The Pearson correlation coefficient, written as r, measures how strongly two quantitative variables are related in a straight-line pattern. It also tells the direction of that relationship, using positive values for variables that tend to increase together and negative values for one variable increasing while the other decreases. Correlation is important because it gives a compact numerical summary of patterns seen in a scatterplot.
It is widely used in science, economics, psychology, and engineering to compare data sets and test for linear trends.
The value of r always lies between -1 and +1. Values near +1 show a strong positive linear relationship, values near -1 show a strong negative linear relationship, and values near 0 show little or no linear relationship. Pearson correlation is based on how each data point varies from the mean of x and the mean of y, then scales that shared variation by the spread of each variable.
A high correlation can support prediction, but it does not prove that one variable causes the other.
Understanding Statistics: The Correlation Coefficient
Correlation is built from distances from the average. For each student or object in a data set, find how far its x value is above or below the x average. Do the same for its y value.
A point that is above average on both variables gives a positive contribution. A point that is below average on both gives a positive contribution too, because two negative distances multiply to a positive result. When one value is above average while the other is below, the contribution is negative.
Adding these contributions reveals whether the variables tend to move in the same direction or in opposite directions. The calculation then divides by the overall spread in each variable.
This scaling makes correlation unit free. The correlation between height in centimetres and mass in kilograms is unchanged if height is converted to metres.
A scatterplot should come before the coefficient. A single number cannot show the full shape of the data. A curved pattern can have a correlation near zero even when the variables are closely connected.
For example, a car's stopping distance may rise much faster as speed increases. The points make a clear curve, yet a straight line is not a good description. Separate groups can cause trouble too.
Combining data from children and adults might produce a strong relationship between shoe size and reading score, even if shoe size has little link to reading within either age group. Looking at groups, curves, gaps, and clusters prevents an overconfident conclusion.
Outliers deserve special attention because correlation can change sharply when only one unusual point is added. Imagine most points form a loose cloud, then one student has an extremely high study time and an extremely high test score. That point may make the correlation seem strong.
Another distant point can weaken a relationship that is otherwise clear. Students should check whether an outlier is a recording mistake, a rare real case, or evidence that the data contain different groups.
Removing a real observation just to improve a result is not good practice. Report what was found and explain how unusual points affect the pattern.
Correlation is often used in everyday claims about health, school results, money, and technology. A study may find that people who sleep more tend to have higher grades. This does not show that extra sleep alone produces higher grades.
Family routines, stress, study habits, or illness could affect both sleep and grades. Sometimes the apparent direction can even be reversed, with poor grades leading to less sleep. Controlled experiments are better for testing causes, though they are not always possible or ethical.
When interpreting a correlation, name the two measured variables, inspect the scatterplot, consider other influences, and state the conclusion carefully. Correlation can support a useful prediction, but predictions still have uncertainty because individual points do not sit exactly on the trend.
Key Facts
- Pearson correlation coefficient r measures the strength and direction of a linear relationship between two quantitative variables.
- The range is -1 ≤ r ≤ 1.
- r = 1 means a perfect positive linear relationship, and r = -1 means a perfect negative linear relationship.
- r = 0 means no linear correlation, but a nonlinear pattern may still exist.
- Formula: r = Σ((xi - x̄)(yi - ȳ)) / sqrt(Σ(xi - x̄)^2 Σ(yi - ȳ)^2).
- The sign of r gives direction, while the magnitude |r| gives strength.
Vocabulary
- Pearson correlation coefficient
- A statistic, written as r, that measures the strength and direction of a linear relationship between two quantitative variables.
- Scatterplot
- A graph that displays paired data values as points on an x-y coordinate plane.
- Positive correlation
- A relationship in which larger values of one variable tend to occur with larger values of the other variable.
- Negative correlation
- A relationship in which larger values of one variable tend to occur with smaller values of the other variable.
- Linear relationship
- A pattern in data that is well described by a straight line.
Common Mistakes to Avoid
- Saying r = 0 means there is no relationship at all is wrong because r only measures linear relationships, so curved patterns can have little linear correlation.
- Treating correlation as causation is wrong because a strong r value does not prove that one variable directly causes changes in the other.
- Using r for categorical data is wrong because Pearson correlation requires paired quantitative measurements.
- Ignoring outliers is wrong because a single extreme point can greatly change the value of r and make the linear relationship look stronger or weaker than it is.
Practice Questions
- 1 A data set has r = 0.82 between hours studied and exam score. Describe the direction and strength of the linear relationship.
- 2 For the paired data (1, 2), (2, 4), (3, 6), and (4, 8), what is the Pearson correlation coefficient r?
- 3 A scatterplot curves upward in a clear U-shape, but its Pearson correlation coefficient is close to 0. Explain why r can be close to 0 even when the variables are related.