – 18 = 18 | (18)2 = 324 |
| Σ x = 144 | Σ (x – x̄)2 = 976 |
Hence,
Statistical sum of squares = Σ (x – x̄)2 =976
Summary
Now after reading the above post, you can easily find the algebraic and statistical sum of squares. You can grab all the basics of the sum of squares from this post. Now you are witnessed that the sum of squares is not a difficult topic just a little effort is required.
You can refer to our most interesting articles
You can check to our a few science articles,
