Spearman’s Rho: Formula, Worked Example and Table

Spearman’s rho was invented by a psychologist, yet the rank formula Charles Spearman printed in 1904 is not the one students learn today. His version added up plain rank differences. The modern formula squares them.

Spearman’s rho is the test psychologists use to find out whether two things go together when the data are ratings, rankings or scores too uneven to trust. Does a higher anxiety rating go with fewer words remembered? Do students who sleep longer say they feel more rested? Does a child’s rank for confidence match their rank for popularity? Each of these asks about a relationship between two co-variables, measured on the same people, and each produces the kind of data Spearman’s rho was built for.

It is the rank-based partner of Pearson’s r, which is explained in our guide to the correlation coefficient. Choosing between them comes down to the questions set out in our guide to choosing the right statistical test: are you looking for a difference or a relationship, and what is the level of measurement of your data? A relationship plus ordinal data leads straight to Spearman’s rho.

The arithmetic is short, but it is easy to lose marks on the details: ranking the two variables together instead of separately, getting tied ranks wrong, forgetting to square d, or applying the “smaller is better” rule from other tests when Spearman’s rho works the other way round. This guide covers all of it, with a full worked example that includes tied ranks, a real AQA exam question where the result just misses significance, a free calculator, and a critical values table from N = 5 to 100. Every printed critical value has been checked against an exact calculation, and the method has been checked against Spearman’s own 1904 example.

What is Spearman’s rho?

Spearman’s rho is a statistic that measures the strength and direction of the relationship between two variables using ranks instead of raw scores. Each person’s score on the first variable is ranked against everyone else’s, the same is done for the second variable, and Spearman’s rho then measures how closely the two sets of ranks agree. Its full name is Spearman’s rank correlation coefficient, which is why it is also called Spearman’s rank or Spearman’s rank order correlation.

The result is a single number between −1 and +1. A value of +1 means the ranks match perfectly: whoever ranks highest on one variable also ranks highest on the other, all the way down. A value of −1 means the ranks are perfectly reversed, so the person who ranks highest on one variable ranks lowest on the other. A value close to 0 means there is no consistent pattern between the two sets of ranks.

Spearman’s rho does two jobs. As a descriptive statistic, it summarises how strong a relationship is and whether it is positive or negative. As an inferential test, it tells you whether a relationship that strong is likely to have turned up by chance in a sample of that size. The first job needs only the formula. The second needs the critical values table, and that is where most exam questions focus.

What does the rho symbol mean?

Rho is the name of the Greek letter ρ, the Greek equivalent of the letter r. Statisticians use Greek letters for values that describe a whole population and Roman letters for values calculated from a sample, so the sample statistic you calculate is usually written r_s, an r with a small s for Spearman. AQA, OCR and Edexcel all name the test Spearman’s rho in their specifications, while the calculated value is conventionally written r_s. You will also see ρ, or simply “rho”, used loosely for the calculated value in textbooks, exam questions and software output. They all refer to the same thing.

Who was Charles Spearman?

Charles Spearman was a British psychologist who came to psychology late. He served as an army officer for almost 15 years, mainly in India, before studying experimental psychology at the Leipzig laboratory founded by Wilhelm Wundt, and in 1904 he still had no formal academic qualification (Lovie and Lovie, 2010). That year he published two papers in the American Journal of Psychology. The first, The Proof and Measurement of Association between Two Things, set out his methods of correlation, including correlation by rank (Spearman, 1904a). The second used those methods to argue for a general factor of intelligence, the idea now known as g that still sits behind debates about what an IQ scale measures (Spearman, 1904b).

So Spearman’s rho was a psychologist’s tool from the very start. Spearman was working with schoolchildren’s examination marks, orders of merit and tests of seeing and hearing, the kind of messy data that rarely looks like a tidy bell curve. He argued that ranking was often the better approach even when measurements existed, because it stopped a few extreme cases from dominating the result and it put two variables with very different distributions on equal terms (Spearman, 1904a). Those are still the reasons for choosing Spearman’s rho today.

When should you use Spearman’s rho?

Use Spearman’s rho when three things are true: you are looking for a relationship rather than a difference, you have two scores from each participant, and your data are at least ordinal. If any of these is not true, a different test is needed.

The three conditions in full

A test of relationship. Spearman’s rho tests a correlational hypothesis, one that predicts two co-variables will rise and fall together, or that one will rise as the other falls. There is no independent variable being manipulated, and no conditions being compared. If your hypothesis says “there will be a difference between”, you need a test of difference instead.

Paired scores. Each participant provides a score on both co-variables, so the data come in pairs. The pairing matters, because the whole calculation compares each person’s rank on one variable with the same person’s rank on the other. If you shuffled one column, the answer would change completely.

