Rank-Order Correlations: Spearman’s Rho and Kendall’s Tau Explained

Not all data in psychology arrives neatly on a ruler-like scale. Researchers routinely collect rankings – who scored highest, which symptom appeared first, how patients ordered their preferences – and this kind of data demands a different statistical approach. When the familiar Pearson’s r isn’t appropriate because your data is ordinal, your distribution is skewed, or your relationship is consistent but not perfectly straight, rank-order correlations step in. The two most widely used are Spearman’s rho (ρ) and Kendall’s tau (τ). Understanding what they measure, how they differ, and when to use each one is an essential part of working with psychological data.

What is ordinal (rank-order) data? 🔗

Before diving into the formulas, it helps to be clear about what makes data “rank-order.” Ordinal data is data measured on a scale where the levels can be ordered – one value is clearly higher or lower than another – but the size of the gap between adjacent levels is not necessarily equal or known. A graduate degree ranks above a bachelor’s degree, which ranks above a high school diploma, but we cannot say the gap between the first two is the same as the gap between the last two. Similarly, if students are ranked 1st, 2nd, and 3rd on a test, we know their order but not by exactly how many marks they differed. This is fundamentally different from interval or ratio data, where equal spacing between values can be assumed.

In ordinal data, the sequence matters, but the exact differences between ranks do not. Many common psychological measurements – Likert-scale ratings, clinical severity rankings, preference orderings, and competency assessments – are ordinal by nature. Using a correlation method that assumes equal intervals or a normal distribution on this kind of data will produce misleading results. That’s why rank-order correlations exist.

Why Pearson’s r isn’t always enough 🔗

Pearson’s correlation requires several conditions to be met: the relationship between variables must be linear, the data must be measured on an interval or ratio scale, and both variables should follow approximately normal distributions. In psychology, these conditions are frequently violated. Behavioral data is often skewed, rating scales are ordinal, and relationships between variables are not always linear. When these assumptions break down, Pearson’s r becomes unreliable or simply inappropriate.

Rank-order correlations sidestep these requirements entirely. Rather than working with raw numbers, they convert data into ranks and assess whether there is a monotonic relationship – one where, as one variable increases, the other consistently increases (or consistently decreases), even if the rate of change is not constant. This makes them suitable for a much wider range of psychological research situations.

Spearman’s rho (ρ): the nonparametric workhorse 🔗

Spearman’s rho, developed by British psychologist and statistician Charles Spearman in 1904, is the most widely used rank-order correlation in psychology. It is technically computed by applying Pearson’s formula to the ranks of the data rather than to the raw scores. The result is a coefficient, denoted ρ or r_s, that ranges from −1 to +1, where +1 indicates a perfect positive monotonic relationship, −1 indicates a perfect negative monotonic relationship, and 0 indicates no monotonic association.

How Spearman’s rho is calculated 🔗

The most common formula for Spearman’s rho when there are no tied ranks is:

ρ = 1 − (6Σd² / n(n²−1))

Here, d is the difference between the ranks assigned to each pair of scores, and n is the number of observations. The calculation involves ranking each variable independently, finding the difference in ranks for each paired observation, squaring those differences, summing them, and plugging the result into the formula. When tied ranks exist – which is common in psychological research, especially with Likert scales – each group of tied observations is assigned the average of the ranks they would have occupied. If three participants tie for 2nd place, each receives rank 3 (the average of 2, 3, and 4), and the next participant receives rank 5.

What makes Spearman’s rho useful in psychology 🔗

Spearman’s rho has three key practical advantages. First, it is robust to outliers: because it operates on ranks rather than raw scores, a single extreme value does not distort the coefficient the way it would in Pearson’s correlation. Second, it requires no distributional assumptions – there is no need for normally distributed data. Third, its output is interpreted exactly like Pearson’s r, so researchers already familiar with standard correlation analysis find it immediately accessible.

