What is a relationship between two variables where if one variable increases the other one also increases?

The correlation requires two scores from the same individuals. These scores are normally identified as X and Y. The pairs of scores can be listed in a table or presented in a scatterplot.

Example: We might be interested in the correlation between your SAT-M scores and your GPA at UNC.

Here are the Math SAT scores and the GPA scores of 13 of the students in this class, and the scatterplot for all 41 students:

The scatterplot has the X values (GPA) on the horizontal (X) axis, and the Y values (MathSAT) on the vertical (Y) axis. Each individual is identified by a single point (dot) on the graph which is located so that the coordinates of the point (the X and Y values) match the individual's X (GPA) and Y (MathSAT) scores.

For example, the student named "Obs5" (in the sixth row of the datasheet) has GPA=2.30 and MathSAT=710. This student is represented in the scatterplot by high-lighted and labled ("5") dot in the upper-left part of the scatterplot. Note that is to the right of MathSAT of 710 and above GPA of 2.30.

Note that the Pearson correlation (explained below) between these two variables is .32.

1. The Direction of a Relationship
The correlation measure tells us about the direction of the relationship between the two variables. The direction can be positive or negative.

1. Positive
: In a positive relationship both variables tend to move in the same direction: If one variable increases, the other tends to also increase. If one decreases, the other tends to also.

In the example above, GPA and MathSAT are positively related. As GPA (or MathSAT) increases, the other variable also tends to increase.

2. Negative
: In a negative relationship the variables tend to move in the opposite directions: If one variable increases, the other tends to decrease, and vice-versa.

The direction of the relationship between two variables is identified by the sign of the correlation coefficient for the variables. Postive relationships have a "plus" sign, whereas negative relationships have a "minus" sign.

2. The Form (Shape) of a Relationship
: The form or shape of a relationship refers to whether the relationship is straight or curved.
1. Linear
: A straight relationship is called linear, because it approximates a straight line. The GPA, MathSAT example shows a relationship that is, roughly, a linear relationship.
2. Curvilinear
: A curved relationship is called curvilinear, because it approximates a curved line. An example of the relationship between the Miles-per-gallon and engine displacement of various automobiles sold in the USA in 1982 is shown below. This is curvilinear (and negative).





In this course we only deal with correlation coefficients that measure linear relationship. There are other correlation coefficients that measure curvilinear relationship, but they are beyond the introductory level.

3. The Degree (Strength) of a Relationship

Finally, a correlation coefficient measures the degree (strength) of the relationship between two variables. The mesures we discuss only measure the strength of the linear relationship between two variables. Two specific strengths are:

1. Perfect Relationship
: When two variables are exactly (linearly) related the correlation coefficient is either +1.00 or -1.00. They are said to be perfectly linearly related, either positively or negatively.
2. No relationship
: When two variables have no relationship at all, their correlation is 0.00.

There are strengths in between -1.00, 0.00 and +1.00. Note, though. that +1.00 is the largest postive correlation and -1.00 is the largest negative correlation that is possible. Here are three examples:

Weight and Horsepower




The relationship between Weight and Horsepower is strong, linear, and positive, though not perfect. The Pearson correlation coefficient is +.92.

Drive Ratio and Horsepower




The relationship between drive ratio and Horsepower is weekly negative, though not zero. The Pearson correlation coefficient is -.59.

Drive Ratio and Miles-Per-Gallon




The relationship between drive ratio and MPG is weekly positive, though not zero. The Pearson correlation coefficient is .42.

1. Prediction
: Correlations can be used to help make predictions. If two variables have been known in the past to correlate, then we can assume they will continue to correlate in the future. We can use the value of one variable that is known now to predict the value that the other variable will take on in the future.

For example, we require high school students to take the SAT exam because we know that in the past SAT scores correlated well with the GPA scores that the students get when they are in college. Thus, we predict high SAT scores will lead to high GPA scores, and conversely.

2. Validity
: Suppose we have developed a new test of intelligence. We can determine if it is really measuring intelligence by correlating the new test's scores with, for example, the scores that the same people get on standardized IQ tests, or their scores on problem solving ability tests, or their performance on learning tasks, etc.

This is a process for validating the new test of intelligence. The process is based on correlation.

3. Reliability
: Correlations can be used to determine the reliability of some measurement process. For example, we could administer our new IQ test on two different occasions to the same group of people and see what the correlation is. If the correlation is high, the test is reliable. If it is low, it is not.
4. Theory Verification
: Many Psychological theories make specific predictions about the relationship between two variables. For example, it is predicted that parents and children's intelligences are positively related. We can test this prediction by administering IQ tests to the parents and their children, and measuring the correlation between the two scores.

By Dr. Saul McLeod, updated 2020

Correlation means association - more precisely it is a measure of the extent to which two variables are related. There are three possible results of a correlational study: a positive correlation, a negative correlation, and no correlation.

A positive correlation is a relationship between two variables in which both variables move in the same direction. Therefore, when one variable increases as the other variable increases, or one variable decreases while the other decreases. An example of positive correlation would be height and weight. Taller people tend to be heavier.

