Sum of Squares

About Sum of Squares

The sum of squares refers to the total of the supplied integers' squares. It is the sum of the squares of a dataset's variation in statistics. To do so, we must find the data's mean and the variation of each data point from the mean, then square and add them. The (a + b)2 identity is used in algebra to find the sum of the squares of two numbers. A formula can also be used to get the sum of squares of the first n natural numbers. The formula can be derived using the mathematical induction technique. These fundamental arithmetic procedures are used in statistics and algebra. There are several methods for calculating the sum of squares of given numbers.

The different sum of squares formulas will be discussed in this article. The sum of squares formula is used to calculate the sum of two or more squares in an equation. The sum of squares formula is sometimes used to describe how effectively a model represents the data being modelled. For a better understanding, let us learn these together with a few solved cases in the following parts. Check out the List of Maths Formulas. 

In statistics, the sum of squares is a method for determining a dataset's dispersion. To calculate this, we add the sum of the squares of each data point's fluctuation. The algebraic identity of (a + b)2 is used to compute the sum of squares of two numbers. In addition, we may compute the sum of squares of n natural numbers using a special formula constructed utilising the mathematical induction concept. Let's look at some formulas for obtaining the sum of squares in various fields of mathematics.

Sum of Squares Formula

In statistics, the sum of squares formula is used to indicate how effectively a model represents the data being modelled. It depicts the dataset's dispersion. The Sum of squares formula is used to calculate the sum of two or more squares in an equation. As a result, here are a few sums of squares formulas:

  • In statistics :∑ni=0(xi - x?)2
  • In algebra : a2 + b2 = (a + b)2 - 2ab
  • Sun of n natural numbers formula: 12 + 22 + 32 + ... + n2 = [n(n+1)(2n+1)] / 6

Where,

  • ∑ - represents the sum
  • xi - each value in the set
  • x? - mean of the values
  • xi – x? - deviation from the mean value
  • (xi – x?)2 - square of the deviation
  • a, b - arbitrary numbers
  • n - number of terms in series

(a+b+c)2 = a2 + b2 + c2 + 2ab + 2bc +2ca

(a+b)2 = a2 + b2 + 2ab

  • In Statistics→Sum of Squares = ∑(xi – x?)2
  • In Algebra→Sum of Squares = a2+b2= (a + b)2- 2ab
  • For "n" Terms→Sum of Squares = 12+22+32.....n2=[n(n + 1)(2n + 1)]/6
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Sum of Squares in Statistics

In statistics, the techniques outlined above assist us in determining the sum of squares. The value indicates how much dispersion there is in a dataset. It calculates the deviation of data points from the mean and aids in the analysis of the data. If the sum of squares is large, it means that the data points are far separated from the mean value. If the value is modest, on the other hand, it indicates that the data has little variance from its mean.

Sum of Squares Error

The sum of squares error (SSE) is the difference between the observed and projected values in statistics. It's also known as the sum of squares residual (SSR) because it's the sum of the squares of the residual or the difference between expected and actual values. The sum of squares error formula is as follows:

SSE = ∑ni=0(yi - f(xi))2, where yi is the ith value of the variable to be predicted, f(xi) is predicted value, & xi is the ith value of the explanatory variable.

The sum of squares error (SSE) can also be calculated by subtracting the sum of squares regression (SSR) from the sum of squares total (SST), i.e. SSE = SST - SSR.

Important Notes on the Sum of Squares

  • SSE = SST - SSR
  • Sum of the squares of n data points = ∑ni=0(xi - x?)2

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Sum of Squares
Sum of Squares

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