A Spearman correlation of 1 results when the two variables being compared are monotonically related, even if their relationship is not linear. This means that all datapoints with greater xvalues than that of a given datapoint will have greater yvalues as well. In contrast, this does not give a perfect Pearson correlation.
When the data are roughly elliptically distributed and there are no prominent outliers, the Spearman correlation and Pearson correlation give similar values.
The Spearman correlation is less sensitive than the Pearson correlation to strong outliers that are in the tails of both samples. That is because Spearman's rho limits the outlier to the value of its rank.
In statistics, Spearman's rank correlation coefficient or Spearman's rho, named after Charles Spearman and often denoted by the Greek letter \rho (rho) or as r_s, is a nonparametric measure of statistical dependence between two variables. It assesses how well the relationship between two variables can be described using a monotonic function. If there are no repeated data values, a perfect Spearman correlation of +1 or −1 occurs when each of the variables is a perfect monotone function of the other.
Spearman's coefficient, like any correlation calculation, is appropriate for both continuous and discrete variables, including ordinal variables.^{[1]}^{[2]} Spearman's \rho and Kendall's \tau can be formulated as special cases of a more general correlation coefficient.
Contents

Definition and calculation 1

Related quantities 2

Interpretation 3

Example 4

Determining significance 5

Correspondence analysis based on Spearman's rho 6

See also 7

References 8

Further reading 9

External links 10
Definition and calculation
The Spearman correlation coefficient is defined as the Pearson correlation coefficient between the ranked variables.^{[3]} For a sample of size n, the n raw scores X_i, Y_i are converted to ranks x_i, y_i, and ρ is computed from:

\rho = {1 \frac {6 \sum d_i^2}{n(n^2  1)}}.
where d_i = x_i  y_i, is the difference between ranks. See the example below. Identical values (rank ties or value duplicates) are assigned a rank equal to the average of their positions in the ascending order of the values. In the table below, notice how the rank of values that are the same is the mean of what their ranks would otherwise be:
Variable X_i

Position in the ascending order

Rank x_i

0.8

1

1

1.2

2

\frac{2+3}{2}=2.5\

1.2

3

\frac{2+3}{2}=2.5\

2.3

4

4

18

5

5

In applications where duplicate values are known to be absent, a simpler procedure can be used to calculate ρ.^{[3]}^{[4]}
This method should not be used in cases where the data set is truncated; that is, when the Spearman correlation coefficient is desired for the top X records (whether by prechange rank or postchange rank, or both), the user should use the Pearson correlation coefficient formula given above.
The standard error of the coefficient (σ) was determined by Pearson in 1907 and Gosset in 1920. It is

\sigma = \frac{ 0.6325 }{ \sqrt{n1} }
Related quantities
There are several other numerical measures that quantify the extent of statistical dependence between pairs of observations. The most common of these is the Pearson productmoment correlation coefficient, which is a similar correlation method to Spearman's rank, that measures the “linear” relationships between the raw numbers rather than between their ranks.
An alternative name for the Spearman rank correlation is the “grade correlation”;^{[5]} in this, the “rank” of an observation is replaced by the “grade”. In continuous distributions, the grade of an observation is, by convention, always one half less than the rank, and hence the grade and rank correlations are the same in this case. More generally, the “grade” of an observation is proportional to an estimate of the fraction of a population less than a given value, with the halfobservation adjustment at observed values. Thus this corresponds to one possible treatment of tied ranks. While unusual, the term “grade correlation” is still in use.^{[6]}
Interpretation
Positive and negative Spearman rank correlations
A positive Spearman correlation coefficient corresponds to an increasing monotonic trend between X and Y.

A negative Spearman correlation coefficient corresponds to a decreasing monotonic trend between X and Y.

The sign of the Spearman correlation indicates the direction of association between X (the independent variable) and Y (the dependent variable). If Y tends to increase when X increases, the Spearman correlation coefficient is positive. If Y tends to decrease when X increases, the Spearman correlation coefficient is negative. A Spearman correlation of zero indicates that there is no tendency for Y to either increase or decrease when X increases. The Spearman correlation increases in magnitude as X and Y become closer to being perfect monotone functions of each other. When X and Y are perfectly monotonically related, the Spearman correlation coefficient becomes 1. A perfect monotone increasing relationship implies that for any two pairs of data values X_{i}, Y_{i} and X_{j}, Y_{j}, that X_{i} − X_{j} and Y_{i} − Y_{j} always have the same sign. A perfect monotone decreasing relationship implies that these differences always have opposite signs.
The Spearman correlation coefficient is often described as being "nonparametric". This can have two meanings. First, the fact that a perfect Spearman correlation results when X and Y are related by any monotonic function can be contrasted with the Pearson correlation, which only gives a perfect value when X and Y are related by a linear function. The other sense in which the Spearman correlation is nonparametric in that its exact sampling distribution can be obtained without requiring knowledge (i.e., knowing the parameters) of the joint probability distribution of X and Y.
Example
In this example, the raw data in the table below is used to calculate the correlation between the IQ of a person with the number of hours spent in front of TV per week.
IQ, X_i

Hours of TV per week, Y_i

106

7

86

0

100

27

101

50

99

28

103

29

97

20

113

12

112

6

110

17

Firstly, evaluate d^2_i. To do so use the following steps, reflected in the table below.

