A perfect positive correlation is indicated when the correlation coefficient, denoted by r, is equal to +1. This signifies a perfect direct linear relationship between two variables, meaning that as one variable increases, the other also increases by a perfectly predictable amount. The mathematical representation is: $$r = 1$$

Perfect Positive Correlation

A zero correlation exists when the correlation coefficient is exactly 0. This signifies that there is no discernible linear relationship between two variables. As a result, knowing the value of one variable provides no basis for predicting the value of the other.

Zero Correlation

A researcher investigates the relationship between the number of hours a student spends playing video games per week and their grade point average (GPA). The study yields a correlation coefficient of r = -0.65. Which statement provides the most accurate analysis of this finding?

Correlations can be described using everyday language that indicates how variables change together. For example, the statement "there is no relationship between height and weight" describes a zero correlation. Statements such as "usually, the taller someone is, the thinner they are" or "usually, the shorter someone is, the heavier they are" describe a negative correlation, where the two variables move in opposite directions. Conversely, a positive correlation is represented by a statement such as "as height increases, typically weight increases," indicating that both variables move in the same direction.

Verbal Descriptions of Correlations

A negative relationship, or negative correlation, occurs when higher scores on one variable tend to be associated with lower scores on another variable. On a scatterplot, this type of relationship is illustrated by data points that generally trend downward from the upper left to the lower right of the graph.

Negative Relationship

A positive relationship, or positive correlation, occurs when higher scores on one variable tend to be associated with higher scores on another variable. On a scatterplot, this is visually represented by data points that generally trend upward from the lower left to the upper right of the graph.

Positive Relationship

The sign of a correlation coefficient, whether positive or negative, indicates the direction of the relationship between variables. Based on this directionality, correlations can be classified into three main types: positive, negative, and zero (no correlation).

Google

University of Chicago

A numerical index, commonly represented as Pearson's $$r$$, that measures the direction and strength of a linear relationship between two quantitative variables. The coefficient is bounded between $$-1.00$$ and $$+1.00$$, where values further from zero indicate a tighter, more predictable association, and values near zero indicate a negligible relationship. In research, this coefficient frequently serves as an effect size measure to evaluate how strongly variables are connected in the population.

Correlation Coefficient

Spielman, R. M., Jenkins, W. J., & Lovett, M. D. (2020, April 22). Psychology (2nd ed.). OpenStax. https://openstax.org/books/psychology-2e/pages/1-introduction

OpenStax Psychology (2nd ed.) Textbook

Types of Correlation

A researcher is examining the statistical association between several pairs of variables. Which of the following numerical values represents the strongest association between two variables?

A correlation matrix is a specific type of table utilized to present the correlation coefficients—typically Pearson’s $$r$$—among several different variables. To avoid redundant information and improve overall readability, only the lower half of the matrix is filled with values, as the upper half would display identical mirror correlations. Furthermore, because any variable's correlation with itself is always a perfect $$+1.00$$, these values along the diagonal are conventionally replaced with dashes.

Correlation Matrix

The computational formula for Pearson's $$r$$ defines it as the mean cross-product of $$z$$ scores. To calculate it, each score for both variables is first transformed into a $$z$$ score. Then, each individual's two $$z$$ scores are multiplied to create a cross-product. The mean of all these cross-products represents Pearson's $$r$$, expressed by the formula: $$r = \frac {\sum(z_xz_y)}{N}$$.

Pearson's r Formula

Restriction of range is a statistical issue that occurs when one or both variables in a correlational study have a limited range of values in the sample relative to the actual population. This artificial limitation can cause a strong overall correlation to appear misleadingly weak or even non-existent. To prevent this, researchers should aim to collect data from subjects spanning a wide range of values for their primary variables.

Restriction of Range

The correlation coefficient indicates the strength of a statistical relationship between two variables. The relationship is considered the weakest when the correlation coefficient is closest to $${}0$$. A coefficient near $${}0$$ signifies a very weak, unpredictable relationship, regardless of whether the sign is positive or negative.

Identifying the Weakest Correlation

Statistical regression is a technique that allows researchers to predict the value of one variable given the value of another variable. While correlation coefficients describe the strength and direction of a relationship, regression utilizes that established relationship to make specific mathematical predictions.

Statistical Regression

The test of the correlation coefficient is the appropriate null hypothesis test for evaluating statistical relationships between quantitative variables when Pearson's $$r$$ is used to describe the strength of those relationships. While it is possible to use the sample correlation coefficient to compute a $$t$$ score and proceed with a standard $$t$$-test, the correlation coefficient is mathematically structured so that it can also be treated directly as its own test statistic to find the $$p$$-value.

Test of the Correlation Coefficient

Scatterplots are highly effective for visually illustrating the varying strengths and directions of Pearson's $$r$$. A perfect negative relationship ($$r = -1.00$$) is depicted as a tight, downward-sloping straight line, whereas a moderate negative relationship ($$r = -0.50$$) shows points loosely clustered around a downward trend. A value of $$r = 0$$ appears as a shapeless cloud of points, indicating no relationship. Conversely, a moderate positive relationship ($$r = +0.50$$) shows a loose upward trend, and a perfect positive relationship ($$r = +1.00$$) features points falling exactly on an upward-sloping diagonal line.

Example of Scatterplots for Pearson's r Values

In psychological research, Cohen provided standard guidelines for interpreting the strength of a statistical relationship measured by Pearson's $$r$$. According to his criteria, $$r$$ values near $$\pm 0.10$$ are considered small, values near $$\pm 0.30$$ are considered medium, and values near $$\pm 0.50$$ are considered large. The positive or negative sign of the coefficient only signifies the direction of the relationship, so an $$r$$ value of $$-0.30$$ is just as strong as a value of $$+0.30$$.

Learn Before

Related

Learn After