Spearman Correlation Coefficient Calculator

Each column is replaced by average ranks, Pearson r is computed on those ranks, and the printed rank table shows how tied values were averaged.

What this calculator answers

Spearman correlation cares about order. The smallest x receives rank 1, the next rank 2, and so on, and the same ranking is done for y. Pearson's formula is then applied to the ranks instead of the raw numbers. A huge raw outlier becomes just the top rank, so it cannot pull the coefficient the way it pulls Pearson r on the original scale. When two values tie, they share the average of the ranks they would have occupied. This page prints that rank table and the resulting coefficient.

The page reports Spearman rho, which is Pearson r computed on average ranks. It also lists the rank of every typed pair. Supporting counts and intermediate terms sit under the headline so a reader can see what was divided by what. When a checkbox or a tail setting changes the divisor or the rejection region, that choice appears in the steps rather than vanishing into a silent default.

The formula, written out

Spearman rank correlation uses this relationship.

rho = Pearson r of the average ranks of x and the average ranks of y

• rank. Position after sorting, with rank 1 at the low end.
• average rank. The shared rank used when values tie.
• rho. Pearson r applied to the two rank columns.
• tie. Two or more equal raw values.

Ranking replaces magnitudes with positions. Average ranks keep the sum of the ranks equal to the sum that untied ranks would have had. Pearson's formula on those ranks is the definition used here. The algebra is the named textbook identity for this worksheet. Intermediate products and quotients appear in order so a calculator and a pencil can agree to the displayed rounding. If a neighbouring page uses a different divisor or a continuity correction, that page says so on its own ticket.

A full worked example

Example loaded in the worksheet

The form is prefilled with a small example so every arithmetic step can be checked by hand. Change one field after you finish the walkthrough and confirm the ticket moves the way the formula predicts.

Sort each column and assign ranks, using the average rank when values tie. Rank table: 1 ranks 1, 3 ranks 1; 2 ranks 2.5, 4 ranks 2; 2 ranks 2.5, 5 ranks 3; 4 ranks 4, 6 ranks 4; 5 ranks 5, 9 ranks 5. This sample has ties, so at least one rank is the average of the positions those tied values occupy.

Pearson r is now computed on the rank columns, not on the raw numbers. The mean x rank is 3 and the mean y rank is 3. Spearman rho is 0.9747.

Rho squared is 0.95, the straight-line share of the rank variance. The raw values were used only to build the ranks.

A second situation

The tied x values are split into 2 and 3

Sort each column and assign ranks, using the average rank when values tie. Rank table: 1 ranks 1, 3 ranks 1; 2 ranks 2, 4 ranks 2; 3 ranks 3, 5 ranks 3; 4 ranks 4, 6 ranks 4; 5 ranks 5, 9 ranks 5. This sample has no ties, so every rank is a whole number. The average-rank rule would apply if two values were equal.

Pearson r is now computed on the rank columns, not on the raw numbers. The mean x rank is 3 and the mean y rank is 3. Spearman rho is 1.

Rho squared is 1, the straight-line share of the rank variance. The raw values were used only to build the ranks.

Splitting a tie into two neighbouring raw values changes the rank table. Rho can move even when the order of the rest of the list is unchanged. The second example changes the feature that this formula is sensitive to, and the ticket shows the new figure beside the same definition. That contrast is the fastest way to see which input owns the movement.

What moves when the input moves

The third y moved from 5 to 9 and the last y moved from 9 to 5, which reshuffles the y ranks. The headline result went from 0.9747 to 0.4617.

The nudged inputs

Sort each column and assign ranks, using the average rank when values tie. Rank table: 1 ranks 1, 3 ranks 1; 2 ranks 2.5, 4 ranks 2; 2 ranks 2.5, 9 ranks 5; 4 ranks 4, 6 ranks 4; 5 ranks 5, 5 ranks 3. This sample has ties, so at least one rank is the average of the positions those tied values occupy.

Pearson r is now computed on the rank columns, not on the raw numbers. The mean x rank is 3 and the mean y rank is 3. Spearman rho is 0.4617.

Rho squared is 0.2132, the straight-line share of the rank variance. The raw values were used only to build the ranks.

How to read the result

Read rho as agreement of order, from minus 1 to plus 1. A value near 1 means the larger x values sit with the larger y values. It is not a slope and it is not a p-value. Say the unit, the sample size or contract terms, and the formula name in the same sentence as the figure. A number without those three pieces invites a later reader to invent a definition that this page never used.

Write the result with its unit, the inputs that produced it, and the name of the formula. Write the result with its unit, the inputs that produced it, and the name of the formula. If the ticket shows a decision sentence, quote the alpha and the tail with it. If the ticket shows a money or dose figure, quote the rate and the schedule that produced it.

Assumptions that have to stay true

• The measurements can be ranked from low to high.
• Ties should share an average rank rather than be broken by typing order.
• You want monotonic association, not specifically a straight line on the raw scale.
• At least three pairs are available.

If an assumption above is false, do not average the discomfort away. Change the worksheet or change the question. A correct arithmetic step on the wrong model is still the wrong answer for the decision you are trying to make.

Mistakes that change the answer

• Ranking only one column and correlating it with the raw other column.
• Breaking ties by whichever row was typed first.
• Reporting Spearman rho as if it were a regression slope.
• Ignoring ties and assigning consecutive ranks anyway.

Where a different calculator is the honest choice

Use the correlation coefficient calculator for Pearson r on the raw numbers. Use the covariance calculator if you need the unit-bearing product. Use the line of best fit calculator when the raw-scale slope is the question. Related links under the essay open pages that keep their own formulas instead of hiding a mode switch inside this ticket.

When the job changes, move to a page built for that job: Correlation Coefficient Calculator, Covariance Calculator, Line of Best Fit Calculator.

Check yourself

Ranks of x are 1, 2, 3 and ranks of y are 1, 2, 3. What is Spearman rho?

The rank columns match exactly, so Spearman rho is 1.

Using the result in a sentence

Ranked preferences, skewed lab values, and ordinal scores are the settings where ranks are the honest input to a correlation. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.

Spearman rho ignores how far apart the raw values are, except for their order. Two clouds with the same ranks share the same rho even when one is much steeper. Pair the headline with the steps before you staff a route, set a grade boundary, or compare two contracts from a single number.