What does the sample correlation coefficient $r$ estimate?

The population correlation coefficient, $\rho$.

What do the sign and magnitude of $r$ indicate?

The sign indicates the direction, and the magnitude indicates the strength of a linear association.

What does $r=0$ establish about two variables?

There is no linear association, but a nonlinear relationship may still exist.

Does the correlation coefficient give the slope of a relationship?

No. It describes the direction and strength of linear association, not the slope.

How is Pearson’s sample correlation related to covariance?

It is covariance standardized by dividing by the product of the variables’ standard deviations.

Business statisticsLesson 1 of 3 in the course Linear Regression and Correlation

Data Structures, Scatterplots, and the Pearson Correlation Coefficient

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

Correlation analysis begins with data on two variables measured together. This lesson introduces common data structures and explains how scatterplots help us inspect whether two variables move together.

A univariate data set records one variable. A bivariate data set records two variables for each observation, such as a person's income and the quantity of a good they purchase. Multivariate data sets contain multiple variables.

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What does the sample correlation coefficient $r$ estimate?: The population correlation coefficient, $\rho$ .
What do the sign and magnitude of $r$ indicate?: The sign indicates the direction, and the magnitude indicates the strength of a linear association.
What does $r=0$ establish about two variables?: There is no linear association, but a nonlinear relationship may still exist.
Does the correlation coefficient give the slope of a relationship?: No. It describes the direction and strength of linear association, not the slope.
How is Pearson’s sample correlation related to covariance?: It is covariance standardized by dividing by the product of the variables’ standard deviations.

Quiz questions

What is the relationship between the sample correlation coefficient $r$ and the population correlation coefficient $\rho$ ?Answer: $r$ is calculated from sample data to estimate $\rho$ .
Which interval contains every possible value of the sample correlation coefficient $r$ ?Answer: $-1 \leq r \leq 1$
If two variables have an exact straight-line relationship with a negative slope, what is their correlation coefficient?Answer: $r=-1$
Two variables lie exactly on a straight line with a positive slope. What is their correlation, regardless of the line’s steepness?Answer: $r=1$
A calculated correlation of zero is found between two variables. Which conclusion is justified?Answer: The variables have no linear association, though a nonlinear relationship may exist.

Based on Introductory Business Statistics 2e by Alexander Holmes, Barbara Illowsky, Susan Dean (OpenStax), licensed CC BY 4.0. Adapted into a summary, flashcards and quizzes.

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