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## Sum of Squares

The **sum of squares** means the sum of the squares of the given numbers. In statistics, it is the sum of the squares of the variation of a dataset. For this, we need to find the mean of the data and find the variation of each data point from the mean, square them and add them. In algebra, the sum of the square of two numbers is determined using the (a + b)^{2} identity. We can also find the sum of squares of the first n natural numbers using a formula. The formula can be derived using the principle of mathematical induction. We do these basic arithmetic operations which are required in statistics and algebra. There are different techniques to find the sum of squares of given numbers.

In this article, we will discuss the different sum of squares formulas. To calculate the sum of two or more squares in an expression, the sum of squares formula is used. Also, the sum of squares formula is used to describe how well the data being modeled is represented by a model. Let us learn these along with a few solved examples in the upcoming sections for a better understanding.

## What is the Sum of Squares?

The sum of squares in statistics is a tool that is used to evaluate the dispersion of a dataset. To evaluate this, we take the sum of the square of the variation of each data point. In algebra, we find the sum of squares of two numbers using the algebraic identity of (a + b)^{2}. Also, in mathematics, we find the sum of squares of n natural numbers using a specific formula which is derived using the principle of mathematical induction. Let us now discuss the formulas of finding the sum of squares in different areas of mathematics.

## Sum of Squares Formula

The sum of squares formula in statistics is used to describe how well the data being modeled is represented by a model. It shows the dispersion of the dataset. To calculate the sum of two or more squares in an expression, the sum of squares formula is used. Thus, a few sums of squares formulas are,

- In statistics : Sum of squares of n data points
^{n}_{i=0}(x_{i}– x̄)^{2} - In algebra : Sum of squares = a
^{2}+ b^{2}= (a + b)^{2}– 2ab - Sum of squares of n natural numbers formula: 1
^{2}+ 2^{2}+ 3^{2}+ … + n^{2}= [n(n+1)(2n+1)] / 6

Where,

- ∑ = represents sum
- x
_{i}= each value in the set - x̄ = mean of the values
- xi – x̄ = deviation from the mean value
- (xi – x̄)
^{2}= square of the deviation - a, b = arbitrary numbers
- n = number of terms in the series

Let a and b be the two numbers. Assuming the squares of a and b are a^{2} and b^{2}. The sum of the squares of a and b is a^{2 }+ b^{2}. We could obtain a formula using the known algebraic identity (a+b)^{2 }= a^{2} + b^{2} + 2ab. Subtracting 2ab from both the sides we can conclude that a^{2} + b^{2} = (a + b)^{2} – 2ab. Let a, b, c be the 3 numbers for which we are supposed to find the sum of squares. The sum of their squares is a^{2 }+ b^{2 }+ c^{2}. Using the known algebraic identity (a+b+c)^{2 }= a^{2 }+ b^{2 }+ c^{2 }+ 2ab + 2bc +2ca, we can evaluate that a^{2 }+ b^{2 }+ c^{2} = (a+b+c)^{2} – 2ab – 2bc – 2ca.

Sum of squares can be calculated using two formulas i.e. by algebra and by the mean. The formula to calculate the sum of the squares of two values are given below,

Sum of Squares Formulas | |
---|---|

In Statistics | Sum of Squares:= Σ(x_{i} + x̄)^{2} |

In Algebra | Sum of Squares of Two Values:= a^{2} + b^{2} = (a + b)^{2} − 2ab |

For “n” Terms | Sum of Squares Formula for “n” numbers= 1^{2} + 2^{2} + 3^{2} ……. n^{2} = [n(n + 1)(2n + 1)] / 6 |

- ∑ = sum
- x
_{i}= each value in the set - x̄ = mean
- x
_{i}– x̄ = deviation - (x
_{i}– x̄)^{2}= square of the deviation - a, b = numbers
- n = number of terms

## Steps to Find Sum of Squares

The total sum of squares can be calculated in statistics using the following steps:

- Step 1: In the dataset, count the number of data points.
- Step 2: Calculate the mean of the data.
- Step 3: Subtract each data point from the mean.
- Step 4: Determine the square of the difference determined in step 3.
- Step 5: Add the squares determined in step 4.

## Sum of Squares in Statistics

The steps discussed above help us in finding the sum of squares in statistics. The value tells us the amount of dispersion in a dataset. It measures the variation of the data points from the mean and helps in studying the data in a better way. If the value of the sum of squares is large, then it implies that there is a high variation of the data points from the mean value. On the other hand, if the value is small, then it implies that there is a low variation of the data from its mean.

## Sum of Squares Error

In statistics, the sum of squares error (SSE) is the difference between the observed value and the predicted value. It is also called the sum of squares residual (SSR) as it is the sum of the squares of the residual, that is, the deviation of predicted values from the actual values. The formula for the sum of squares error is given by,

SSE = ∑^{n}_{i=0} (y_{i} – f(x_{i}))^{2}, where y_{i} is the i^{th} value of the variable to be predicted, f(x_{i}) is the predicted value, and x_{i} is the i^{th} value of the explanatory variable.

