How to Use the Sum of Squares Calculator
This calculator helps you find the statistical sum of squares, Σ(xᵢ − x̄)² or algebraic sum of squares, Σxi², for any set of data. To use the calculator:
- Select the type of sum of squares you want to calculate
- Enter the values in the data input field. You can separate the numbers using commas, spaces, tabs, or line breaks. Alternatively, you can paste values directly from text documents, Excel, or even Google Sheets
- Click calculate
The calculator will return the correct type of sum of squares and show you exactly how to find the value. This makes this tool useful for students learning variance and standard deviation.
What Is Sum of Squares?
Sum of squares is a statistical method used to calculate how much a dataset’s individual values vary from the overall mean. However, in algebra, it involves squaring each individual value and getting the sum of these squared values.
Sometimes, the statistical sum of squares is also called the sum of squared deviations from the mean.
Sum of Squares Formula
The statistical sum of squares formula is: SS = Σ(xᵢ − x̄)²
Where:
- SS is the sum of squares
- xi is each individual data point
- x̄ is the mean of the data
- xᵢ − x̄ represents the deviation of each value from the mean
However, for the algebraic sum of squares (raw sum of squares), the formula is: SS = Σxi²
Where:
- SS is the sum of squares
- xi is each individual data point
- xi² denotes the square of each individual observation
How to Find the Statistical Sum of Squares
To find the statistical sum of squares manually:
- Find the mean of the data
- Subtract the mean from each observation and square these deviations
- Sum all the squared deviations
Example 1
An AP Statistics teacher records the number of questions answered correctly by five students on a short SAT Math practice quiz. The scores are: 80, 87, 75, 76, 90, 68, 79, 81.
Find the statistical sum of squares for the scores.
Solution
To find the statistical sum of squares for the dataset, follow these steps:
Step 1: Find the mean of the data
By definition, the mean formula is: x̄ = Σx/n
Substituting the values into the formula, we get:
x̄ = (80+87+75+76+90+68+79+81)/8
= 636/8
= 79.5
Want to learn more about calculating the sample or population mean? Use the mean calculator.
Step 2: Subtract the mean from each observation and square these deviations
The table below shows the actual steps from subtracting the mean from each observation and squaring the deviations
| Score, xi | Deviation (xi−x̄) | Squared Deviation (xi – x̄)2 |
|---|---|---|
| 80 | (80-79.5) = 0.5 | (0.5)2 = 0.25 |
| 87 | (87-79.5) = 7.5 | (7.5)2 = 56.25 |
| 75 | (75-79.5) = -4.5 | (-4.5)2 = 20.25 |
| 76 | (76-79.5) = -3.5 | (-3.5)2 = 12.25 |
| 90 | (90-79.5) = 10.5 | (10.5)2 = 110.25 |
| 68 | (68-79.5) = -11.5 | (-11.5)2 = 132.25 |
| 79 | (79-79.5) = -0.5 | (-0.5)2 = 0.25 |
| 81 | (81-79.5) = 1.5 | (1.5)2 = 2.25 |
Step 3: Sum all the squared deviations
By definition, the statistical sum of squares formula is SS= Σ(xᵢ − x̄)². In this case, we add all the values in column (xᵢ - x̄)2
Thus, Σ(xᵢ − x̄)² = 0.25 + 56.25 + 20.25 + 12.25 + 110.25 + 132.25 + 0.25 + 2.25
= 334
Therefore, the statistical sum of squares is: SS = 334.
You can verify these results using the sum of squares calculator as follows:
- Select the statistical sum of squares
- Copy and paste the values into the data input field
- Click calculate
The calculator returns the sum of squares as: Σ(xᵢ − x̄)² = 334
How to Find the Algebraic Sum of Squares
To find the algebraic sum of squared values:
- Square each value in the dataset
- Sum all the squared values
Example 2
An AP Statistics teacher records the scores of six students on a quiz. The scores are: 72, 84, 91, 88, 95, 79
Find the algebraic sum of squares for the scores.
Solution
To find the algebraic sum of squares for the dataset, follow these steps:
Step 1: Square each value in the dataset
The table below shows the data point and the square of each of the values
| Score, x | Squared Value, x2 |
|---|---|
| 72 | (72)2 = 5,184 |
| 84 | (84)2 = 7,056 |
| 91 | (91)2 = 8,281 |
| 88 | (88)2 = 7,744 |
| 95 | (95)2 = 9,025 |
| 79 | (79)2 = 6,241 |
Step 2: Sum all the squared values
By definition, the algebraic sum of squares formula is: SS = Σxi²
Adding all the squared values gives Σxi²
That is: Σxi² = 5184+7056+8281+7744+9025+6241
= 43531
Therefore, the algebraic sum of squares is: SS = 43531
Alternatively, you can quickly verify these results using the sum of squares calculator and following these steps:
- Select the Algebraic sum of squares option
- Copy and paste the values into the data input field
- Click calculate
The calculator instantly returns Σx² = 43531 (similar to the manual results)
