Wize University Statistics Textbook > Scatterplots, Association, and Correlation

Solving for Correlation

Coefficient of DeterminationNext Section

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Solving for Correlation

r=r=r= correlation, where −1≤r≤1-1\le r\le1−1≤r≤1

r=1n−1∑(xi−xˉSx)(yi−yˉSy)=∑XY−(∑X)(∑Y)n(n−1)SxSy=(SSxy)2(SSxx)(SSyy)\displaystyle\boxed{ \large r=\frac{1}{n-1}\sum\Big(\frac{x_i-\bar{x}}{S_x}\Big)\Big(\frac{y_i-\bar{y}}{S_y}\Big)=\frac{\sum XY-\frac{(\sum X)(\sum Y)}{n}}{(n-1)S_xS_y}=\sqrt{\frac{\color{green}(SS_{xy})\color{black}^2}{\color{blue}(SS_{xx})\color{Red}(SS_{yy})}}}r=n−11​∑(Sx​xi​−xˉ​)(Sy​yi​−yˉ​​)=(n−1)Sx​Sy​∑XY−n(∑X)(∑Y)​​=(SSxx​)(SSyy​)(SSxy​)2​​​  

where
Sx=∑(xi−xˉ)2n−1=SSxxn−1\large S_x=\sqrt\frac{\sum(x_i-\bar{x})^2}{n-1}=\sqrt{\frac{SS_{xx}}{n-1}}Sx​=n−1∑(xi​−xˉ)2​​=n−1SSxx​​​

Sy=∑(yi−yˉ)2n−1=SSyyn−1\large S_y=\sqrt{\frac{\sum(y_i-\bar{y})^2}{n-1}}=\sqrt{\frac{SS_{yy}}{n-1}}Sy​=n−1∑(yi​−yˉ​)2​​=n−1SSyy​​​

SSxx=∑(xi−xˉ)2=∑xi2−(∑xi)2n\color{blue}SS_{xx}=\sum(x_i-\bar{x})^2=\sum x^2_i-\frac{(\sum x_i)^2}{n}SSxx​=∑(xi​−xˉ)2=∑xi2​−n(∑xi​)2​

SSyy=∑(yi−yˉ)2=∑yi2−(∑yi)2n\color{red}SS_{yy}=\sum(y_i-\bar{y})^2=\sum y^2_i-\frac{(\sum y_i)^2}{n}SSyy​=∑(yi​−yˉ​)2=∑yi2​−n(∑yi​)2​

SSxy=∑(xi−xˉ)(yi−yˉ)=∑xiyi−(∑xi)(∑yi)n\color{green}SS_{xy}=\sum(x_i-\bar{x})(y_i-\bar{y})=\sum x_iy_i-\frac{(\sum x_i)(\sum y_i)}{n}SSxy​=∑(xi​−xˉ)(yi​−yˉ​)=∑xi​yi​−n(∑xi​)(∑yi​)​

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Wize Concept
We can also rewrite the the following formula:
r=∑XY−(∑X)(∑Y)n(n−1)SxSy=∑XY−n(∑X)n(∑Y)n(n−1)SxSy=∑XY−nx‾ y‾(n−1)SxSy\displaystyle{r=\frac{\sum XY-\frac{(\sum X)(\sum Y)}{n}}{(n-1)S_xS_y}=\frac{\sum_{ }^{ }XY-{n}\frac{(\sum_{ }^{ }X)}{n}\frac{(\sum_{ }^{ }Y)}{n}}{(n-1)S_xS_y}=\frac{\sum_{ }^{ }XY-n\overline{x}\ \overline{y}}{(n-1)S_xS_y}}r=(n−1)Sx​Sy​∑XY−n(∑X)(∑Y)​​=(n−1)Sx​Sy​∑​XY−nn(∑​X)​n(∑​Y)​​=(n−1)Sx​Sy​∑​XY−nx y​​

Exam Tip
There are several ways calculate rrr depending on what information you are given:

