How to Calculate Correlation Between Two Interval Levels

Correlation measures the statistical relationship between two interval-level variables. This guide explains how to calculate and interpret different types of correlation coefficients, with practical examples and an interactive calculator.

What is Correlation?

Correlation is a statistical measure that examines the relationship between two variables. When variables are correlated, changes in one variable are associated with changes in the other variable. Correlation does not imply causation, meaning that just because two variables are correlated doesn't mean one causes the other.

Correlation coefficients range from -1 to 1. A value of 1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no correlation.

Types of Correlation

There are several types of correlation coefficients, each suitable for different types of data:

Pearson Correlation Coefficient - Measures linear correlation between two continuous variables
Spearman's Rank Correlation - Measures monotonic relationships between two variables
Kendall's Tau - Measures ordinal association between two variables

This guide focuses on Pearson and Spearman correlation, which are most commonly used for interval-level data.

Pearson Correlation Coefficient

The Pearson correlation coefficient (r) measures the linear relationship between two continuous variables. The formula is:

r = Σ[(X - X̄)(Y - Ȳ)] / √[Σ(X - X̄)²Σ(Y - Ȳ)²]

Where:

X and Y are the variables
X̄ and Ȳ are the means of X and Y
Σ represents the sum of all values

The Pearson coefficient is most appropriate when:

Both variables are normally distributed
The relationship between variables is linear
There are no outliers in the data

Spearman's Rank Correlation

Spearman's rank correlation (ρ) measures the monotonic relationship between two variables. It's based on the ranked values rather than the actual values. The formula is:

ρ = 1 - [6Σd² / n(n² - 1)]

Where:

d is the difference between ranks of corresponding variables
n is the number of pairs

Spearman's correlation is appropriate when:

The relationship between variables is not strictly linear
Data is ordinal or not normally distributed
There are outliers in the data

How to Interpret Correlation Results

Interpreting correlation coefficients requires understanding the strength and direction of the relationship:

0.00 to 0.19 - Very weak correlation
0.20 to 0.39 - Weak correlation
0.40 to 0.59 - Moderate correlation
0.60 to 0.79 - Strong correlation
0.80 to 1.00 - Very strong correlation

The sign of the coefficient indicates the direction:

Positive coefficient (+) indicates a positive relationship
Negative coefficient (-) indicates a negative relationship

Remember that correlation does not imply causation. Just because two variables are correlated doesn't mean one causes the other.

Example Calculation

Let's calculate the Pearson correlation between hours studied (X) and exam scores (Y) for 5 students:

Student	Hours Studied (X)	Exam Score (Y)
1	2	85
2	4	90
3	3	88
4	5	92
5	1	80

Using the Pearson formula, we calculate:

r = Σ[(X - X̄)(Y - Ȳ)] / √[Σ(X - X̄)²Σ(Y - Ȳ)²]

Calculating the means: X̄ = 3, Ȳ = 87.4

Calculating the numerator: Σ[(X - X̄)(Y - Ȳ)] = 15.6

Calculating the denominator: √[Σ(X - X̄)²Σ(Y - Ȳ)²] = 10.2

Final result: r ≈ 0.63

This indicates a moderate positive correlation between hours studied and exam scores.

FAQ

What is the difference between Pearson and Spearman correlation?

Pearson correlation measures linear relationships between continuous variables, while Spearman correlation measures monotonic relationships based on ranked values. Pearson is more appropriate for normally distributed data with linear relationships, while Spearman works better with ordinal data or non-linear relationships.

How do I know which correlation coefficient to use?

Choose Pearson correlation if your data is continuous and normally distributed with a linear relationship. Use Spearman correlation if your data is ordinal or has a non-linear relationship. For small sample sizes, consider consulting with a statistician.

What does a correlation coefficient of 0.5 mean?

A correlation coefficient of 0.5 indicates a moderate positive relationship between the two variables. This means that as one variable increases, the other tends to increase as well, but the relationship isn't extremely strong.