As a non-parametric measure, Spearman’s rho is the appropriate choice when variables are intrinsically ordinal, when continuous data is heavily skewed or contains extreme outliers, or when the researcher simply wants to assess the strength and direction of a monotonic – rather than strictly linear – relationship. According to simulation research comparing rank-order correlation methods, when data contains tied values, Spearman’s rho tends to return coverage rates closer to the desired levels than Kendall’s tau, which gives it a practical edge in many psychological data sets where tied responses are common.

Kendall’s tau (τ): the concordance-based alternative 🔗

Kendall’s tau, developed by Maurice Kendall in 1938, approaches rank correlation from a fundamentally different angle. Rather than computing differences between ranks and applying a formula, it systematically examines every possible pair of observations and asks a simple question: do the two variables agree or disagree in their ordering for this pair?

Concordant and discordant pairs 🔗

Two observations are called concordant if both variables rank them in the same relative order – for instance, if participant A scores higher than participant B on both Variable X and Variable Y. They are discordant if one variable ranks A above B while the other ranks B above A. Kendall’s tau is then calculated as:

τ = (C − D) / (C + D)

where C is the number of concordant pairs and D is the number of discordant pairs. Like Spearman’s rho, the result ranges from −1 to +1. A τ of +1 means every pair is concordant (perfect agreement in ordering), −1 means every pair is discordant (perfect reversal), and 0 means concordant and discordant pairs are equally frequent. The most widely used version in practice is tau-b, which applies a correction for tied ranks and is built into most statistical software packages, including SPSS.

What makes Kendall’s tau distinct 🔗

Kendall’s tau has a particularly clean probabilistic interpretation: it directly estimates the difference between the probability that any two randomly selected observations are in the same rank order and the probability that they are in the opposite order. Penn State’s statistics program notes that because tau-b estimates a population parameter directly, many statisticians consider it conceptually superior to Spearman’s rho for formal inference.

Kendall’s tau is especially recommended when the dataset is small and contains many tied ranks. Laerd Statistics describes Kendall’s tau-b as a preferred alternative to Spearman’s rho specifically in small-sample scenarios with frequent ties. Additionally, Kendall’s tau is less sensitive to outliers than Spearman’s rho and is often seen as easier to interpret in communicating findings to non-statistical audiences, because its value can be explained directly in terms of the proportion of agreeing versus disagreeing pairs.

One numerical note worth keeping in mind: for the same data set, Kendall’s tau typically produces a numerically smaller coefficient than Spearman’s rho. This does not mean the relationship is weaker – it reflects a difference in how the two statistics are scaled. The two are not directly interchangeable numerically, but they generally lead to the same substantive conclusions about the direction and significance of a relationship.

Monotonic relationships: the shared requirement 🔗

Both Spearman’s rho and Kendall’s tau are designed specifically to detect monotonic relationships. A monotonic relationship is one where the variables consistently move in the same direction – both increase together, or as one increases the other decreases – but not necessarily at a constant rate. This is less restrictive than linearity, which requires a constant rate of change. As Laerd Statistics explains, monotonicity is “less restrictive” than a linear relationship, which is precisely why rank correlations can capture associations that Pearson’s r would miss or underestimate.

However, it is important to note what rank-order correlations cannot do: they will not detect a non-monotonic relationship, such as a U-shaped curve, where the association changes direction. If a scatterplot suggests the relationship is not consistently increasing or decreasing throughout its range, neither Spearman’s rho nor Kendall’s tau is the right tool, and a different method should be considered.

When to use rank-order correlations in practice 🔗

Three core situations make rank-order correlations the right statistical choice:

Ordinal data. When variables are measured on an ordinal scale – Likert ratings, clinical severity rankings, preference orderings, performance categories – Pearson’s correlation is inappropriate because the intervals between categories are not necessarily equal. Both Spearman’s rho and Kendall’s tau are built for this type of data. Statistics Solutions confirms that both coefficients assess statistical associations based purely on the relative ordering of data, making them the natural fit for ordinal measurement scales.