At least ordinal data. Ordinal data can be put in order but the gaps between values are not known to be equal. A rating of 4 on a 1 to 5 scale is more than a 2, but nobody can say it is twice as much. Spearman’s rho only uses the order of the scores, so it is safe with ordinal data. It also works with interval or ratio data, which are simply ranked first. It will not work with nominal data, because categories such as “left-handed” and “right-handed” have no order to rank.

Is Spearman’s rho parametric or non-parametric?

Spearman’s rho is a non-parametric test. A parametric test such as Pearson’s r or a t-test makes assumptions about the population the data came from, in particular that the scores are interval level and roughly normally distributed, which is judged partly from the mean and standard deviation. Spearman’s rho makes no assumption about the shape of the distribution, because once the scores are replaced by ranks, their original shape no longer matters. Ranks 1 to 10 always have the same spread, whatever the raw scores looked like.

That is also why Spearman’s rho is described as robust to outliers. A participant who scores 95 when everyone else scores between 10 and 20 would drag Pearson’s r around. In Spearman’s rho, that participant simply receives the top rank, one place above the next person, and has no more influence than anyone else.

Spearman’s rho or Pearson’s r?

Choose Pearson’s r when both variables are interval or ratio data, roughly normally distributed, and related in a straight line. Choose Spearman’s rho when either variable is ordinal, when the data are skewed or contain outliers, or when the relationship is consistent in direction but curved. The technical word for that last case is monotonic: as one variable goes up, the other keeps going up (or keeps going down), but not necessarily at a steady rate.

A simple example shows the difference. If reaction time falls quickly with the first few hours of practice and then levels off, the relationship is curved. Pearson’s r, which measures how well the points fit a straight line, would understate it. Spearman’s rho only asks whether more practice always goes with a faster time, and if it does, it returns a value close to −1. In fact, Spearman’s rho is exactly Pearson’s r calculated on the ranks instead of the raw scores, which is why the two often give similar answers on well-behaved data.

The Spearman’s rho formula explained

The Spearman’s rho formula is r_s = 1 − 6Σd² / n(n² − 1). This is the version printed in the Pearson Edexcel formulae and statistical tables, and it is the one A level students use by hand.

The logic sits in Σd². If the two sets of ranks agree perfectly, every d is zero, Σd² is zero, and the formula gives 1 − 0 = +1. The more the ranks disagree, the bigger the squared differences become. When the ranks are completely reversed, Σd² reaches its largest possible value, which is n(n² − 1) / 3, and the formula gives 1 − 2 = −1. Everything else lands in between. Squaring does two useful things: it stops positive and negative differences cancelling out, and it makes large disagreements count for much more than small ones.

The 6 and the n(n² − 1) are what you get when you write out Pearson’s r for two sets of ranks running from 1 to n and simplify. Because ranks from 1 to n always have the same mean and the same spread, most of Pearson’s formula cancels, leaving only the squared rank differences. That shortcut is exact whenever there are no tied ranks. With ties it becomes a close approximation, which is covered in its own section below.

What Spearman actually printed in 1904

Spearman’s 1904 paper recommended two ways of correlating ranks. The main one was to take Pearson’s product moment method and apply it to ranks instead of measurements, which Spearman described as using precisely the same calculation with ranks substituted for the scores (Spearman, 1904a). That is the statistic we now call Spearman’s rho. The squared-difference shortcut is simply a faster way of getting the same answer.

The formula Spearman actually printed for his quick “method of rank differences” was different: R = 1 − 3Σd / (n² − 1), where Σd is the sum of the rank differences without squaring them (Spearman, 1904a). He illustrated it with five people ranked for seeing and hearing. Their rank differences were 0, 2, 2, 1 and 3, so Σd = 8, and R = 1 − 24 / 24 = 0. Spearman concluded there was no correlation.

Put the same five pairs of ranks through the modern formula and the answer is slightly different. Squaring the differences gives 0 + 4 + 4 + 1 + 9 = 18, so r_s = 1 − (6 × 18) / (5 × 24) = 1 − 108 / 120 = 0.1. That is a very weak positive correlation, and with only five people it is nowhere near significant: the exact two-tailed probability of a result this strong is 0.95. The two methods tell the same story, but they are not the same statistic, and only the squared version is Spearman’s rho as it is taught and tested today.

How to calculate Spearman’s rho: a worked example

To calculate Spearman’s rho by hand, rank each variable separately, find the difference between each person’s two ranks, square those differences and add them up, then put the total into the formula and compare the answer with the critical value. The example below uses invented data so every step can be checked, and it deliberately includes tied scores on both variables, because that is where most mistakes happen.