∑X,∑Y,Sx,Sy,xˉ,yˉ,n,SSxx,SSyy,SSxy,∑(xi−xˉ)2,∑(yi−yˉ)2\sum X,\sum Y,S_x,S_y,\bar{x},\bar{y},n,SS_{xx},SS_{yy},SS_{xy},\sum(x_i-\bar{x})^2,\sum(y_i-\bar{y})^2∑X,∑Y,Sx​,Sy​,xˉ,yˉ​,n,SSxx​,SSyy​,SSxy​,∑(xi​−xˉ)2,∑(yi​−yˉ​)2

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Example: Solving for Correlation

Find the correlation coefficient and interpret it.

∑x=30\sum x=30∑x=30
∑y=25\sum y=25∑y=25
∑xy=122\sum xy=122∑xy=122
∑x2=220\sum x^2=220∑x2=220
∑y2=159\sum y^2=159∑y2=159

                     r=1n−1∑(xi−xˉSx)(yi−yˉSy)=∑XY−(∑X)(∑Y)n(n−1)SxSy=(SSxy)2(SSxx)(SSyy)\displaystyle{ \large r=\frac{1}{n-1}\sum\Big(\frac{x_i-\bar{x}}{S_x}\Big)\Big(\frac{y_i-\bar{y}}{S_y}\Big)=\frac{\sum XY-\frac{(\sum X)(\sum Y)}{n}}{(n-1)S_xS_y}=\sqrt{\frac{\color{green}(SS_{xy})\color{black}^2}{\color{blue}(SS_{xx})\color{Red}(SS_{yy})}}}r=n−11​∑(Sx​xi​−xˉ​)(Sy​yi​−yˉ​​)=(n−1)Sx​Sy​∑XY−n(∑X)(∑Y)​​=(SSxx​)(SSyy​)(SSxy​)2​​  

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SSxx=∑(xi−xˉ)2=∑xi2−(∑xi)2n\color{blue}SS_{xx}=\sum(x_i-\bar{x})^2=\sum x^2_i-\frac{(\sum x_i)^2}{n}SSxx​=∑(xi​−xˉ)2=∑xi2​−n(∑xi​)2​

SSxx=220−(30)25=40SS_{xx}=220-\frac{\left(30\right)^2}{5}=40SSxx​=220−5(30)2​=40

SSyy=∑(yi−yˉ)2=∑yi2−(∑yi)2n\color{red}SS_{yy}=\sum(y_i-\bar{y})^2=\sum y^2_i-\frac{(\sum y_i)^2}{n}SSyy​=∑(yi​−yˉ​)2=∑yi2​−n(∑yi​)2​

SSyy=159−(25)25=34SS_{yy}=159-\frac{\left(25\right)^2}{5}=34SSyy​=159−5(25)2​=34

SSxy=∑(xi−xˉ)(yi−yˉ)=∑xiyi−(∑xi)(∑yi)n\color{green}SS_{xy}=\sum(x_i-\bar{x})(y_i-\bar{y})=\sum x_iy_i-\frac{(\sum x_i)(\sum y_i)}{n}SSxy​=∑(xi​−xˉ)(yi​−yˉ​)=∑xi​yi​−n(∑xi​)(∑yi​)​

SSxy=122−(30)(25)5=−28SS_{xy}=122-\frac{\left(30\right)\left(25\right)}{5}=-28SSxy​=122−5(30)(25)​=−28

r=(SSxy)2(SSxx)(SSyy) \large r=\sqrt{\frac{\color{green}(SS_{xy})\color{black}^2}{\color{blue}(SS_{xx})\color{Red}(SS_{yy})}}r=(SSxx​)(SSyy​)(SSxy​)2​​

r=−(28)2(40)(34)=−0.5765=−0.7593r=-\sqrt{\frac{\left(28\right)^2}{\left(40\right)\left(34\right)}}=-\sqrt{0.5765}=-0.7593r=−(40)(34)(28)2​​=−0.5765​=−0.7593

Exam Tip
There are multiple ways to solve for rrr. Use the formula based on what variables are given.

