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Correlation

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Wize University Statistics Textbook > Scatterplots, Association, and Correlation

Solving for Correlation

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Find the correlation coefficient and interpret it.



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


r=1n−1∑(xi−xˉSx)(yi−yˉSy)=∑XY−(∑X)(∑Y)n(n−1)SxSy=(SSxy)2(SSxx)(SSyy) \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})}}


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}
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}


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}



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

More Solving for Correlation Questions:
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 rr :
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 rr 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=0.3x
Correlations
Which of the following is likely to be the value of the correlation coefficient rr 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 rr 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 rr :
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
∑y=235\sum_{ }^{ }y=235
Correlation
∑X=33,115\sum X=33,115
∑Y=14,263,500\sum Y=14,263,500
∑XY=15,157,905,000\sum 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}
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
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
∑y2=687\sum_{ }^{ }y^2=687