An Introduction to Rank Correlations: Spearman’s Rho and Kendall’s Tau

When psychologists want to measure the relationship between two variables, they reach for correlation analysis. But not all data behaves the same way. Some variables are neatly measured on continuous scales, normally distributed, and perfectly suited for the workhorse of correlation statistics – Pearson’s r. Many variables in psychology, however, don’t play by those rules. They come as rankings, Likert-scale ratings, or skewed distributions that violate the assumptions Pearson’s correlation demands. That’s precisely where rank correlations come in – specifically, Spearman’s rho (ρ) and Kendall’s tau (τ).

What are rank correlations?
Why Pearson’s correlation isn’t always enough
Understanding monotonic relationships
Spearman’s rho (ρ): the rank-based Pearson
When to use Spearman’s rho
Kendall’s tau (τ): concordance and discordance
Numerical values: why Kendall’s tau is typically smaller
Spearman’s rho vs. Kendall’s tau: which should you use?
Practical applications in psychology
The bigger picture: choosing the right correlation

What are rank correlations?

Rank correlations are non-parametric statistical methods used to measure the strength and direction of association between two variables. Rather than working with raw data values, these methods first convert the data into ranks – ordering observations from smallest to largest – and then assess how consistently those ranks correspond across the two variables.

According to the University of Virginia Library, correlation coefficients from these methods range from −1 to +1, where −1 indicates a perfect negative relationship, +1 a perfect positive one, and 0 signals no detectable association. This range is the same as Pearson’s, which makes the output immediately familiar – even though the underlying calculation is very different.

The key advantage of rank correlations is that they sidestep the strict assumptions of parametric tests. Statistics Solutions notes that both Spearman’s rho and Kendall’s tau are particularly useful when data does not meet the requirements for parametric tests, such as normally distributed data or interval-level measurement.

Why Pearson’s correlation isn’t always enough

Pearson’s correlation coefficient is the most widely used measure of association, but it carries specific requirements. The data must be interval or ratio scaled, the relationship between variables must be linear, and both variables should follow a roughly normal distribution. In practice, psychological research frequently violates one or more of these conditions.

A tutorial published in Anesthesia and Analgesia (Schober et al., 2018) explains that for non-normally distributed continuous data, ordinal data, or data with meaningful outliers, a Spearman rank correlation is the appropriate alternative for measuring a monotonic association. In other words, when the linearity or normality assumption breaks down, Pearson’s coefficient may give a misleading picture – and rank correlations offer a more honest one.

Consider a study examining the relationship between socioeconomic status (ranked from low to high) and levels of reported life satisfaction on a 5-point scale. Neither variable is truly continuous or normally distributed, and the interval between response categories can’t be assumed to be equal. Pearson’s r would be inappropriate here. Rank correlations are built precisely for this situation.

Understanding monotonic relationships

A central concept for rank correlations is the monotonic relationship. Unlike a linear relationship – where one variable increases at a constant rate as the other increases – a monotonic relationship simply requires that both variables move in the same direction, but not necessarily at a steady rate.

Laerd Statistics explains that a monotonic relationship does one of two things: as one variable increases, the other also increases; or as one variable increases, the other decreases. The change in direction never reverses. This is what makes monotonic relationships “less restrictive” than linear ones. For example, the link between years of professional experience and job competency might follow a monotonic pattern – competency generally improves with experience, but not at a perfectly constant, linear rate.

Rank correlations are designed to detect these monotonic patterns. They will return near-zero values for relationships that genuinely reverse direction (non-monotonic), so it’s worth verifying – ideally via a scatter plot – that the relationship you’re examining is at least roughly monotonic before interpreting the coefficient.

Spearman’s rho (ρ): the rank-based Pearson

Spearman’s rho, also written as rs, is the older and more widely used of the two rank correlation methods. Laerd Statistics describes it as the non-parametric version of the Pearson product-moment correlation – it works by converting raw data into ranks and then applying the Pearson correlation formula to those ranks. Because it operates on ranks rather than raw scores, the assumption of normality no longer applies.

Spearman’s rho is calculated using the differences between the paired ranks of two variables. The formula is straightforward when there are no tied ranks. When ties exist – two or more observations sharing the same rank – a slightly adjusted version is used, or the full Pearson formula is applied to the ranked data directly.

When to use Spearman’s rho

Spearman’s rho is the right choice in several common research situations:

Ordinal data: When variables are measured on an ordinal scale – such as Likert-type ratings, ranked preferences, or educational level (high school, bachelor’s, postgraduate) – Pearson’s correlation is not appropriate because the distances between categories cannot be assumed equal. Statistics Solutions confirms that Spearman’s rank correlation is appropriate when variables are measured on a scale that is at least ordinal.
Non-normal distributions: When data is skewed or contains outliers that would distort a parametric correlation, Spearman’s rho provides more stable results. Research published in the OPTIMA Journal of Guidance and Counseling found that Spearman’s correlation provides more stable results for non-normally distributed data or data containing outliers compared to Pearson’s.
Monotonic but non-linear relationships: When two variables tend to move together but not at a constant rate, Spearman’s rho captures the relationship where Pearson’s would underestimate it.

The coefficient is interpreted the same way as Pearson’s r: values closer to +1 or −1 indicate stronger associations, and the sign indicates direction. This familiarity makes it accessible to researchers who are already comfortable with Pearson’s correlation.

Kendall’s tau (τ): concordance and discordance

Kendall’s tau takes a different approach to measuring rank correlation. Rather than computing differences between ranks, it examines all possible pairs of observations and classifies each pair as either concordant or discordant.