A negative correlation is a relationship between two variables in which an increase in one variable is associated with a decrease in the other. An example of negative correlation would be height above sea level and temperature. As you climb the mountain (increase in height) it gets colder (decrease in temperature).

Correlation Definitions, Examples & Interpretation

A zero correlation exists when there is no relationship between two variables. For example there is no relationship between the amount of tea drunk and level of intelligence.

Scattergrams

A correlation can be expressed visually. This is done by drawing a scattergram (also known as a scatterplot, scatter graph, scatter chart, or scatter diagram).

A scattergram is a graphical display that shows the relationships or associations between two numerical variables (or co-variables), which are represented as points (or dots) for each pair of score.

A scattergraph indicates the strength and direction of the correlation between the co-variables.

When you draw a scattergram it doesn't matter which variable goes on the x-axis and which goes on the y-axis.

Remember, in correlations we are always dealing with paired scores, so the values of the 2 variables taken together will be used to make the diagram.

Decide which variable goes on each axis and then simply put a cross at the point where the 2 values coincide.

Some uses of Correlations

Prediction

• If there is a relationship between two variables, we can make predictions about one from another.

Validity

• Concurrent validity (correlation between a new measure and an established measure).

Reliability

• Test-retest reliability (are measures consistent).
• Inter-rater reliability (are observers consistent).

Theory verification

Correlation Coefficients: Determining Correlation Strength

Instead of drawing a scattergram a correlation can be expressed numerically as a coefficient, ranging from -1 to +1. When working with continuous variables, the correlation coefficient to use is Pearson’s r.

The correlation coefficient (r) indicates the extent to which the pairs of numbers for these two variables lie on a straight line. Values over zero indicate a positive correlation, while values under zero indicate a negative correlation.

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

There is no rule for determining what size of correlation is considered strong, moderate or weak. The interpretation of the coefficient depends on the topic of study.

When studying things that are difficult to measure, we should expect the correlation coefficients to be lower (e.g. above 0.4 to be relatively strong). When we are studying things that are more easier to measure, such as socioeconomic status, we expect higher correlations (e.g. above 0.75 to be relatively strong).)

In these kinds of studies, we rarely see correlations above 0.6. For this kind of data, we generally consider correlations above 0.4 to be relatively strong; correlations between 0.2 and 0.4 are moderate, and those below 0.2 are considered weak.

When we are studying things that are more easily countable, we expect higher correlations. For example, with demographic data, we we generally consider correlations above 0.75 to be relatively strong; correlations between 0.45 and 0.75 are moderate, and those below 0.45 are considered weak.

Correlation vs Causation

Causation means that one variable (often called the predictor variable or independent variable) causes the other (often called the outcome variable or dependent variable).

Experiments can be conducted to establish causation. An experiment isolates and manipulates the independent variable to observe its effect on the dependent variable, and controls the environment in order that extraneous variables may be eliminated.

A correlation between variables, however, does not automatically mean that the change in one variable is the cause of the change in the values of the other variable. A correlation only shows if there is a relationship between variables.

While variables are sometimes correlated because one does cause the other, it could also be that some other factor, a confounding variable, is actually causing the systematic movement in our variables of interest.

Correlation does not always prove causation as a third variable may be involved. For example, being a patient in hospital is correlated with dying, but this does not mean that one event causes the other, as another third variable might be involved (such as diet, level of exercise).

Summary

"Correlation is not causation" means that just because two variables are related it does not necessarily mean that one causes the other.

A correlation identifies variables and looks for a relationship between them. An experiment tests the effect that an independent variable has upon a dependent variable but a correlation looks for a relationship between two variables.

This means that the experiment can predict cause and effect (causation) but a correlation can only predict a relationship, as another extraneous variable may be involved that it not known about.

Strengths of Correlations

1. Correlation allows the researcher to investigate naturally occurring variables that maybe unethical or impractical to test experimentally. For example, it would be unethical to conduct an experiment on whether smoking causes lung cancer.


2. Correlation allows the researcher to clearly and easily see if there is a relationship between variables. This can then be displayed in a graphical form.

Limitations of Correlations

1. Correlation is not and cannot be taken to imply causation. Even if there is a very strong association between two variables we cannot assume that one causes the other.

For example suppose we found a positive correlation between watching violence on T.V. and violent behavior in adolescence. It could be that the cause of both these is a third (extraneous) variable - say for example, growing up in a violent home - and that both the watching of T.V. and the violent behavior are the outcome of this.


2. Correlation does not allow us to go beyond the data that is given. For example suppose it was found that there was an association between time spent on homework (1/2 hour to 3 hours) and number of G.C.S.E. passes (1 to 6). It would not be legitimate to infer from this that spending 6 hours on homework would be likely to generate 12 G.C.S.E. passes.

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How to reference this article:

McLeod, S. A. (2018, January 14). Correlation definitions, examples & interpretation. Simply Psychology. www.simplypsychology.org/correlation.html

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