Spearman Rank Correlation Calculator

StatisticsLast updated: August 22, 2026

The Spearman rank correlation coefficient (ρ, rho) is a nonparametric measure of association computed from the rank numbers of two variables rather than their raw values. It assumes neither linearity nor normality; it only looks for a relationship that moves in one direction (a monotonic one). That makes it far more dependable than the Pearson correlation for Likert-type scales, ranking data, small samples, and data sets that contain extreme values.

This calculator converts your data pairs to ranks, assigns the average rank to tied values, sums the squared rank differences to obtain rho, and tests whether the coefficient differs from zero in the population using a t approximation with n − 2 degrees of freedom. The output shows the rank transformation table, the scatter plots, the Fisher z confidence interval, and the Pearson r for the same data side by side, so you can decide which coefficient to report with full information.

Each value belongs to one observation (one participant). You can separate them with commas, spaces, or line breaks.
Each value must match the observation in the same position in the X field; both fields must contain the same number of values.
Sets the false positive risk. In the social sciences 5% is the common choice.
The probability that the interval covers the true parameter.

Spearman Rank Correlation Formulas

Shortcut (no ties):  ρ = 1 − 6·Σdᵢ² / [n(n² − 1)]
dᵢ = rank(xᵢ) − rank(yᵢ)
With ties:           ρ = Σ(Rxᵢ − R̄x)(Ryᵢ − R̄y) / √[Σ(Rxᵢ − R̄x)² · Σ(Ryᵢ − R̄y)²]
Significance:        t = ρ·√[(n − 2) / (1 − ρ²)] ,  df = n − 2
Critical coefficient: ρ_critical = t_critical / √(df + t_critical²)
Fisher z:            z = ½·ln[(1 + ρ)/(1 − ρ)] = artanh(ρ)
Confidence interval: tanh( z ± z_(1−α/2)·√(1.06 / (n − 3)) )

Tied values are given the average rank. When ties are present the shortcut formula is only approximate; in that case the calculator applies the Pearson formula to the ranks and reports the tie-corrected rho — which is also what SPSS and the R function cor(method="spearman") return.

How to Calculate

  1. Paste the X and Y values into the two fields in the same order; each row must belong to the same observation (the same participant), and both fields must contain the same number of values.
  2. Choose the significance level (α) and the confidence level; α = 0.05 with 95% confidence is the common choice in the social sciences.
  3. Study the rank transformation table: it shows how the values were converted to ranks, which average rank was assigned to tied values, and the Σd² total.
  4. Read rho, t, the degrees of freedom, the critical value, p, and the confidence interval together in the results card; the decision line states directly whether the relationship is significant.
  5. Compare Pearson r with Spearman rho; if the two coefficients diverge noticeably, look at the scatter plot to check for curvature or extreme values.
  6. Report the APA line (e.g. ρ(8) = 0.95; p < 0.001) together with the confidence interval.

Worked Examples

Study time and exam score

For 10 students, the weekly study time and the exam score give ρ = 0.952 with Σd² = 8. Because t(8) = 8.749 exceeds the critical value of 2.306, the relationship is significant at p < 0.001, and the 95% confidence interval runs from 0.795 to 0.989. Pearson r = 0.957 is very close, so the relationship is both monotonic and linear.

Spearman rho (ρ): 0.952 · Effect size label: very strong (|ρ| = 0.952) · ρ² (shared rank variance): 0.905 (90.5%)

Likert scale (tied ranks)

A 5-point Likert item produces many tied values; the tie-corrected ρ = 0.876 while the shortcut formula gives 0.8829 — the gap is exactly the effect of the tie correction. With Σd² = 33.5 and t(10) = 5.743 the relationship is significant at p < 0.001; the 95% confidence interval is 0.595 to 0.966 and Pearson r = 0.883 is very close.

Spearman rho (ρ): 0.876 · Effect size label: very strong (|ρ| = 0.876) · ρ² (shared rank variance): 0.767 (76.7%)

Curved monotonic relationship (Pearson misleads)

In an exponentially increasing relationship, Σd² = 2 gives ρ = 0.988, while Pearson r stops at 0.839: the relationship is almost perfectly monotonic but not linear. Spearman rho is the coefficient to report for this data; t(8) = 18.000 with p < 0.001.

Spearman rho (ρ): 0.988 · Effect size label: very strong (|ρ| = 0.988) · ρ² (shared rank variance): 0.976 (97.6%)

Frequently Asked Questions

What is the difference between Spearman rho and Pearson r?
Pearson r measures the LINEAR relationship between the raw values and assumes an interval or ratio scale with approximate normality. Spearman rho converts the values to ranks and applies the same formula, so it looks only for a MONOTONIC relationship, assumes no normality, and is barely affected by extreme values. When the relationship is linear the two coefficients come out very close; when it is curved but monotonic, rho is noticeably higher.
When should I use Spearman and when Pearson?
Ordinal scales, Likert items, ranking data, pronounced outliers, small samples, and skewed distributions all call for Spearman. When the variables are continuous, the relationship is linear, and the distributions are approximately normal, Pearson is more powerful (more likely to detect a true relationship in the same sample). If you are unsure, look at the scatter plot: if the cloud of points does not follow a straight line, prefer Spearman.
How do tied values affect the result?
Tied values are assigned the average rank; for example two equal observations in positions 3 and 4 both receive rank 3.5. With ties the shortcut formula 1 − 6Σd²/[n(n²−1)] is no longer exact, because the variance of the ranks shrinks. The calculator therefore applies the Pearson formula to the ranks and reports the tie-corrected rho; if the proportion of ties is very high, Kendall tau-b is the better choice.
How many observations does a Spearman correlation need?
The calculation can be run with as few as 4 pairs, but at that size the p-value and the confidence interval are not trustworthy. For the t approximation to work reasonably, n ≥ 10 is usually recommended, and n ≥ 20–30 for a stable estimate. This calculator produces the Fisher z confidence interval only when n ≥ 6, because for smaller n the variance of the transformation is not even approximately right.
Can rho squared be interpreted like r² in Pearson?
Not quite. ρ² gives the proportion of variance shared at the RANK level, not the proportion of variance explained in the raw values. So rather than saying "86% of the variance of the variable is explained", it is more accurate to say "there is strong agreement between the two rankings". As an effect size, rho itself is what gets reported.
Do the results match SPSS, R, and Excel?
The coefficient is identical to the output of SPSS (Correlate → Bivariate → Spearman) and of the R function cor.test(x, y, method="spearman"). For the p-value this calculator uses the t approximation with n − 2 degrees of freedom, the same one SPSS uses; because R uses the exact permutation distribution in small samples, small differences can appear in the third decimal. Excel has no built-in Spearman function: you have to produce the ranks with RANK.AVG and then apply CORREL.
Does a significant rho show causation?
No. Like Pearson r, Spearman rho only measures co-variation; it cannot tell you which variable affects the other or whether the relationship is driven by a third variable. To remove the effect of a third variable you need partial correlation, and to make predictions you need regression analysis.