We can also evaluate the sum of squares error (SSE) by subtracting the sum of squares regression (SSR) from the sum of squares total (SST), that is, SSE = SST – SSR

**Important Notes on Sum of Squares**

- The sum of squares in statistics is a tool that is used to evaluate the dispersion of a dataset.
- SSE = SST – SSR
- Sum of squares of n data points
^{n}_{i=0}(x_{i}– x̄)^{2}

## Sum of Squares Examples

**Example 1:** Using the sum of squares formula, find the value of 4^{2} + 6^{2}?

**Solution:** To find : value of 4^{2} + 6^{2}

Given: a = 4, b = 6

Using sum of squares formula a^{2} + b^{2} = (a + b)^{2} − 2ab, we have

4^{2} + 6^{2} = (4 + 6)^{2} − 2(4)(6)

= 100 − 2(24)

= 100 − 48

= 52

**Answer: **The value of 4^{2} + 6^{2} is 52.

**Example 2 :** Calculate the sum of the following series 1^{2} + 2^{2} + 3^{2} ……. 100^{2}

**Solution:**

To Find: Sum of the series

Using sum of squares formula for n terms, 1^{2} + 2^{2} + 3^{2} + … + n^{2} = [n(n+1)(2n+1)] / 6

Given: n =100

= [100(100+1)(2×100+1)] / 6

= (100 × 101 × 201) / 6

= 338350

**Answer: **The sum of the given series is 338350.

**Example 3:** A dataset has points 4, 5, 6, 12, 9, 10. Find the sum of squares for the given data.

**Solution:** We have 6 data points and their sum is 4 + 5 + 6 + 12 + 9 + 10 = 46. The mean of the given data is given by,

Mean, x̄ = Sum / Number of data points

= 46 / 6

= 7.67

So, the sum of squares is given by,

∑^{n}_{i=0} (x_{i} – x̄)^{2} = (4 – 7.67)^{2} + (5 – 7.67)^{2} + (6 – 7.67)^{2} + (12 – 7.67)^{2} + (9 – 7.67)^{2} + (10 – 7.67)^{2}

= (-3.67)^{2} + (-2.67)^{2 }+ (-1.67)^{2 }+ (4.33)^{2 }+ (1.33)^{2 }+ (2.33)^{2}

= 13.4689 + 7.1289 + 2.7889 + 18.7489 + 1.7689 + 5.4289

= 49.3334

**Answer:** The sum of squares for the given data is 49.3334

**Question:** What is the value of 4^{2} + 10^{2}?

**Solution:**

The formula for sum of squares is: a^{2} + b^{2} = (a + b)^{2} − 2ab

From the given equation: a=4 and b=10

4^{2 }+ 10^{2} = (4 + 10)^{2} − 2(4)(10)

= 196 − 2(40)

= 196 − 80

= 116

## FAQs on Sum of Squares

### What is the Sum of Squares?

The sum of squares in statistics is a tool that is used to evaluate the dispersion of a dataset. To evaluate this, we take the sum of the square of the variation of each data point.

### What is the Sum of Squares Formula?

We use the sum of squares in different areas of mathematics. A few sums of squares formulas are,

- In statistics : Sum of squares of n data points
^{n}_{i=0}(x_{i}– x̄)^{2} - In algebra : Sum of squares = a
^{2}+ b^{2}= (a + b)^{2}– 2ab - Sum of squares of n natural numbers formula = 1
^{2}+ 2^{2}+ 3^{2}+ … + n^{2}= [n(n+1)(2n+1)] / 6

### What is Total Sum of Squares?

The total sum of squares (TSS), also called just the sum of squares (SS), is the sum of squares of the difference of each data point from the mean. Its formula is given by, ∑^{n}_{i=0} (x_{i} – x̄)^{2}

### What is Sum of Squares Error?

The Sum of squares error, also known as the residual sum of squares, is the difference between the actual value and the predicted value of the data.

### What Is the Expansion of Sum of Squares Formula?

a^{2} + b^{2} formula is known as the sum of squares formula in algebra and it is read as a square plus b square. Its expansion is expressed as a^{2} + b^{2} = (a + b)^{2} – 2ab.

### What Is the Sum of Squares Formula in Algebra?

The sum of squares formula is one of the important algebraic identities. It is represented by a^{2} + b^{2} and is read as a square plus b square. The sum of squares (a^{2 }+ b^{2}) formula is expressed as a^{2} + b^{2} = (a + b)^{2} – 2ab

### How To Calculate Sum of Squares?

The total sum of squares can be calculated in statistics using the following steps:

- Step 1: In the dataset, count the number of data points.
- Step 2: Calculate the mean of the data.
- Step 3: Subtract each data point from the mean.
- Step 4: Determine the square of the difference determined in step 3.
- Step 5: Add the squares determined in step 4.

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