Pearson vs Spearman Correlation: Choose Based on the Relationship, Not One Normality Test

Researchers are often given a simple rule for Pearson vs Spearman: use Pearson when the variables are normally distributed and Spearman when they are not. That rule is too crude.

Relationship form, measurement scale, unusual observations, and the type of inference being made therefore matter alongside distributional assumptions. (Moore et al., 2014, 2021; Adams & Lawrence, 2018; Lovric, 2011).

A normality test on each variable cannot answer all of those questions. Even marginally normal variables can have a relationship for which Pearson’s r is a poor summary, while non-normal variables can still exhibit a clear linear pattern. The scatterplot and the scientific meaning of the coefficient should come before a mechanical “normal/non-normal” switch. (Moore et al., 2014, 2021; Tabachnick & Fidell, 2013).

What Does Correlation Measure?

A correlation coefficient reduces the relationship between two variables to a number describing its direction and strength. But different correlation coefficients summarize different aspects of that relationship. (Devore & Berk, 2012; Moore et al., 2014).

The word linear is essential. Moore et al. (2014) describe Pearson’s r as measuring the direction and strength of linear, straight-line association and caution that a correlation can be calculated for any scatterplot even though the coefficient summarizes only linear relationships. Devore and Berk (2012) similarly show that a value near zero may reflect the absence of a linear relationship rather than the absence of a relationship altogether.

Linear and Monotonic Are Not the Same Thing

A linear relationship can be represented adequately by a straight line. Pearson correlation is constructed to summarize this form of association. If the relationship is strong but substantially curved, Pearson’s r may understate or otherwise poorly represent it. (Moore et al., 2014, 2021; Tabachnick & Fidell, 2013).

A monotonic relationship preserves ordering: as one variable increases, the other generally moves in one direction rather than repeatedly reversing direction. The rate of change does not have to be constant. A relationship can therefore be strongly ordered while clearly curved. Spearman correlation can remain informative in this situation because it operates on ranks rather than requiring the raw observations to align around a straight line. (Adams & Lawrence, 2018; Lovric, 2011).

Start With the Scatterplot

Before calculating Pearson’s r, Adams and Lawrence (2018) recommend graphing the relationship. Moore et al. (2014, 2021) make the same point more broadly: always plot the data and examine the form, direction, strength, and unusual observations before relying on a correlation coefficient.

For two quantitative variables, inspect the scatterplot for:

• an approximately straight-line pattern;
• a consistently increasing or decreasing but curved pattern;
• reversals in direction, such as U-shaped or inverted-U relationships;
• separate clusters or subgroups;
• restricted ranges;
• isolated observations that may dominate the coefficient. (Moore et al., 2014, 2021; Adams & Lawrence, 2018; Tabachnick & Fidell, 2013).

A coefficient should summarize a pattern that actually exists in the plot. It should not substitute for looking at that pattern.

Four Synthetic Relationship Shapes

Measurement Scale Matters

Pearson’s r is formulated for quantitative variables whose numerical values can meaningfully enter calculations involving means, standard deviations, and cross-products. Moore et al. (2014, 2021) explicitly define Pearson correlation for two quantitative variables. (Moore et al., 2014, 2021).

Spearman’s rho is naturally applicable when the variables themselves are ordinal or when the research question concerns relative ordering. Adams and Lawrence (2018) define Spearman’s rho as a correlation between ordinal variables and show that its interpretation concerns correspondence between ranks. (Adams & Lawrence, 2018).

Outliers Are a Relationship Problem, Not Just a Normality Problem

A marginal histogram can reveal an extreme observation, but the important issue for correlation is often how a case sits in the two-variable configuration. A point may be unusual because of one extreme value or because its combination of values is unusual relative to the rest of the sample. (Tabachnick & Fidell, 2013).

Pearson’s r can be strongly affected by such points because it is not resistant. A single observation can strengthen, weaken, or even reverse the apparent linear association. This is another reason a scatterplot is essential. (Moore et al., 2014, 2021; Tabachnick & Fidell, 2013).

Do not automatically delete an outlier. Adams and Lawrence (2018) recommend first considering whether it reflects data-entry error, misunderstanding, or a genuine extreme response. A valid extreme observation may be scientifically important rather than disposable. Any exclusion or alteration needs a defensible rationale. (Adams & Lawrence, 2018).