A student wants to know whether anxiety goes with poorer memory. Ten participants rate how anxious they feel on a scale of 1 (very calm) to 10 (very anxious), then learn a list of 20 words and recall as many as they can. The anxiety rating is ordinal, since a rating of 6 is not known to be exactly twice as anxious as a 3, so Spearman’s rho is the right test.

Step 1: State the hypotheses

The student expects higher anxiety to go with fewer words recalled, which predicts the direction of the relationship. That makes it a directional hypothesis and a one-tailed test.

• Alternative (directional) hypothesis: there will be a negative correlation between anxiety rating and the number of words recalled.
• Null hypothesis: there will be no correlation between anxiety rating and the number of words recalled.

Notice that a correlational hypothesis talks about a correlation between co-variables, not a difference between conditions, and it names both variables in a way that could be measured.

Step 2: Rank the first variable

Rank the anxiety ratings on their own, giving rank 1 to the lowest score. J has the lowest rating (1), so J gets rank 1. A (2) gets rank 2, F (3) gets rank 3 and C (4) gets rank 4. Then there is a tie: E and H both rated their anxiety as 5. They would have taken ranks 5 and 6, so each receives the average of the two, (5 + 6) ÷ 2 = 5.5. The next score, I’s rating of 6, carries on at rank 7, because ranks 5 and 6 have been used up. B, G and D take ranks 8, 9 and 10.

Step 3: Rank the second variable separately

Now rank the recall scores on their own, again giving rank 1 to the lowest. D recalled the fewest words (8) and gets rank 1, followed by B (9) on rank 2, H (10) on rank 3 and G (11) on rank 4. E and I both recalled 12 words, so they share ranks 5 and 6 and each receives 5.5. C (14) continues at rank 7, then F (15) at 8, J (16) at 9 and A (17) at 10.

The most common mistake at this stage is ranking all 20 scores together as one list. That is what the Mann-Whitney U test does, and it is wrong here. Spearman’s rho compares each person’s position within the first variable with their position within the second, so each variable must be ranked separately. It does not matter whether you rank from lowest to highest or highest to lowest, as long as you rank both variables the same way.

Step 4: Find d and d² for each participant

For each participant, take the rank on the second variable away from the rank on the first to find d, then square it. The sign of d does not matter once it is squared, but it is worth keeping on the working so you can check it.

There is a built-in check here. The d values, before squaring, must always add up to zero, because both columns of ranks add up to the same total (55 for ten participants). If your d column does not sum to zero, there is a ranking error somewhere, and it is worth finding before going any further.

Step 5: Add up the squared differences

Σd² = 64 + 36 + 9 + 81 + 0 + 25 + 25 + 6.25 + 2.25 + 64 = 312.5.

Step 6: Put the values into the formula

With n = 10, n² − 1 = 99, so n(n² − 1) = 990.

r_s = 1 − (6 × 312.5) / 990 = 1 − 1,875 / 990 = 1 − 1.894 = −0.894 (to three decimal places).

The most common arithmetic slip is to work out 1 − 6 first and then multiply. The formula means take the whole fraction away from 1, so calculate 6Σd² ÷ n(n² − 1) first and subtract that from 1 as the very last step. A value of −0.894 describes a strong negative correlation: the higher a participant rated their anxiety, the fewer words they tended to recall.

Step 7: Find the critical value

To find the critical value you need three things: N, the significance level and whether the test is one-tailed or two-tailed. Here N = 10 (the number of pairs of scores), the level is the conventional p ≤ 0.05, and the hypothesis is directional, so it is a one-tailed test. The table gives a critical value of 0.564.

Step 8: Compare and state the conclusion

For Spearman’s rho, the result is significant if the calculated value, ignoring the minus sign, is equal to or greater than the critical value. Here 0.894 is greater than 0.564, and the correlation is negative, which is the direction the hypothesis predicted. So the result is significant.

In a write-up: the calculated value of r_s = −0.894 is greater than the critical value of 0.564 for N = 10 at p ≤ 0.05 for a one-tailed test, so the null hypothesis can be rejected and the alternative hypothesis accepted. There was a significant negative correlation between anxiety rating and the number of words recalled. It would even pass at the stricter p ≤ 0.01 level, where the one-tailed critical value is 0.745.

What it does not show is that anxiety caused the poorer recall. It is a correlation, so it is equally possible that people who knew they had a poor memory felt anxious about the test, or that a third variable such as tiredness raised anxiety and lowered recall at the same time. That point is often worth a mark in its own right.

