A correlation coefficient indicates the and the of the relationship between two variables

PPA 696 RESEARCH METHODS

CORRELATION

CORRELATION

Correlation coefficients are statistics which can help to describe data sets which contain variables measured at the interval and ratio levels. Correlation coefficients are measures of association between two (or more) variables.

Correlation is a measure of association that tests whether a relationship exists between two variables. It indicates both the strength of the association and its direction (direct or inverse). The Pearson product-moment correlation coefficient, written as r, can describe a linear relationship between two variables.

For example is there a relationship between:
the budget of the police department and the crime rate?
the hours of batting practice and a player's batting average?

The value of r can range from 0.0, indicating no relationship between the two variables, to positive or negative 1.0, indicating a strong linear relationship between the two variables.
 

SCATTERPLOTS

It is useful to obtain a plot of the joint distribution of the values of the two variables, X and Y. These are called scatterplots. The values of X are displayed on the lower, or horizontal axis (called the X-axis) and the values of Y are displayed on the upper or vertical axis (called the Y-axis).

If small values of X are associated with small values for Y, and large values of X are associated with large values of Y, then the data will stretch from the lower left hand corner of the plot to the upper right hand corner of the plot. This indicates a positive relationship.

If small values of X are associated with large values for Y, and large values of X are associated with small values of Y, then the data will stretch from the upper left hand corner of the plot to the lower right hand corner of the plot. This indicates an inverse relationship.

If there is no discernible pattern to the distribution, then the two variables probably are not related in a linear fashion. There may be a strong, non-linear relationship between the two variables (for example, think of the normal curve) but it cannot be detected by r.

When there are only a few data points, it is fairly easy to estimate the strength of the relationship by eyeballing the data. However, with many data points statistics are needed to summarize the strength and direction of the relationship.

The Pearson r assumes that the variables are measured at the interval or ratio level. If the variables are measured at the ordinal level, however (for example, a Likert-type scale), then the Spearman rank correlation can be used. Neither Pearson nor Spearman are designed for use with variables measured at the nominal level; instead, use the point-biserial correlation (for one nominal variable) or phi (for two nominal variables).

The formula for r is as follows:

What Is a Correlation Coefficient?

A correlation coefficient, often expressed as r, indicates a measure of the direction and strength of a relationship between two variables. When the r value is closer to +1 or -1, it indicates that there is a stronger linear relationship between the two variables.

Correlational studies are quite common in psychology, particularly because some things are impossible to recreate or research in a lab setting. Instead of performing an experiment, researchers may collect data to look at possible relationships between variables. From the data they collect and its analysis, researchers then make inferences and predictions about the nature of the relationships between variables.

A correlationis a statistical measurement of the relationship between two variables. Remember this handy rule: The closer the correlation is to 0, the weaker it is. The closer it is to +/-1, the stronger it is.

Types of Correlation

Correlation strength ranges from -1 to +1.

Positive Correlation

A correlation of +1 indicates a perfect positive correlation, meaning that both variables move in the same direction together.

Negative Correlation

A correlation of –1 indicates a perfect negative correlation, meaning that as one variable goes up, the other goes down.

Zero Correlation

A zero correlation suggests that the correlation statistic does not indicate a relationship between the two variables. This does not mean that there is no relationship at all; it simply means that there is not a linear relationship. A zero correlation is often indicated using the abbreviation r = 0.

Scatter Plots and Correlation

Scatter plots (also called scatter charts, scattergrams, and scatter diagrams) are used to plot variables on a chart to observe the associations or relationships between them. The horizontal axis represents one variable, and the vertical axis represents the other.

Each point on the plot is a different measurement. From those measurements, a trend line can be calculated. The correlation coefficient is the slope of that line. When the correlation is weak (r is close to zero), the line is hard to distinguish. When the correlation is strong (r is close to 1), the line will be more apparent.

Strong vs. Weak Correlations

Correlations can be confusing, and many people equate positive with strong and negative with weak. A relationship between two variables can be negative, but that doesn't mean that the relationship isn't strong.

A weak positive correlation indicates that, although both variables tend to go up in response to one another, the relationship is not very strong. A strong negative correlation, on the other hand, indicates a strong connection between the two variables, but that one goes up whenever the other one goes down.

For example, a correlation of -0.97 is a strong negative correlation, whereas a correlation of 0.10 indicates a weak positive correlation. A correlation of +0.10 is weaker than -0.74, and a correlation of -0.98 is stronger than +0.79.

Correlation Does Not Equal Causation

Correlation does not equal causation. Just because two variables have a relationship does not mean that changes in one variable cause changes in the other. Correlations tell us that there is a relationship between variables, but this does not necessarily mean that one variable causes the other to change.

An oft-cited example is the correlation between ice cream consumption and homicide rates. Studies have found a correlation between increased ice cream sales and spikes in homicides. However, eating ice cream does not cause you to commit murder. Instead, there is a third variable: heat. Both variables increase during summertime.

Illusory Correlations

An illusory correlation is the perception of a relationship between two variables when only a minor relationship—or none at all—actually exists. An illusory correlation does not always mean inferring causation; it can also mean inferring a relationship between two variables when one does not exist.

For example, people sometimes assume that, because two events occurred together at one point in the past, one event must be the cause of the other. These illusory correlations can occur both in scientific investigations and in real-world situations.

Stereotypes are a good example of illusory correlations. Research has shown that people tend to assume that certain groups and traits occur together and frequently overestimate the strength of the association between the two variables.

For example, suppose someone holds the mistaken belief that all people from small towns are extremely kind. When they meet a very kind person, their immediate assumption might be that the person is from a small town, despite the fact that kindness is not related to city population.

A Word From Verywell

Psychology research makes frequent use of correlations, but it's important to understand that correlation is not the same as causation. This is a frequent assumption among those not familiar with statistics and assumes a cause-effect relationship that might not exist.

Frequently Asked Questions

• How do you find the correlation coefficient?

  You can calculate the correlation coefficient in a few different ways, with the same result. The general formula is rXY=COVXY/(SX SY), which is the covariance between the two variables, divided by the product of their standard deviations:

• How do you calculate a correlation coefficient in Excel?

  In the cell in which you want the correlation coefficient to appear, enter =CORREL(A2:A7,B2:B7), where A2:A7 and B2:B7 are the variable lists to compare. Press Enter.

• How do you find a linear correlation coefficient?

  Finding the linear correlation coefficient requires a long, difficult calculation, so most people use a calculator or software such as Excel or a statistics program.

• How do you interpret a correlation coefficient?

  Correlations range from -1.00 to +1.00. The correlation coefficient (expressed as r ) shows the direction and strength of a relationship between two variables. The closer the r value is to +1 or -1, the stronger the linear relationship between the two variables is.
  

• What is the difference between correlation and causation?

  Correlations indicate a relationship between two variables, but one doesn't necessarily cause the other to change.

Verywell Mind uses only high-quality sources, including peer-reviewed studies, to support the facts within our articles. Read our editorial process to learn more about how we fact-check and keep our content accurate, reliable, and trustworthy.

Additional Reading

• Correlation and regression. In: Swinscow TDV. Statistics at Square One. The BMJ.

By Kendra Cherry
Kendra Cherry, MS, is an author and educational consultant focused on helping students learn about psychology.

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