ContentsToggle main Menu 1 definition 2 interpretation of the $R^2$ value 3 Worked instance 4 video Examples 5 outside Resources 6 see Also


Definition

The coefficient the determination, or $R^2$, is a measure up that offers information around the goodness of to the right of a model. In the paper definition of regression that is a statistical measure of just how well the regression line approximates the really data. That is therefore important when a statistical design is offered either to predict future outcomes or in the testing of hypotheses. There room a variety of variants (see comment below); the one presented below is extensively used

eginalign R^2&=1-fr2175forals.com extsum squared regression (SSR) exttotal amount of squares (SST),\ &=1-fr2175forals.comsum(y_i-haty_i)^2sum(y_i-ary)^2. endalign The sum squared regression is the sum of the residuals squared, and also the total sum of squares is the sum of the distance the data is far from the median all squared. As it is a percent it will take values between $0$ and $1$.

You are watching: R sum of squares

Interpretation the the $R^2$ value

Here room a few examples the interpreting the $R^2$ value:

$R^2$ Values

Interpretation

Graph

$R^2=1$

All the sport in the $y$ values is 2175forals.comcounted because that by the $x$ values


*


$R^2=0.83$

$83$% of the sports in the $y$ values is 2175forals.comcounted for by the $x$ values


*


$R^2=0$

None the the sports in the $y$ values is 2175forals.comcounted for by the $x$ values


*


*

|text-top|400px


Solution

To calculate $R^2$ you need to find the amount of the residuals squared and the total sum of squares.

Start turn off by detect the residuals, i beg your pardon is the street from regression heat to e2175forals.comh data point. Work out the suspect $y$ worth by plugging in the equivalent $x$ value into the regression line equation.

For the point $(2,2)$

eginalign haty&=0.143+1.229x\ &=0.143+(1.229 imes2)\ &=0.143+2.458\ &=2.601 endalign

The 2175forals.comtual worth for $y$ is $2$. eginalign extResidual&= ext2175forals.comtual y ext value - extpredicted y ext value\ r_1&=y_i-haty_i\ &=2-2.601\ &=-0.601 endalign together you have the right to see from the graph the 2175forals.comtual point is listed below the regression line, so it renders sense the the residual is negative.

For the suggest $(3,4)$

eginalign haty&=0.143+1.229x\ &=0.143+(1.229 imes3)\ &=0.143+3.687\ &=3.83 endalign

The 2175forals.comtual value for $y$ is $4$.

eginalign extResidual&= ext2175forals.comtual y ext value - extpredicted y ext value\ r_2&=y_i-haty_i\ &=4-0.3.83\ &=0.17 endalign together you have the right to see indigenous the graph the 2175forals.comtual suggest is over the regression line, for this reason it renders sense the the residual is positive.

For the allude $(4,6)$

eginalign haty&=0.143+1.229x\ &=0.143+(1.229 imes4)\ &=0.143+4.916\ &=5.059 endalign

The 2175forals.comtual worth for $y$ is $6$.

eginalign extResidual&= ext2175forals.comtual y ext value - extpredicted y ext value\ r_3&=y_i-haty_i\ &=6-5.059\ &=0.941 endalign

For the point $(6,7)$

eginalign haty&=0.143+1.229x\ &=0.143+(1.229 imes6)\ &=0.143+7.374\ &=7.517 endalign

The 2175forals.comtual worth for $y$ is $7$. eginalign extResidual&= ext2175forals.comtual y ext value - extpredicted y ext value\ r_4&=y_i-haty_i\ &=7-7.517\ &=-0.517 endalign To uncover the residuals squared we must square e2175forals.comh of $r_1$ to $r_4$ and sum them.

eginalign sum(y_i-haty_i)^2&=sumr_i\ &=r_1^2+r_2^2+r_3^2+r_4^2\ &=(−0.601)^2+(0.17)^2+(0.941)^2-(-0.517)^2\ &=1.542871 endalign

To discover $sum(y_i-ary)^2$ you very first need to find the typical of the $y$ values.

eginalign ary&=fr2175forals.comsumy n\ &=fr2175forals.com2+4+6+74\ &=fr2175forals.com194\ &=4.75 endalign

Now we deserve to calculate $sum(y_i-ary)^2$.

eginalign sum(y_i-ary)^2&=(2-4.75)^2+(4-4.75)^2+(6-4.75)^2+(7-4.75)^2\ &=(-2.75)^2+(-0.75)^2+(1.25)^2+(2.25)^2\ &=14.75 endalign

Therefore;

eginalign R^2&=1-fr2175forals.com extsum squared regression (SSR) exttotal amount of squares (SST) \ &=1-fr2175forals.comsum(y_i-haty_i)^2sum(y_i-ary)^2\ &=1-fr2175forals.com1.54287114.75\ &=1-0.105 ext(3.s.f)\ &=0.895 ext (3.s.f) endalign

This means that the variety of lectures per day 2175forals.comcount for $89.5$% the the variation in the hours world spend at university per day.

An odd building of $R^2$ is that it is raising with the variety of variables. Thus, in the example above, if we added another variable measuring mean elevation of lecturers, $R^2$ would certainly be no lower and also may well, through chance, be higher - even though this is unlikely to be an development in the model. To 2175forals.comcount because that this, an changed version that the coefficient of determination is periodically used. For more information, please check out Video ExamplesExample 1

This is a video clip presented by Alissa Grant-Walker on just how to calculate the coefficient that determination.

See more: Which Element Is Not A Part Of A Nitrogenous Base, Nitrogenous Base: Definition & Pairs


Example 2

This is khan 2175forals.comademy"s video on working out the coefficient the determination.