Non-normal or skewed continuous data. Even when data is continuous, rank-order correlations are preferred if the distribution departs substantially from normality, or if there are significant outliers. Converting raw scores to ranks reduces the influence of extreme values and removes the normality assumption entirely.

Small sample sizes. With small samples, Pearson’s correlation is sensitive to violations of its assumptions. Kendall’s tau in particular tends to produce more accurate p-values with smaller samples, and the exact probability distributions used to test its significance are well-defined even for very small n.

Spearman’s rho vs. Kendall’s tau: which one to choose? 🔗

In most situations, both methods lead to equivalent conclusions. Research on both coefficients confirms that their interpretations are typically very similar and lead to the same inferences. However, some practical differences guide the choice:

Choose Spearman’s rho when your sample is larger, when tied ranks are prevalent (as Spearman handles ties more robustly under those conditions), or when you want a coefficient that is directly interpretable in the same way as Pearson’s r. It is by far the more commonly reported measure in published psychological research.

Choose Kendall’s tau when your sample is small, when you need a coefficient with a clean probabilistic interpretation (probability of concordance minus probability of discordance), or when you want a statistic with more stable distributional properties for significance testing. Tau-b, which adjusts for tied ranks, is the variant to use in most psychological research contexts.

Ultimately, both are legitimate and well-validated tools. The choice should be guided by your data’s characteristics – sample size, degree of tied ranks, and the audience for your findings – rather than by a preference for higher-looking numbers.

Applications in psychological research 🔗

Rank-order correlations appear across many branches of psychology. In personality research, investigators use them to examine whether ordinal ratings on one trait (such as conscientiousness level) are associated with rankings on another (such as academic persistence). In clinical psychology, they are used to assess whether ranked severity of symptoms at intake correlates with ranked treatment outcomes at discharge. In educational psychology, a researcher might ask whether students who rank higher on test anxiety also tend to rank lower on exam performance – a question ideally suited to Spearman’s rho, since the relationship may be monotonic without being perfectly linear. In organizational research, interviewers’ independent ranking of job candidates can be compared using Kendall’s tau to assess how well two evaluators agree – and the coefficient’s concordance-discordance logic makes the result easy to explain to decision-makers who are not statisticians.

Historically, these methods were introduced specifically because psychological and social science data so frequently fails the conditions required for parametric tests. Early applications of Spearman’s rank correlation were rooted in psychology and the social sciences, and the method has remained central to the field’s statistical toolkit ever since.

What do you think? When you encounter a Likert-scale variable in a psychology study, what would lead you to choose Kendall’s tau over Spearman’s rho – or would you consider reporting both? And if a relationship between two psychological variables is monotonic but clearly not linear, does that change how you would interpret the strength of a rank-order correlation compared to a Pearson’s r?

References

1. https://www.statisticssolutions.com/free-resources/directory-of-statistical-analyses/correlation-pearson-kendall-spearman/
2. https://statistics.laerd.com/statistical-guides/spearmans-rank-order-correlation-statistical-guide.php
3. https://www.jove.com/science-education/v/13646/spearman-s-rank-correlation-test
4. https://scales.arabpsychology.com/stats/spearmans-rho/
5. https://www.sciencedirect.com/science/article/abs/pii/S0003347215000196
6. https://en.wikipedia.org/wiki/Kendall_rank_correlation_coefficient
7. https://online.stat.psu.edu/stat509/lesson/18/18.3
8. https://statistics.laerd.com/spss-tutorials/kendalls-tau-b-using-spss-statistics.php
9. https://www.statisticshowto.com/kendalls-tau/
10. https://www.statisticssolutions.com/free-resources/directory-of-statistical-analyses/kendalls-tau-and-spearmans-rank-correlation-coefficient/
11. https://numiqo.com/tutorial/kendalls-tau
12. https://www.numberanalytics.com/blog/guide-spearmans-rank-correlation-analysis