Spearman’s rho calculator and critical values table

The calculator below ranks both variables for you, handles tied ranks, shows every rank and d, and checks the answer against the critical values table. Choose a two-tailed hypothesis if you did not predict a direction, or pick the direction you predicted for a one-tailed test. The placeholder numbers are the worked example above, so you can check it gives r_s = −0.894.

For small samples with no tied ranks (N up to 12), the calculator also works out the exact probability by counting every possible way the ranks could have been arranged. For larger samples, or when there are ties, it uses the t approximation described below.

A second worked example: an AQA question that is not significant

A result can be quite a sizeable correlation and still not be significant, and AQA’s 2021 A level Paper 2 (7182/2) turned that into a whole sequence of questions (AQA, 2021a). A researcher randomly selected 18 first-year university students, asked each one how many hours they had slept the night before, and asked them to rate how well rested they felt on a scale of 1 to 5. The researcher predicted a positive correlation, chose the 5% level of significance, and used Spearman’s rho. The calculated value was 0.395.

Why Spearman’s rho? The mark scheme gives two reasons, each worth two marks: the hypothesis is correlational, because the researcher is looking for a relationship between hours slept and feeling rested, and the data are ordinal, because the 1 to 5 rating is an arbitrary scale that can be ranked but is not a standardised measure (AQA, 2021b).

Finding the critical value. There are 18 participants, so N = 18. The hypothesis predicts a positive correlation, so it is directional and one-tailed. The level is 0.05. The paper printed a short extract of the table, and the row for N = 18 gives 0.401 in the one-tailed 0.05 column. The mark scheme awards one mark each for the value 0.401, for recognising the test is one-tailed, for N = 18 and for the 0.05 level (AQA, 2021b).

The conclusion. The calculated value of 0.395 is less than the critical value of 0.401, so the result is not significant and the researcher’s hypothesis should not be accepted. The null hypothesis is retained. That is the answer the mark scheme requires (AQA, 2021b).

Working the probability out exactly shows how close it was. With 18 participants there are more than six million billion possible ways the ranks could be arranged, and the proportion of them that produce a value of 0.395 or higher by chance is 0.053. The researcher missed the 0.05 cut-off by less than half of one percentage point. That is precisely why the paper went on to ask about a Type II error: a real relationship between sleep and feeling rested may well exist, and a sample of 18 was simply not large enough to show it reliably. Our guide to statistical significance, p-values and errors explains why a larger sample is the usual cure.

The same extract also shows how quickly the critical values fall as N grows. At N = 20, the one-tailed critical value is 0.380, and 0.395 would have been significant. Two more participants would have changed the conclusion, even if the strength of the relationship had stayed exactly the same.

How to read the Spearman’s rho critical values table

To read the table, find the row for N, the number of pairs of scores, then the column for your significance level and for a one-tailed or two-tailed test. The number where they meet is the critical value. Your calculated r_s, ignoring any minus sign, must be equal to or greater than it for the result to be significant.

Why r_s must be equal to or greater than the critical value

Spearman’s rho works the opposite way round from several other tests on the A level specification. For the sign test, the Wilcoxon signed-rank test and the Mann-Whitney U test, a small calculated value is the impressive one, so the value must be equal to or less than the critical value. For Spearman’s rho, as for Pearson’s r and the chi-squared test, the impressive result is a large one, so it must be equal to or greater than the critical value.

The reason is easy to remember once you think about what the number means. A correlation of 0 is exactly what you would expect if there were no relationship at all. The further r_s moves from 0, towards +1 or −1, the less likely it is to have happened by chance. The critical value marks the point beyond which a result is unlikely enough, in a sample of that size, to count as significant. There is a simple memory aid for which way the rule goes: if the test’s name contains an R, such as Spearman’s rho, Pearson’s r or chi-squared, the calculated value must be gReater than or equal to the critical value.

One-tailed or two-tailed?

Use a one-tailed test when your hypothesis predicted the direction of the correlation, positive or negative. Use a two-tailed test when it only predicted that there would be a correlation of some kind. The table has two header rows for this reason: each column shows a one-tailed level on top and the matching two-tailed level underneath, because a one-tailed test at 0.05 uses the same critical value as a two-tailed test at 0.10.

A one-tailed test carries an extra condition that the table cannot show. The result only counts if it runs in the direction you predicted. If you predicted a negative correlation and got r_s = +0.894, the result is not significant, however large it is, because it contradicts the hypothesis. Always check the sign before you check the size.

What N means

N is the number of participants, which is the same as the number of pairs of scores. In the worked example, ten participants produced twenty scores, but N = 10. Unlike the sign test and the Wilcoxon test, nobody is dropped for a zero difference: a participant whose two ranks match simply contributes d = 0 to the total.

