The sum of the squares is a method used to find the sum of the square of the numbers. It is widely used in mathematics and statistics. A sum of the square is used to find the squared summation of the given numbers.
In statistics, the sum of the squares is used to find the variance, standard deviation, and other topics. While in mathematics, the sum of squares is used to calculate the summation of the given numbers.
In this post, we will learn how to calculate the sum of the squares algebraically and statistically with a lot of examples.
What is the sum of the squares?
The sum of the squares is a way to calculate various terms of mathematics and statistics. In mathematics, the sum of squares is used frequently in arithmetic. While in statistics, the sum of squares is used to find the summation square of the differences from the mean.
Some different techniques and formulas are used for the calculation of the sum of squares. We can find the sum of the squares of different terms such as:
- A2+ B2 + C2
- (A1)2+(A2)2 + (A3)2 + … + (An)2
- Σ (x – x¯)2Σ (x – x¯)2
Some other terms from algebra can also be used to find the sum of squares. For the calculation of the sum of squares, either algebraically or statistically, you can use the sum of squares calculator.
How to calculate the sum of squares?
For the calculations of the sum of squares algebraically or statistically, let’s take some examples.
Example 1
Find the sum of squares algebraically and statistically for the given data, 2, 9, 19, 23, 35, 41, and 44?
Solution
Step 1: First of all, take the given set of data.
2, 9, 19, 23, 35, 41, 44
Step 2: Now take squares of each number and put a sum sign among them to find the sum of the square algebraically.
Algebraic sum of squares = (2)2 + (9)2 + (19)2 + (23)2 + (35)2 + (41)2 + (44)2
Algebraic sum of squares = 4 + 81 + 361 + 529 + 1225 + 1681 + 1936
Algebraic sum of squares = 5817
Step 3: Now find the mean of the given data.
Sum of the data = 2 + 9 + 19 + 23 + 35 + 41 + 44
Sum of the data = 173
Total numbers = 7
Statistical mean = Sum of the data / Total numbers
Statistical mean = 173/7
Statistical mean = 24.71
Step 4: Now find the statistical sum of squares.
| Statistical data (x) | (x – x¯) | (x – x¯)2 |
| 2 | 2 – 24.71 = -22.71 | (-22.71)2 = 515.7 |
| 9 | 9 – 24.71 = -15.71 | (-15.71)2 = 1246.8 |
| 19 | 19 – 24.71 = -5.71 | (-5.71)2 = 32.60 |
| 23 | 23 – 24.71 = -1.71 | (-1.71)2 = 2.924 |
| 35 | 35 – 24.71 = 10.29 | (10.29)2 = 105.9 |
| 41 | 41 – 24.71 = 16.29 | (16.29)2 = 265.4 |
| 44 | 44 – 24.71= 19.29 | (19.29)2 = 372.1 |
| Σ x = 173 | Σ (x – x¯)2 = 1541 |
Hence, the statistical sum of squares is 1541.
Example 2
Find the sum of squares algebraically and statistically for the given data, 12, 17, 19, 25, 42, 51, 59, and 64?
Solution
Step 1: First of all, take the given set of data.
12, 17, 19, 25, 42, 51, 59, 64
Step 2: Now take squares of each number and put a sum sign among them to find the sum of the square algebraically.
Algebraic sum of squares = (12)2 + (17)2 + (19)2 + (25)2 + (42)2 + (51)2 + (59)2 + (64)2
Algebraic sum of squares = 144 + 289 + 361 + 625 + 1764 + 2601 + 3481 + 4096
The algebraic sum of squares = 13361
Step 3: Now find the mean of the given data.
Sum of the data = 12 + 17 + 19 + 25 + 42 + 51 + 59 + 64
Sum of the data = 289
Total numbers = 8
Statistical mean = Sum of the data / Total numbers
Statistical mean = 289/8
Statistical mean = 36.13
Step 4: Now find the statistical sum of squares.
| Statistical data (x) | (x – x¯) | (x – x¯)2 |
| 12 | 12 – 36.13 = -24.13 | (-24.13)2 = 582.3 |
| 17 | 17 – 36.13 = -19.13 | (-19.13)2 = 366 |
| 19 | 19 – 36.13 = -17.13 | (-17.13)2 = 293.4 |
| 25 | 25 – 36.13 = -11.13 | (-11.13)2 = 123.9 |
| 42 | 42 – 36.13 = 5.87 | (5.87)2 = 34.46 |
| 51 | 51 – 36.13 = 14.87 | (14.87)2 = 221.1 |
| 59 | 59 – 36.13 = 22.87 | (22.87)2 = 523 |
| 64 | 64 – 36.13 = 27.87 | (27.87)2 = 776.7 |
| Σ x = 289 | Σ (x – x¯)2 = 2921 |
Hence, the statistical sum of squares is 2921.
Example 3
Find the sum of squares algebraically and statistically for the given data, 1.2, 2.7, 3.9, 4.5, 5.2, 6.1, 6.9, and 7.4?
Solution
Step 1: First of all, take the given set of data.
1.2, 2.7, 3.9, 4.5, 5.2, 6.1, 6.9, 7.4
Step 2: Now take squares of each number and put a sum sign among them to find the sum of the square algebraically.
Algebraic sum of squares = (1.2)2 + (2.7)2 + (3.9)2 + (4.5)2 + (5.2)2 + (6.1)2 + (6.9)2 + (7.4)2
Algebraic sum of squares = 1.44 + 7.29 + 15.21 + 20.25 + 27.04 + 37.21 + 47.61 + 54.76
The algebraic sum of squares = 210.81
Step 3: Now find the mean of the given data.
Sum of the data = 1.2 + 2.7 + 3.9 + 4.5 + 5.2 + 6.1 + 6.9 + 7.4
Sum of the data = 37.9
Total numbers = 8
Statistical mean = Sum of the data / Total numbers
Statistical mean = 37.9/8
Statistical mean = 4.737
Step 4: Now find the statistical sum of squares.
| Statistical data (x) | (x – x¯) | (x – x¯)2 |
| 1.2 | 1.2 – 4.737 = -3.537 | (-3.537)2 = 12.51 |
| 2.7 | 2.7 – 4.737 = -2.037 | (-2.037)2 = 4.149 |
| 3.9 | 3.9 – 4.737 = -0.837 | (-0.837)2 = 0.7006 |
| 4.5 | 4.5 – 4.737 = -0.237 | (-0.237)2 = 0.05617 |
| 5.2 | 5.2 – 4.737 = 0.463 | (0.463)2 = 0.2144 |
| 6.1 | 6.1 – 4.737 = 1.363 | (1.363)2 = 1.858 |
| 6.9 | 6.9 – 4.737 = 2.163 | (2.163)2 = 4.679 |
| 7.4 | 7.4 – 4.737 = 2.663 | (2.663)2 = 7.092 |
| Σ x = 289 | Σ (x – x¯)2 = 31.26 |
Hence, the statistical sum of squares is 31.26.
Summary
The sum of squares is very beneficial for solving the problems related to math and stat. By following the above examples, you can find the sum of squares easily. The sum of squares is used in formulas for the calculation of different terms.
