How To Calculate Correlation Between Two Variables

Calculate the Pearson, Spearman, or Kendall correlation between two variables

How to Calculate Correlation Between Two Variables: A Comprehensive Guide

Correlation measures the statistical relationship between two continuous variables. Understanding how to calculate and interpret correlation is fundamental in statistics, research, and data analysis. This guide explains the different types of correlation coefficients, their calculation methods, and practical applications.

What is Correlation?

Correlation quantifies the degree to which two variables are related. It indicates:

• Direction: Positive (both increase together) or negative (one increases as the other decreases)
• Strength: Ranges from -1 (perfect negative) to +1 (perfect positive), with 0 indicating no relationship
• Linearity: Pearson correlation measures linear relationships specifically

Types of Correlation Coefficients

1. Pearson Correlation (r)

Measures linear relationships between normally distributed variables. Formula:

r = Σ[(X_i – X̄)(Y_i – Ȳ)] / √[Σ(X_i – X̄)² Σ(Y_i – Ȳ)²]

When to use: Both variables are continuous and normally distributed, with a linear relationship.

2. Spearman Rank Correlation (ρ)

Measures monotonic relationships (not necessarily linear) using ranked data. Formula:

ρ = 1 – [6Σd_i² / n(n² – 1)]

When to use: Variables are ordinal, or the relationship isn’t linear but consistent in direction.

3. Kendall Tau (τ)

Measures ordinal association based on the number of concordant vs. discordant pairs. Formula:

τ = (C – D) / √[(C + D)(C + D + T)]

When to use: Small datasets or when many tied ranks exist.

National Institute of Standards and Technology (NIST)

The NIST Engineering Statistics Handbook provides authoritative guidance on correlation analysis, including detailed explanations of Pearson, Spearman, and Kendall methods with real-world examples.

NIST Handbook on Correlation →

Step-by-Step Calculation Process

1. Data Collection

Gather paired observations (X, Y) for your variables. Example dataset:

Observation	X (Study Hours)	Y (Exam Score)
1	2	50
2	4	60
3	6	70
4	8	80
5	10	90

2. Pearson Correlation Calculation

1. Calculate means: X̄ = (2+4+6+8+10)/5 = 6; Ȳ = (50+60+70+80+90)/5 = 70
2. Compute deviations: (X_i – X̄) and (Y_i – Ȳ)
3. Multiply deviations: (X_i – X̄)(Y_i – Ȳ)
4. Sum products: Σ[(X_i – X̄)(Y_i – Ȳ)] = 280
5. Sum squared deviations:
   • Σ(X_i – X̄)² = 40
   • Σ(Y_i – Ȳ)² = 1000
6. Apply formula: r = 280 / √(40 × 1000) = 280 / 200 = 0.997

3. Interpretation

Correlation Strength	Absolute Value Range
Very weak	0.00 – 0.19
Weak	0.20 – 0.39
Moderate	0.40 – 0.59
Strong	0.60 – 0.79
Very strong	0.80 – 1.00

In our example, r = 0.997 indicates an almost perfect positive linear relationship between study hours and exam scores.

Statistical Significance Testing

To determine if the observed correlation is statistically significant:

1. State hypotheses:
   • H₀: ρ = 0 (no correlation)
   • H_a: ρ ≠ 0 (correlation exists)
2. Calculate test statistic:

   t = r√[(n – 2)/(1 – r²)]

   For our example: t = 0.997√[(5-2)/(1-0.997²)] ≈ 28.7
3. Determine critical value: For α = 0.05 (two-tailed) and df = n-2 = 3, critical t = ±3.182
4. Compare: |28.7| > 3.182 → reject H₀

UCLA Statistical Consulting

The UCLA Institute for Digital Research and Education offers comprehensive tutorials on correlation analysis, including how to perform calculations in R, Stata, and SPSS with sample datasets.

UCLA Correlation Analysis Guide →

Common Mistakes to Avoid

• Assuming causation: Correlation ≠ causation. A third variable may influence both.
• Ignoring nonlinearity: Pearson’s r only detects linear relationships. Use Spearman’s ρ for monotonic relationships.
• Outliers: Extreme values can artificially inflate or deflate correlation coefficients.
• Restricted range: Limited data ranges may underestimate true correlations.
• Ecological fallacy: Group-level correlations don’t necessarily apply to individuals.

Practical Applications

1. Finance: Correlation between stock returns to diversify portfolios (assets with r ≈ 0)
2. Medicine: Relationship between risk factors (e.g., smoking) and health outcomes
3. Marketing: Correlation between ad spend and sales revenue
4. Education: Relationship between study time and academic performance
5. Psychology: Validating survey scales (item-total correlations)

Advanced Topics

Partial Correlation

Measures the relationship between two variables after controlling for one or more additional variables. Formula:

r_xy.z = (r_xy – r_xzr_yz) / √[(1 – r_xz²)(1 – r_yz²)]

Semipartial Correlation

Similar to partial correlation but only removes the influence of the control variable from one of the primary variables.

Nonparametric Alternatives

For non-normal data or small samples:

• Spearman’s ρ: Rank-based Pearson correlation
• Kendall’s τ: Based on concordant/discordant pairs
• Hoeffding’s D: Measures general dependence

Software Implementation

Most statistical software can compute correlations:

• Excel: =CORREL(array1, array2) for Pearson
• R: cor(x, y, method="pearson")
• Python:

  from scipy.stats import pearsonr, spearmanr, kendalltau
  r, p = pearsonr(x, y)  # Returns (correlation, p-value)

• SPSS: Analyze → Correlate → Bivariate

Real-World Example: Height vs. Weight

A classic example in biostatistics examines the relationship between height and weight in adults. A study of 1000 individuals might yield:

Statistic	Value	Interpretation
Pearson r	0.72	Strong positive linear relationship
Spearman ρ	0.71	Consistent with Pearson (linear relationship)
p-value	< 0.001	Statistically significant
R-squared	0.52	52% of weight variance explained by height

National Center for Health Statistics (NCDC)

The NCHS provides national health statistics where correlation analyses are frequently applied, such as in growth charts and health indicator relationships. Their methodological guidelines are considered gold standards for health data analysis.

NCHS Health Statistics →

Frequently Asked Questions

Can correlation be greater than 1 or less than -1?

No. The mathematical properties of correlation coefficients constrain them to the [-1, 1] range. Values outside this range indicate calculation errors.

What’s the difference between correlation and regression?

Correlation measures the strength/direction of a relationship. Regression models the relationship to predict one variable from another. Correlation is symmetric (r_xy = r_yx); regression is not (predicting Y from X ≠ predicting X from Y).

How many data points are needed for reliable correlation?

Minimum recommendations:

• Pearson: At least 20-30 observations for stable estimates
• Spearman/Kendall: Can work with as few as 5-10 observations

More data improves reliability. For publication-quality results, aim for ≥100 observations.

What does a correlation of 0.4 mean?

A correlation of 0.4 indicates a moderate positive relationship. The coefficient of determination (r² = 0.16) means 16% of the variance in one variable is explained by the other. While statistically significant with sufficient sample size, practical significance depends on the context.

How do I report correlation results in APA format?

Example: “Study time and exam scores were strongly positively correlated, r(8) = .997, p < .001, 95% CI [0.98, 1.00]." Include:

• Correlation coefficient (r, ρ, or τ)
• Degrees of freedom (n-2)
• Exact p-value
• Confidence interval (recommended)