Why small samples need such high values

With five participants, only a perfect correlation of 1.000 can be significant for a two-tailed test at p ≤ 0.05. There are only 120 ways to arrange five ranks, and even a near-perfect match turns up by chance too often to trust. As N grows, the number of possible arrangements explodes, chance agreements become proportionally rarer, and the critical value falls. By N = 30, a correlation of 0.362 is enough at the same level. This is why a moderate correlation in a small class project so often fails to reach significance, as it did in the AQA question.

A note on the printed table

The critical values from N = 5 to 30 in the table above are the ones printed by Pearson Edexcel in its formulae and statistical tables (Pearson Education, 2024), and the extract AQA used in its 2021 question is identical for N = 16 to 20. For this guide, every value up to N = 19 was also recalculated exactly, by counting every possible arrangement of the ranks. Almost every value matches the exact calculation when rounded to three decimal places. The exceptions are all at N = 5, 17 and 19, and they are worth knowing about.

• N = 17, two-tailed 0.05 (one-tailed 0.025). The table prints 0.485. The smallest value that is exactly significant at this level is 0.488. A result of 0.485 has an exact one-tailed probability of 0.0252, a fraction over 0.025.
• N = 17, two-tailed 0.01 (one-tailed 0.005). The table prints 0.615, where the exact value is 0.618, the same kind of tiny generosity.
• N = 17 and 19, the strictest column. Here the table leans the other way, printing 0.662 and 0.628 where the exact values are 0.659 and 0.626, so it is very slightly harder to pass than it needs to be.
• N = 5 at the strictest levels. The table prints 1.000 in the right-hand columns, but even a perfect correlation with five participants has a one-tailed probability of 1 ÷ 120 = 0.0083, or 0.017 two-tailed. In practice, no result with five participants can reach a two-tailed 0.01 level.

None of these cells affects the worked examples on this page, and none is an error that should worry you in an exam, where you should always use the value printed on the paper. They are a reminder that printed tables are rounded summaries of an exact calculation, which is why the calculator reports the probability alongside the table verdict.

Beyond N = 30: the t approximation

For larger samples, r_s can be converted into a t value with t = r_s × √((N − 2) / (1 − r_s²)), which is then checked against the t distribution with N − 2 degrees of freedom. The rows from N = 35 to 100 in the table, shown in grey, were calculated this way. At N = 30 the approximation comes out up to 0.005 lower than the printed values, depending on the column, so a result that only just clears a starred row deserves a little caution. For a result well clear of the critical value, the difference does not matter.

How to deal with tied ranks in Spearman’s rho

When two or more scores on the same variable are equal, give each of them the average of the ranks they would have occupied, then carry on ranking from the next unused rank. Two scores sharing ranks 5 and 6 both get 5.5, and the next score gets 7. Three scores sharing ranks 3, 4 and 5 all get 4, and the next score gets 6. The last rank should always equal N, which is a quick way to check nothing has gone wrong.

Ties are shared within a variable, never across the two variables. In the worked example, E and H tied on anxiety and E and I tied on recall. Those are two separate ties, handled separately, and it does not matter that E appears in both.

Why ties make the formula approximate

The shortcut formula 1 − 6Σd² / n(n² − 1) is exact only when there are no ties. Its derivation assumes both sets of ranks are exactly the whole numbers 1 to n, and averaged ranks break that assumption slightly. Spearman himself noted in 1904 that bracketing individuals at the same rank makes a rank formula slightly incorrect, but usually by too little to worry about (Spearman, 1904a).

The exact tie-corrected value is found by calculating Pearson’s r on the ranks themselves, which is what statistics software does. For the worked example, the shortcut gives −0.894 and Pearson’s r on the ranks gives −0.905. The difference is small, and it does not change the conclusion. Methods textbooks also give a longer correction formula for ties (Siegel and Castellan, 1988), but for A level work the shortcut formula with averaged ranks is what examiners expect. The calculator on this page shows both values whenever there are ties.

Ties become a real problem only when there are a great many of them, for example when 30 people answer a 1 to 5 rating scale and most choose 3 or 4. With so few distinct values, the ranks carry little information and the correlation becomes unreliable. The better fix is at the design stage: use a wider scale or a more sensitive measure.

How to interpret Spearman’s rho

Interpreting Spearman’s rho means reading three separate things from the result: its direction, its strength and its significance. They answer different questions, and a good interpretation mentions all three.

Direction: positive or negative

The sign of r_s gives the direction. A positive value means high ranks on one variable go with high ranks on the other, as in the AQA sleep study, where more sleep was expected to go with feeling more rested. A negative value means high ranks on one go with low ranks on the other, as in the worked example, where higher anxiety went with lower recall. The minus sign says nothing about strength: −0.894 is exactly as strong as +0.894.