Computing Pearson’s r Is Not the Same as Making Inference About It

This distinction resolves much of the confusion around Pearson correlation assumptions.

For an observed sample of paired quantitative values, Pearson’s sample correlation can be calculated directly from those observations. Devore and Berk (2012) define r as a descriptive sample measure of how strongly the two variables are linearly related in the observed data. Computing that coefficient is conceptually different from using the sample value to test or estimate a population correlation. (Devore & Berk, 2012).

Inference introduces additional assumptions because the goal has changed. You are no longer merely describing the sample: you are using the sample correlation as evidence about an underlying population parameter. Devore and Berk (2012), for example, derive small-sample inferential procedures for the population correlation under a bivariate normal model for the paired observations.

That is a joint assumption about X and Y, not simply two disconnected statements that “X passed a normality test” and “Y passed a normality test.” Tabachnick and Fidell (2013) make the more general point that univariate normality of individual variables does not guarantee multivariate normality; all relevant linear combinations need not be normally distributed merely because each variable is individually normal.

Why Marginal Normality Alone Is Not a Complete Decision Rule

Suppose a researcher runs a normality test on X and another on Y.

Even if both look approximately normal, that result does not establish that their relationship is linear, that there are no influential observations, or that the joint distribution satisfies the assumptions of a particular inferential procedure. Tabachnick and Fidell (2013) explicitly note that univariate normality does not guarantee multivariate normality and recommend using bivariate scatterplots to assess linearity. (Tabachnick & Fidell, 2013).

Conversely, if one marginal distribution is skewed, that fact alone does not describe whether the observed X–Y relationship is approximately linear. Pearson’s r remains a calculable descriptive measure of linear association; whether a particular inferential procedure based on it is justified is a separate issue. (Devore & Berk, 2012).

Pearson vs Spearman Decision Matrix

Data pattern	Pearson consideration	Spearman consideration	Diagnostic to inspect
Approximately straight-line relationship between quantitative variables	Directly targets linear direction and strength.	Can describe rank ordering but changes the emphasis from raw-value linear association to ranks.	Scatterplot; unusual points; range of both variables.
Strong increasing or decreasing relationship with noticeable curvature	May incompletely summarize the relationship because r captures only the linear component.	Consider when the scientifically relevant pattern is consistent ordering rather than straight-line change.	Scatterplot with attention to whether direction reverses.
U-shaped or inverted-U relationship	A value near zero can coexist with a strong nonlinear relationship.	Do not assume Spearman fixes the problem; a nonmonotonic relationship can also be poorly summarized by one rank coefficient.	Scatterplot; check for reversals in direction.
One or a few extreme observations dominate the pattern	Use cautiously because Pearson’s r is not resistant.	Ranking can change the impact of raw magnitudes, but the observation still requires investigation rather than automatic dismissal.	Scatterplot; data-entry verification; sensitivity analysis with transparent justification.
Two genuinely quantitative variables but skewed marginal distributions	Skewness alone does not determine whether the relationship is linear; separate descriptive calculation from inferential assumptions.	Consider only if rank-based association matches the research question or other conditions motivate ranking.	Scatterplot plus distribution plots; evaluate assumptions for the specific inferential procedure.
Ordinal variables	Pearson may impose a quantitative interpretation that the scale does not support.	Natural candidate because Spearman analyzes correspondence between ranks.	Measurement definition, category ordering, number of ties.
Many tied ranks	Not a Pearson-versus-Spearman decision based on normality; first determine whether quantitative measurement is defensible.	Ties deserve explicit attention because extensive ranking/ties can reduce information and sensitivity.	Frequency tables and tied-rank pattern.
Marginals appear normal	This does not establish linearity, absence of outliers, or joint normality.	Normality alone is not a reason to reject a rank-based coefficient if rank association is the actual target.	Scatterplot; joint pattern; inferential assumptions.
Marginals fail a normality test	Do not automatically discard Pearson; determine relationship form and distinguish sample description from inference.	Do not automatically select Spearman solely because of the test result.	Scatterplot, outliers, scale, shape, and the assumptions of the planned inference.