Pretty strong, negative correlation.

We want to predict a person’s body fat based on their age. Age and percentage body fat were measured in 18 adult males.

∑x=702∑y=461∑xy=19,871∑x2=31,396∑y2=13,449\begin{array}{ll}\sum x=702\\\sum y=461\\\sum xy=19,871\\\sum x^2=31,396\\\sum y^2=13,449\end{array} ∑x=702∑y=461∑xy=19,871∑x2=31,396∑y2=13,449​

(i) What is the average age in the sample?

[Enter answer as a whole number, e.g. 31]

I don't know

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Example: Solving for Correlation
Solve for the correlation coefficient.


∑x=42\sum_{ }^{ }x=42∑​x=42
∑y=235\sum_{ }^{ }y=235∑​y=235
∑xy=1911\sum_{ }^{ }xy=1911∑​xy=1911
Sx=3.3466S_x=3.3466Sx​=3.3466
Sy=18.7341S_y=18.7341Sy​=18.7341


                     r=1n−1∑(xi−xˉSx)(yi−yˉSy)=∑XY−(∑X)(∑Y)n(n−1)SxSy=(SSxy)2(SSxx)(SSyy) \displaystyle{\large r=\frac{1}{n-1}\sum\Big(\frac{x_i-\bar{x}}{S_x}\Big)\Big(\frac{y_i-\bar{y}}{S_y}\Big)=\frac{\sum XY-\frac{(\sum X)(\sum Y)}{n}}{(n-1)S_xS_y}=\sqrt{\frac{\color{green}(SS_{xy})\color{black}^2}{\color{blue}(SS_{xx})\color{Red}(SS_{yy})}}}r=n−11​∑(Sx​xi​−xˉ​)(Sy​yi​−yˉ​​)=(n−1)Sx​Sy​∑XY−n(∑X)(∑Y)​​=(SSxx​)(SSyy​)(SSxy​)2​​  

There are multiple ways to solve for r. Based on the variables are given in this question, we use this formula:

r=∑XY−(∑X)(∑Y)n(n−1)SxSy\large r=\frac{\sum XY-\frac{(\sum X)(\sum Y)}{n}}{(n-1)S_xS_y}r=(n−1)Sx​Sy​∑XY−n(∑X)(∑Y)​​

r=1911−(42)(235)6(6−1)(3.3466)(18.7341)=266313.4815=0.8485r=\frac{1911-\frac{\left(42\right)\left(235\right)}{6}}{\left(6-1\right)\left(3.3466\right)\left(18.7341\right)}=\frac{266}{313.4815}=0.8485r=(6−1)(3.3466)(18.7341)1911−6(42)(235)​​=313.4815266​=0.8485

Strong, positive correlation.

∑X=33,115\sum X=33,115∑X=33,115
∑Y=14,263,500\sum Y=14,263,500∑Y=14,263,500
∑XY=15,157,905,000\sum XY=15,157,905,000∑XY=15,157,905,000

Solve for the correlation coefficient.

Solve for the correlation coefficient. [Enter answer with 2 decimal places, e.g. 0.86]

I don't know

Extra Practice

Correlation Coefficient

Suppose that in a sample of 20 boys and 20 fathers, each father is 15 cm taller than his son.  Which of the following is most likely the value of r?	

Correlation: Changing Units of X Practice Question

Problem
A researcher correctly calculated the correlation between the shoe size of his subjects, in inches, and the weight of his subjects, in pounds. It turned out to be 0.69.  He then decides to change the shoe size units into centimetres.  Which of the following is the new value of rrr  :

Correlation Coefficient

Suppose that in a sample of 20 boys and 20 fathers, each father is 15 cm taller than his son.  Which of the following is most likely the value of r?	

Correlation

Which of the following is most likely the correlation coefficient between the number of text messages you receive and your daily commute time?