Strength: how close to 1

The size of r_s, ignoring the sign, gives the strength. There is no single agreed scale. One widely used set of benchmarks, originally proposed for Pearson’s r, treats about 0.1 as a small effect, 0.3 as medium and 0.5 as large (Cohen, 1988), and many textbooks describe values above about 0.7 as strong. These are rules of thumb, not thresholds, and what counts as strong depends on the field: a correlation of 0.3 between a personality score and a real-world behaviour can be meaningful, while 0.9 would be expected between two versions of the same test.

Significance: could it be chance?

Significance says whether a correlation of that size is unlikely to have arisen by chance in a sample of that size. It is not the same as strength. A weak correlation can be significant in a large sample, and a fairly strong one can fail in a small sample. With 100 participants, r_s = 0.2 is significant at p ≤ 0.05 two-tailed. With 8 participants, even r_s = 0.7 is not significant at that level. Always report the value of r_s as well as whether it was significant.

What a correlation cannot tell you

A significant Spearman’s rho shows that two variables are related. It does not show that one causes the other. The relationship could run in either direction, or both variables could be driven by a third, unmeasured variable. This limitation belongs to the correlational design, not to Spearman’s rho itself, and it is covered in more depth in our guide to the correlation coefficient.

How to write up a Spearman’s rho result

A good write-up states the test, the calculated value, N, the critical value, the level of significance, whether it was one-tailed or two-tailed, the decision about the hypotheses, and what it means in plain words. In an A level answer, a full statement for the worked example looks like this:

A Spearman’s rho test found a calculated value of r_s = −0.894. This is greater than the critical value of 0.564 for N = 10 at p ≤ 0.05 (one-tailed), so the result is significant. The null hypothesis is rejected and the alternative hypothesis is accepted: there is a significant negative correlation between anxiety rating and the number of words recalled.

In a formal report written in APA style, the result is condensed into one line, with the degrees of freedom (N − 2) in brackets and no zero before the decimal point, because a correlation can never be larger than 1 (American Psychological Association, 2020). If the tie-corrected value from software is reported, the worked example becomes: r_s(8) = −.91, p < .001, one-tailed. Our guides to the standard error and standard deviation cover the descriptive statistics that usually sit alongside it in a results section.

How to calculate Spearman’s rho in Excel and SPSS

Excel has no single Spearman’s rho function, but two built-in functions do the job. RANK.AVG ranks a score within a list and gives tied scores their average rank (Microsoft, n.d.), and CORREL then calculates Pearson’s r on the two columns of ranks, which is Spearman’s rho.

1. Put the first variable in column A and the second in column B, one participant per row.
2. In column C, rank the first variable with =RANK.AVG(A2,A$2:A$11,1). The final 1 ranks from lowest to highest. Fill it down.
3. In column D, rank the second variable in the same way with =RANK.AVG(B2,B$2:B$11,1).
4. In any empty cell, enter =CORREL(C2:C11,D2:D11). The answer is Spearman’s rho.

Because CORREL works on the ranks directly, Excel gives the tie-corrected value, so for the worked example it returns −0.905 rather than the −0.894 from the hand formula. In SPSS, the route is Analyze, then Correlate, then Bivariate, where you tick Spearman instead of Pearson. SPSS also reports the tie-corrected value along with a probability.

Spearman’s rho vs Pearson’s r, Kendall’s tau and chi-squared

Spearman’s rho is one of several ways to measure association. The right choice depends on the level of measurement, the shape of the relationship and whether you are counting categories or measuring people.

Kendall’s tau is the other common rank correlation. Instead of squaring rank differences, it counts how many pairs of participants are ranked in the same order on both variables and how many are ranked in opposite orders (Kendall, 1938). It usually gives a smaller number than Spearman’s rho for the same data, so the two are not interchangeable, but it is not on any A level specification. The chi-squared test is sometimes confused with a correlation because it also looks for an association, but it works on counts of people in categories, not on pairs of scores.

Strengths and limitations of Spearman’s rho

Evaluating the choice of test is a regular exam question, and Spearman’s rho has a clear set of strengths and weaknesses.

Strengths

• It works with ordinal data. Ratings, rankings and questionnaire scores are common in psychology, and Spearman’s rho can analyse them without pretending the gaps between points are equal.
• It makes no assumption about the distribution. Skewed data and data from small samples, which cannot be shown to be normal, are handled safely.
• It resists outliers. One extreme score becomes just the highest rank and cannot dominate the result.
• It detects curved relationships. Any relationship that consistently rises or consistently falls is picked up, even if it is not a straight line.
• It is quick to calculate by hand. With a small sample, ranking and one formula are all that is needed.

