Learn here on sum of squares formula. The sum of squares is the total mean value of the squares. The total sum of squares (TSS) is defined as the combination of the explained sum of the squares(ESS) and the residual sum of the squares(RSS). The total sum of squares is the sum, the over all observations of differences of the each observation from the over all mean. Based on the statistical linear models the total sum of squares formula is defined as the sum of squares of the difference of dependent values or variables and its mean value.

 

 

The Total Sum of Squares Formula :

The total sum of squares formula for the 1 to n natural numbers is,

1^2+2^2+3^2+.....+n^2=(n*(n+1)*(2n+1))/6

The total sum of cubes formula for the 1 to n natural numbers is,

`1^3+2^3+3^3+4^3+......+n^3=(n^2 * (n+1)^2)/4`

The following formula is used to find the total sum of squares.

`sum_(i=1)^n (y_i -bary)^2.`

Here `y_i=a+bx_i.`

i is the observed index value from 1 to n and a,b are the coefficient of the estimated values.

Another formula is ,

TSS=ESS+RSS

        `= sum_(i=1)^n (haty_i -hat(bary))^2+ sum_(i=1)^n (y_i -f(x_i))^2.`

Here the value of ESS is `hat(y_i)=hata+hatb x_i` and a,b terms are the the coefficients .

Here in the RSS is `y_i=a +bx_i+e_i and f(x_i)=a+bx_i` and the term a,b are the coefficients, x and y term are the regresand and regressor, e is the error values.

So the total sum of squares formula is rewritten as,

`TSS=sum_(i)^n(( hata+hatb x_i)-( hat(bary))^2+ sum_(i=1)^n((a +bx_i+e_i)-(a+bx_i))^2`

Sum of Squares Formula - Examples

Find the total sum of squares of the following numbers 2, 3, 4,5,6,7.

Solution:

Given, 2, 3,4,5,6,7.

The squares of the following number are

`2^2,3^2,4^2,5^2,6^2,7^2 = 4,9,16,25,36,49.`

The total sum of squares =`4+9+16+25+36+49=136.`

The result of the total sum of squares is 136.

Example 2 for total sum of squares formula:

Find the total sum of squares of the following numbers 1,2,3,4,5,6,7,8,9,10.

Solution:

Method 1:

The formula is ,

The total sum of squares formula for the 1 to n natural numbers is,
`1^2+2^2+3^2+.....+n^2=(n*(n+1)*(2n+1))/6`

The squares of the following number are

`1^2,2^2,3^2,4^2,5^2,6^2,7^2,9^2,10^2.`

Total sum of squares formula =` (n*(n+1)*(2n+1))/6`

Here n=10, therefore

Total sum of squares formula=`(10*(10+1)*(2(10)+1))/6=385.`

 

 

Method 2:

Total sum of squares formula =`1^2+2^2+3^2+4^2+5^2+6^2+7^2+9^2+10^2`

                                                `=1+4+9+16+25+36+49+64+81+100=385.`

The result of the total sum of squares is `385.`