High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize University Statistics Textbook > Scatterplots, Association, and Correlation

Correlation

ScatterplotsPrevious Section
Solving for CorrelationNext Section
Popular Courses
Find My Course
0:00 / 0:00

Correlation

The correlation coefficient r\colorFour{r} measures the direction and strength of the linear relationship between two quantitative variables:
  1. Direction (positive, negative, neither)
  2. Strength (strong, moderate, weak, no) r=r= correlation where −1≤r≤1-1\le r\le1

r=1n−1Σ(xi−x‾Sx)(yi−y‾Sy)\displaystyle\boxed{r=\frac{1}{n-1}\Sigma\left(\frac{x_i-\overline{x}}{S_x}\right)\left(\frac{y_i-\overline{y}}{S_y}\right)}


Exam Tip
There are multiple ways to calculate rr. (See: Solving for Correlation)


PAGE BREAK

Range of Correlation Values

The correlation coefficient rr must be a value between −1-1 and +1+1, inclusive.
  • The closer rr is to −1-1, the stronger the negative linear relationship between XX and YY
  • The closer rr is to +1+1, the stronger the positive linear relationship between XX and YY
  • The closer rr is to 00, the weaker the linear relationship between XX and YY
  • When r=0r=0, that means there is no linear correlation.


Credit: https://www.mathsisfun.com/data/correlation.html


PAGE BREAK

Characteristics of Correlation
  1. We can only correlate quantitative data.
  2. If you switch X and Y, the correlation value does not change.
  3. Correlation coefficient has no units.
  4. Correlation is based on linear models only.
  5. Correlation does not imply causation!
Answer the following questions.

(i) 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?
Extra Practice
The correlation between the number of hours of operation and sales in dollars at Pino's Pizza is 0.43. Which of the following is true?
Correlation and Regression Line
If r=−1.00r=-1.00, then 100% of the data points fall exactly on the regression line and the slope cannot be equal or greater than 0.
Correlation
The correlation between the number of hours of operation and sales in dollars at Pino's Pizza is 0.43. Which of the following is true?
Correlation
Numbers in a data set are in dollar units ($). Which of the following measures will not be in dollar units?
Correlation
Which of the following statements is/or TURE? (select all that apply)
Correlation
True or False: A correlation coefficient of 0.6 means that 60% of the data points in the scatter plot have a linear relationship.
Predictions and Residual Plots
Suppose the following summary data was given:
x-variable: μx=10, sx=2\mu_x=10,\ s_x=2
y-variable: μy=250, sy=10\mu_y=250,\ s_y=10
Correlation
If we double all the YY data and without changing the XX data, the correlation will not change but the slope will get steeper.
Correlation
If R2=0.86R^2=0.86, then there must be a strong, positive correlation between XX and YY.
Correlation
The correlation between the number of hours of operation and sales in dollars at Pino's Pizza is 0.43. Which of the following is true?
Hypothesis Testing: Correlation
For the following statements, select true or false. Enter T for True, F for False.
Correlation
If we swap xx and yy, what will happen to the correlation coefficient rr?
Correlation
The correlation between midterm grades and final exam grades is r=0.88r=0.88. The correlation between assignment grades and final exam grades is r=0.44r=0.44.
Which of the following is false?
Correlation
In a plot, if we swap xx and yy, rr will not change.
Correlation
r>0r>0; large values of xx are associated with large values of yy.
Correlation
r=−0.91r=-0.91; the correlation is very weak.
Correlation
r<0r<0; increasing xx will cause yy to decrease.
Correlation
r=−1.00r=-1.00; the correlation is perfectly negative.
Correlation
r=2.00r=2.00; the correlation is very strong.
Correlation
r=0.33r=0.33; the correlation is positive, but weak.
Correlation: Coefficient of Determination
True or false?
About 81% of the variation of quantities sold can be explained by price. The correlation between price and sales revenue is r=0.9r=0.9.
Simple Linear Regression: Correlation
True or false? Enter T if True, F if False. (Use capital letters: T, F)
1. If 𝒓 = −𝟏. 𝟎𝟎, then 100% of the data points fall exactly on the regression line and the slope cannot be equal or greater than 0.
T
Correlation
The correlation is 0.56 when the unit for xx is years. What will the correlation coefficient be after xx is changed from years to months of experience? [Enter your answer with 2 decimal places, e.g. 0.36]
Correlation
Correlation rr has no units unless xx and yy have the same units.