Limitations

• It throws information away. Replacing interval scores with ranks discards the size of the gaps, so when the data genuinely meet Pearson’s assumptions, Pearson’s r is the more powerful test and more likely to detect a real relationship.
• It misses relationships that change direction. A U-shaped or inverted-U relationship, such as performance rising and then falling with arousal, can produce a Spearman’s rho close to zero even though the variables are strongly related.
• Heavy ties weaken it. The formula becomes approximate, and short rating scales can leave too few distinct values to rank meaningfully.
• Small samples need very high values. With fewer than about 10 participants, only very strong correlations can reach significance, which raises the risk of a Type II error.
• It cannot show cause and effect. Like every correlation, it shows that variables are related, not why.

Common Spearman’s rho mistakes in exams

Most lost marks on Spearman’s rho come from a small number of predictable slips, and every one of them is avoidable.

1. Using the wrong comparison rule. Spearman’s rho must be equal to or greater than the critical value. Carrying over the “less than” rule from the sign test, Wilcoxon or Mann-Whitney reverses every conclusion.
2. Ranking both variables together. Each variable is ranked separately.
3. Forgetting to square d. Without squaring, the differences add up to zero every time.
4. Doing 1 − 6 first. Work out the whole fraction, then take it from 1.
5. Counting scores instead of participants. N is the number of pairs, not the number of scores.
6. Ignoring the direction on a one-tailed test. A strong correlation in the wrong direction is not significant.
7. Choosing Spearman’s rho for a difference. If the hypothesis compares conditions, a test of difference is needed. Our guide to choosing the right statistical test sets out the full decision.
8. Claiming causation. “Anxiety reduces recall” is not supported. “Anxiety is negatively correlated with recall” is.

What each exam board requires

All three major A level Psychology specifications in England include Spearman’s rho, but they ask for different things.

For AQA students, that means the questions most likely to appear are the ones in the 2021 paper: justify the choice of test, find the critical value, reach a conclusion and discuss Type I and Type II errors. Edexcel students may also be asked to calculate it, since the formula is printed in their tables. For everyone, the worked example and the table-reading steps on this page cover what is needed.

Conclusion

Spearman’s rho measures how closely two sets of ranks agree, and it is the test to use when you are looking for a relationship between two co-variables and at least one of them is ordinal. Rank each variable separately, give tied scores the average rank, square and add the rank differences, and put the total into 1 − 6Σd² / n(n² − 1). The answer lies between −1 and +1, and its sign shows the direction while its size shows the strength.

To test significance, compare r_s, ignoring the sign, with the critical value for N. It must be equal to or greater than the critical value, and for a one-tailed test it must also run in the predicted direction. As the AQA sleep question shows, a respectable correlation in a small sample can still fall short, which is a reason to collect more data rather than to stop looking. And however strong the result, a correlation shows that two things go together, not that one causes the other.

Frequently Asked Questions

What is Spearman’s rho used for?

Spearman’s rho is used to measure and test the relationship between two variables recorded for the same people, when the data are ranks, ratings or scores that cannot be trusted to follow a normal distribution. Psychologists use it for questions such as whether self-rated stress goes with hours of sleep, or whether two judges rank the same drawings in a similar order. It gives both the strength and the direction of the relationship.

Is Spearman’s rho the same as Spearman’s rank?

Yes. Spearman’s rho, Spearman’s rank, Spearman’s rank correlation coefficient and Spearman’s rank order correlation are all names for the same statistic. Psychology specifications tend to say Spearman’s rho, while geography and biology courses more often say Spearman’s rank, but the formula and the critical values are identical.

Is Spearman’s rho parametric or non-parametric?

Spearman’s rho is non-parametric. It works on ranks, so it does not require interval data or a normal distribution, which are the assumptions behind parametric tests such as Pearson’s r. The trade-off is that it can be slightly less powerful than Pearson’s r when those assumptions genuinely hold.

Can Spearman’s rho be negative?

Yes. Spearman’s rho is negative when high ranks on one variable go with low ranks on the other, and it can fall anywhere down to −1, a perfectly reversed order. A negative value is just as strong as a positive value of the same size. When checking significance, ignore the minus sign, but for a one-tailed test make sure a negative correlation was what you predicted.

What is the symbol for Spearman’s rho?

The calculated value is usually written r_s, a lower-case r with a subscript s. The Greek letter ρ (rho) gives the test its name and is sometimes used for the value too, especially in software output. In APA style, the value is reported as r_s with the degrees of freedom in brackets.

What are the assumptions of Spearman’s rho?