The matrix reflects a broader principle across the sources: choose the statistic for the relationship it is intended to summarize and then check the assumptions relevant to the interpretation you plan to make. (Moore et al., 2014, 2021; Devore & Berk, 2012; Adams & Lawrence, 2018; Tabachnick & Fidell, 2013).

How Interpretation Changes

A Pearson coefficient should be interpreted explicitly as a linear correlation. A positive value means higher values of one variable tend to accompany higher values of the other in a straight-line pattern; its magnitude describes how closely the observations follow that pattern. (Moore et al., 2014, 2021).

A Spearman coefficient should be interpreted as a rank-order association. A positive coefficient indicates that higher ranks on one variable tend to accompany higher ranks on the other; a negative coefficient indicates opposite ordering. Its interpretation concerns concordance in ordering rather than a constant raw-unit increase in one variable for a unit increase in the other. (Adams & Lawrence, 2018; Lovric, 2011).

A Practical Pearson-or-Spearman Workflow

When deciding Pearson vs Spearman, use this sequence:

1. Define the research question. Are you interested in linear association between quantitative values or correspondence in ordering/ranks? (Moore et al., 2021; Adams & Lawrence, 2018).
2. Check the measurement scale. Decide whether the observed values support quantitative arithmetic or represent ordered categories/ranks. (Moore et al., 2021; Adams & Lawrence, 2018).
3. Plot the paired data. Examine form, direction, strength, clusters, and unusual points before calculating a summary coefficient. (Moore et al., 2014, 2021).
4. Classify the relationship shape. Distinguish approximately linear, monotonic curved, and nonmonotonic patterns. Pearson directly addresses the first; Spearman can address consistent rank ordering, but neither is automatically sufficient for a relationship that reverses direction. (Devore & Berk, 2012; Adams & Lawrence, 2018; Tabachnick & Fidell, 2013).
5. Investigate outliers rather than merely detecting them. Verify data quality and determine whether an unusual case is genuine and influential. (Adams & Lawrence, 2018; Tabachnick & Fidell, 2013).
6. Separate description from inference. Computing a sample coefficient and testing a population correlation are different tasks with different assumptions. (Devore & Berk, 2012).
7. Check the assumptions of the particular inferential procedure. Do not reduce this step to marginal normality testing. Joint distributional conditions, linearity, sampling, and other features can matter. (Devore & Berk, 2012; Tabachnick & Fidell, 2013).
8. Interpret the coefficient you actually estimated. Call Pearson a linear correlation and Spearman a rank-order correlation; do not treat them as interchangeable estimates of exactly the same feature. (Moore et al., 2021; Adams & Lawrence, 2018).

Common Mistakes

Bottom Line

For correlation with non normal data, do not make a Pearson-versus-Spearman decision from one normality test.

Use Pearson when the target is the strength and direction of an appropriately measured linear relationship, while remaining alert to outliers and to the additional assumptions needed for inference. Use Spearman when a rank-order relationship is the meaningful target, including settings involving ordinal measurements or consistent ordering that is not adequately described as straight-line change. (Moore et al., 2014, 2021; Adams & Lawrence, 2018; Lovric, 2011).

Most importantly, distinguish the observed coefficient from the inference made about a population. Pearson’s r can be computed from paired quantitative observations without first passing two marginal normality tests; particular hypothesis tests or confidence procedures can impose stronger distributional assumptions. Under the small-sample correlation inference developed by Devore and Berk (2012), that assumption concerns the joint bivariate distribution. (Devore & Berk, 2012; Tabachnick & Fidell, 2013).

References

Adams, K. A., & Lawrence, E. K. (2018). Research methods, statistics, and applications (2nd ed.). SAGE Publications.

Devore, J. L., & Berk, K. N. (2012). Modern mathematical statistics with applications (2nd ed.). Springer.

Lovric, M. (Ed.). (2011). International encyclopedia of statistical science. Springer. https://doi.org/10.1007/978-3-642-04898-2

Moore, D. S., McCabe, G. P., & Craig, B. A. (2014). Introduction to the practice of statistics (8th ed.). W. H. Freeman.

Moore, D. S., McCabe, G. P., & Craig, B. A. (2021). Introduction to the practice of statistics (10th ed.). Macmillan Learning.

Tabachnick, B. G., & Fidell, L. S. (2013). Using multivariate statistics (6th ed.). Pearson.