Correlation

The correlation coefficient $\colorFour{r}$ measures the direction and strength of the linear relationship between two quantitative variables:

Direction (positive, negative, neither)
Strength (strong, moderate, weak, no) $r=$ correlation where $-1\le r\le1$

$\displaystyle\boxed{r=\frac{1}{n-1}\Sigma\left(\frac{x_i-\overline{x}}{S_x}\right)\left(\frac{y_i-\overline{y}}{S_y}\right)}$

Range of Correlation Values

The correlation coefficient $r$ must be a value between $-1$ and $+1$ , inclusive.

The closer $r$ is to $-1$ , the stronger the negative linear relationship between $X$ and $Y$
The closer $r$ is to $+1$ , the stronger the positive linear relationship between $X$ and $Y$
The closer $r$ is to $0$ , the weaker the linear relationship between $X$ and $Y$
When $r=0$ , that means there is no linear correlation.

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

Characteristics of Correlation

We can only correlate quantitative data.
If you switch X and Y, the correlation value does not change.
Correlation coefficient has no units.
Correlation is based on linear models only.
Correlation does not imply causation!

Extra Practice

Correlation

Correlation $r$ has no units unless $x$ and $y$ have the same units.

Correlation

The correlation is 0.56 when the unit for $x$ is years. What will the correlation coefficient be after $x$ is changed from years to months of experience? [Enter your answer with 2 decimal places, e.g. 0.36]

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.

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 $r=0.9$ .

Correlation

$r=0.33$ ; the correlation is positive, but weak.

Correlation

$r=2.00$ ; the correlation is very strong.

Correlation

$r=-1.00$ ; the correlation is perfectly negative.

Correlation

$r<0$ ; increasing $x$ will cause $y$ to decrease.

Correlation

$r=-0.91$ ; the correlation is very weak.

Correlation

$r>0$ ; large values of $x$ are associated with large values of $y$ .

Correlation

In a plot, if we swap $x$ and $y$ , $r$ will not change.

Correlation

The correlation between midterm grades and final exam grades is $r=0.88$ . The correlation between assignment grades and final exam grades is $r=0.44$ .

Which of the following is false?

Correlation

If we swap $x$ and $y$ , what will happen to the correlation coefficient $r$ ?

Hypothesis Testing: Correlation

For the following statements, select true or false. Enter T for True, F for False.

Correlation

Numbers in a data set are in dollar units ($). Which of the following measures will not be in dollar units?

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

If $R^2=0.86$ , then there must be a strong, positive correlation between $X$ and $Y$ .

Correlation and Regression Line

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.

Correlation

If we double all the $Y$ data and without changing the $X$ data, the correlation will not change but the slope will get steeper.

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?

Predictions and Residual Plots

Suppose the following summary data was given:

x-variable: $\mu_x=10,\ s_x=2$
y-variable: $\mu_y=250,\ s_y=10$

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.

Correlation

Which of the following statements is/or TURE? (select all that apply)

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

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Correlation

Range of Correlation Values

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Extra Practice