Spearman’s rho has only a few assumptions. The data must be at least ordinal, so they can be ranked. Each participant must provide a pair of scores, one on each variable. The pairs should be independent of one another, meaning each participant appears once. And the relationship should be monotonic, consistently rising or consistently falling. It does not assume normality or a straight-line relationship.

Is Spearman’s rho descriptive or inferential?

Spearman’s rho is both. The coefficient itself is a descriptive statistic, summarising how strongly and in which direction two sets of ranks are related. Comparing it with a critical value, or calculating a p-value, turns it into an inferential test of whether the relationship is likely to exist beyond the sample.

How do you calculate Spearman’s rho in Excel?

1. Enter each variable in its own column, one participant per row.
2. Rank each column with RANK.AVG, using an order argument of 1 for both.
3. Apply CORREL to the two columns of ranks.

The result is the tie-corrected Spearman’s rho, which can differ slightly from a hand calculation when there are tied scores.

Does Spearman correlation assume linearity?

No. Spearman correlation assumes a monotonic relationship, not a linear one. The variables must move consistently in one direction together, but not at a constant rate, so a curve that keeps rising is fine. What Spearman correlation cannot capture is a relationship that rises and then falls, which can give a value near zero.

References

• American Psychological Association. (2020). Publication manual of the American Psychological Association (7th ed.). American Psychological Association.
• AQA. (2021a). A-level Psychology 7182/2, Paper 2: Psychology in context. Question paper. AQA.
• AQA. (2021b). A-level Psychology 7182/2, Paper 2: Psychology in context. Mark scheme. AQA.
• AQA. (2021c). AS and A-level Psychology specification (7181, 7182) (Version 1.2). AQA.
• Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates.
• Kendall, M. G. (1938). A new measure of rank correlation. Biometrika, 30(1/2), 81–93.
• Lovie, S., and Lovie, P. (2010). Commentary: Charles Spearman and correlation: A commentary on “The proof and measurement of association between two things”. International Journal of Epidemiology, 39(5), 1151–1153.
• Microsoft. (n.d.). RANK.AVG function. Microsoft Support.
• OCR. (2026). A Level Psychology H567 specification (Version 1.5). OCR.
• Pearson Education. (2024). Pearson Edexcel Level 3 GCE Psychology: Formulae and statistical tables, May–June 2026 assessment window and beyond. Pearson Education.
• Pearson Education. (2026). Pearson Edexcel Level 3 Advanced GCE in Psychology specification (Issue 4). Pearson Education.
• Siegel, S., and Castellan, N. J. (1988). Nonparametric statistics for the behavioral sciences (2nd ed.). McGraw-Hill.
• Spearman, C. (1904a). The proof and measurement of association between two things. American Journal of Psychology, 15(1), 72–101.
• Spearman, C. (1904b). “General intelligence,” objectively determined and measured. American Journal of Psychology, 15(2), 201–292.

Further Reading and Research

Recommended Articles

• What Is a Correlation Coefficient? Pearson’s r and Psychology Statistics
• Statistical Significance: P-values, Errors and Effect Sizes Guide
• Levels of Measurement: Nominal, Ordinal and Interval

Suggested Books

• Coolican, H. (2019). Research Methods and Statistics in Psychology (7th ed.). Routledge.
  • A standard undergraduate and A level reference that covers correlation, rank tests and significance, with plenty of psychology examples.
• Siegel, S., and Castellan, N. J. (1988). Nonparametric Statistics for the Behavioral Sciences (2nd ed.). McGraw-Hill.
  • The classic reference for rank-based tests, including Spearman’s rho, Kendall’s tau and the correction for tied ranks.
• Field, A. (2024). Discovering Statistics Using IBM SPSS Statistics (6th ed.). Sage.
  • A readable guide to running and reporting correlations in software once you move beyond hand calculation.

Recommended Websites

• AQA A-level Psychology (7182)
  • The specification, past papers and mark schemes, including the 2021 Paper 2 used as the second worked example here.
• Pearson Edexcel A level Psychology
  • The specification and the formulae and statistical tables that print the Spearman’s rho formula and critical values.
• Laerd Statistics
  • A step-by-step guide to the assumptions and interpretation of Spearman’s rank-order correlation, with worked output.

Kathy Brodie

Kathy Brodie is an Early Years Professional, Trainer and Author of multiple books on Early Years Education and Child Development. She is the founder of Early Years TV and the Early Years Summit.

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To cite this article please use:

Early Years TV Spearman’s Rho: Formula, Worked Example and Table. Available at: https://www.earlyyears.tv/spearmans-rho-formula-worked-example-table/ (Accessed: 2 October 2026).