Correlation

Which of the following is most likely the correlation coefficient between the number of cousins you have and the mark you get on your midterm?

Correlation

Calculate the correlation coefficient between time in hours and distance in kilometers given the following data:
sx2=64  sy=10   b = 0.3

Correlation

Which of the following is likely to be the value of the correlation coefficient rrr between the  variables (1) program of study (such as engineering, sciences, social studies) and (b) amount of homework in hours

Correlations

Calculate the correlation coefficient between time in hours and distance in kilometers given the following data:
sx  2=64,  sy=10,  y^=0.3xs_x^{\ \ 2}=64,\ \ s_y=10,\ \ \hat y=0.3xsx  2​=64,  sy​=10,  y^​=0.3x

Correlations

Which of the following is likely to be the value of the correlation coefficient rrr between the variables (1) program of study (such as engineering, sciences, social studies) and (b) amount of homework in hours

Correlation Coefficient

Suppose that in a sample of 20 boys and 20 fathers, each father is 15 cm taller than his son.  Which of the following is most likely the value of r?	

Correlation

Which of the following is likely to be the value of the correlation coefficient rrr between the  variables (1) program of study (such as engineering, sciences, social studies) and (b) amount of homework in hours

Correlation: Perfect Linear Correlation

Suppose that in a sample of 20 boys and 20 fathers, each father is 15 cm taller than his son.  Which of the following is most likely the value of r?	   

Correlation: Changing Units of X Practice Question

Problem
A researcher correctly calculated the correlation between the shoe size of his subjects, in inches, and the weight of his subjects, in pounds. It turned out to be 0.69.  He then decides to change the shoe size units into centimetres.  Which of the following is the new value of rrr  :

Solving for Correlation

The following data is obtained to assess the relationship between height in cm and shoe size. 
The coefficient of correlation is?

Correlation

Solve for the correlation coefficient.
∑x=42\sum_{ }^{ }x=42∑​x=42
∑y=235\sum_{ }^{ }y=235∑​y=235

Correlation

Find the correlation coefficient and interpret it.
∑x=30\sum x=30∑x=30
∑y=25\sum y=25∑y=25

Correlation

∑X=33,115\sum X=33,115∑X=33,115
∑Y=14,263,500\sum Y=14,263,500∑Y=14,263,500
∑XY=15,157,905,000\sum XY=15,157,905,000∑XY=15,157,905,000

Correlation

We want to predict a person’s body fat based on their age. Age and percentage body fat were measured in 18 adult males.
∑x=702∑y=461∑xy=19,871∑x2=31,396∑y2=13,449\begin{array}{ll}\sum x=702\\\sum y=461\\\sum xy=19,871\\\sum x^2=31,396\\\sum y^2=13,449\end{array} ∑x=702∑y=461∑xy=19,871∑x2=31,396∑y2=13,449​

Correlation

(ii) Solve for the correlation coefficient.

Correlation

Answer the following questions.

Practice: Correlation
Answer the following questions.

Correlation and Regression Line

Suppose we want to see if there is a relationship between the number of times a student attended their labs in the entire semester (x) and their final exam grade (y). We randomly sample 8 students and recorded our results.   
∑(xi−x‾)(yi−y‾)=264.6\sum_{ }^{ }\left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)=264.6∑​(xi​−x)(yi​−y​)=264.6

Correlation

Paul wants to find the correlation between iPhone models (i.e. iPhone 3, 4, 5, 6, 7, 8, X, etc.) and battery life. What is wrong here?

Correlation and Residual Plots

Pamela and Tim are trying to be healthy by taking the stairs instead of the elevator. We want to see there is a relationship between the floor you live on and the time it takes to get to your floor from the lobby. Pamela lives on the 8th floor. Tim lives in the penthouse on the 20th floor.
∑x2=258\sum_{ }^{ }x^2=258∑​x2=258
∑y2=687\sum_{ }^{ }y^2=687∑​y2=687

Wize University Statistics Textbook > Scatterplots, Association, and Correlation

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