A pair is concordant if both variables move in the same direction – that is, whichever observation ranks higher on variable X also ranks higher on variable Y. A pair is discordant if the ranks move in opposite directions. Kendall’s tau is then calculated as the difference between the proportion of concordant and discordant pairs. Statistics Solutions highlights that this gives Kendall’s tau a particularly direct probabilistic interpretation: a positive tau means concordant pairs outnumber discordant pairs, and vice versa.

There are several variants of Kendall’s tau – tau-a, tau-b, and tau-c. Tau-b is the most commonly reported version, as it accounts for tied ranks by adjusting the formula accordingly. StatPlus documentation notes that Kendall’s tau approaches a normal distribution more rapidly than Spearman’s rho as sample size increases, and is more accurate when ties are present.

Numerical values: why Kendall’s tau is typically smaller

One thing to be aware of when comparing the two methods: Kendall’s tau typically produces a numerically smaller coefficient than Spearman’s rho for the same dataset. Statistics Solutions confirms this clearly – Spearman’s rho values tend to be larger than Kendall’s tau values. This is not an error or a sign that one is more accurate; it is simply a consequence of the different computational approaches. The two coefficients are measuring related but not identical things.

Despite this numerical difference, the two measures generally lead to the same substantive conclusions – the direction and significance of the relationship will usually agree, even if the exact coefficient values differ.

Spearman’s rho vs. Kendall’s tau: which should you use?

Both coefficients measure monotonic association, both are non-parametric, and both are appropriate for ordinal or non-normally distributed data. The choice between them depends on the specifics of the data and what the researcher prioritises.

A simulation study published in Animal Behaviour (Puth et al., 2015) found that for data without tied values, Kendall’s tau produces consistently narrower confidence intervals, which could make it preferable in those contexts. However, when data contains ties, Spearman’s rho returns coverage rates closer to the desired level, while Kendall’s tau can become less accurate as the number of tied values increases.

Research on psychiatric rating scales found that Kendall’s tau-b maintained better control of Type I errors and produced tighter confidence intervals compared to both Pearson’s r and Spearman’s rho – particularly valuable when working with smaller clinical samples. Julius AI’s statistical guide summarises the practical trade-offs neatly: Kendall’s tau offers better statistical properties in its distribution and more accurate p-values with smaller sample sizes, while Spearman’s rho tends to be more widely reported and more intuitive for researchers already familiar with Pearson’s correlation.

A useful rule of thumb: use Spearman’s rho when you have a larger dataset, the results need to be compared against Pearson’s values, or there is a moderate amount of tied data. Use Kendall’s tau when sample sizes are small, precise p-values matter, and tied values are minimal.

Practical applications in psychology

Rank correlations are widely used across many areas of psychological research. In personality psychology, researchers examining the relationship between extraversion ratings and leadership effectiveness scores – both measured on ordinal scales – rely on Spearman’s rho or Kendall’s tau to capture associations that may not be perfectly linear. In clinical psychology, rating scales for symptom severity (such as depression or anxiety inventories) produce ordinal data where Pearson’s assumptions are routinely violated.

In educational psychology, studying the relationship between students’ ranked class performance and their ranked levels of academic self-efficacy is a natural fit for rank correlations. The data is inherently ordinal, the relationship is expected to be monotonic rather than linear, and the sample size may be modest – exactly the conditions these methods were designed for.

Even in survey-based research, SurveyMonkey’s research resources note that Spearman’s correlation is ideal for Likert-scale questions – whether examining whether higher satisfaction ratings align with higher loyalty scores, or whether preference rankings for product features correspond to purchase intent rankings. Because Spearman’s rho is based on ranks, it remains stable even when a small number of extreme responses might distort a Pearson correlation.

The bigger picture: choosing the right correlation

Rank correlations – Spearman’s rho and Kendall’s tau – exist because real data is messy. Psychological variables rarely conform to the tidy assumptions of parametric statistics. Attitudes, emotions, social rankings, clinical symptoms, and performance on psychological tasks all generate data that is better suited to rank-based analysis.

Choosing between Pearson’s, Spearman’s, or Kendall’s methods is not just a technical detail – it directly affects whether your findings are valid. A Pearson correlation applied to ordinal Likert data can produce distorted coefficients; a rank correlation on the same data produces a trustworthy measure of association. Understanding what each method assumes, and what each one actually measures, is foundational to conducting and interpreting psychological research accurately.

Both Spearman’s rho and Kendall’s tau measure the same underlying concept – the tendency for two variables to move together in a consistent direction – but through different computational lenses. In most practical cases, they will agree. When they don’t, the discrepancy itself can be informative about the nature of the data.

What do you think? If you were designing a psychology study that used Likert-scale questionnaires, how would you decide whether to use Spearman’s rho or Kendall’s tau – and what features of your data would guide that choice? And more broadly, does knowing the difference between a monotonic and a linear relationship change how you think about measuring associations in psychological research?

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Table of Contents

What are rank correlations? 🔗

Why Pearson’s correlation isn’t always enough 🔗

Understanding monotonic relationships 🔗

Spearman’s rho (ρ): the rank-based Pearson 🔗

When to use Spearman’s rho 🔗

Kendall’s tau (τ): concordance and discordance 🔗

Numerical values: why Kendall’s tau is typically smaller 🔗

Spearman’s rho vs. Kendall’s tau: which should you use? 🔗

Practical applications in psychology 🔗

The bigger picture: choosing the right correlation 🔗