Sort the data by the first column (X_i). Create a new column x_i and assign it the ranked values 1,2,3,...n.

Next, sort the data by the second column (Y_i). Create a fourth column y_i and similarly assign it the ranked values 1,2,3,...n.

Create a fifth column d_i to hold the differences between the two rank columns (x_i and y_i).

Create one final column d^2_i to hold the value of column d_i squared.
IQ, X_i

Hours of TV per week, Y_i

rank x_i

rank y_i

d_i

d^2_i

86

0

1

1

0

0

97

20

2

6

−4

16

99

28

3

8

−5

25

100

27

4

7

−3

9

101

50

5

10

−5

25

103

29

6

9

−3

9

106

7

7

3

4

16

110

17

8

5

3

9

112

6

9

2

7

49

113

12

10

4

6

36

With d^2_i found, add them to find \sum d_i^2 = 194. The value of n is 10. These values can now be substituted back into the equation : \rho = 1 {\frac {6 \sum d_i^2}{n(n^2  1)}}. to give

\rho = 1 {\frac {6\times194}{10(10^2  1)}}
which evaluates to ρ = 29/165 = −0.175757575... with a Pvalue = 0.627188 (using the t distribution)
This low value shows that the correlation between IQ and hours spent watching TV is very low, although the negative value suggests that the longer the time spent watching television the lower the IQ. In the case of ties in the original values, this formula should not be used; instead, the Pearson correlation coefficient should be calculated on the ranks (where ties are given ranks, as described above).
Determining significance
One approach to test whether an observed value of ρ is significantly different from zero (r will always maintain −1 ≤ r ≤ 1) is to calculate the probability that it would be greater than or equal to the observed r, given the null hypothesis, by using a permutation test. An advantage of this approach is that it automatically takes into account the number of tied data values there are in the sample, and the way they are treated in computing the rank correlation.
Another approach parallels the use of the Fisher transformation in the case of the Pearson productmoment correlation coefficient. That is, confidence intervals and hypothesis tests relating to the population value ρ can be carried out using the Fisher transformation:

F(r) = {1 \over 2}\ln{1+r \over 1r} = \operatorname{artanh}(r).
If F(r) is the Fisher transformation of r, the sample Spearman rank correlation coefficient, and n is the sample size, then

z = \sqrt{\frac{n3}{1.06}}F(r)
is a zscore for r which approximately follows a standard normal distribution under the null hypothesis of statistical independence (ρ = 0).^{[7]}^{[8]}
One can also test for significance using

t = r \sqrt{\frac{n2}{1r^2}}
which is distributed approximately as Student's t distribution with n − 2 degrees of freedom under the null hypothesis.^{[9]} A justification for this result relies on a permutation argument.^{[10]}
pvrank^{[11]} is a very recent R package that computes rank correlations and their pvalues with various options for tied ranks. It is possible to compute exact Spearman coefficient test pvalues for n ≤ 26.
A generalization of the Spearman coefficient is useful in the situation where there are three or more conditions, a number of subjects are all observed in each of them, and it is predicted that the observations will have a particular order. For example, a number of subjects might each be given three trials at the same task, and it is predicted that performance will improve from trial to trial. A test of the significance of the trend between conditions in this situation was developed by E. B. Page^{[12]} and is usually referred to as Page's trend test for ordered alternatives.
Correspondence analysis based on Spearman's rho
Classic correspondence analysis is a statistical method that gives a score to every value of two nominal variables. In this way the Pearson correlation coefficient between them is maximized.
There exists an equivalent of this method, called grade correspondence analysis, which maximizes Spearman's rho or Kendall's tau.^{[13]}
See also
References

^ Scale types

^ Lehman, Ann (2005). Jmp For Basic Univariate And Multivariate Statistics: A Stepbystep Guide. Cary, NC: SAS Press. p. 123.

^ ^{a} ^{b} Myers, Jerome L.; Well, Arnold D. (2003). Research Design and Statistical Analysis (2nd ed.). Lawrence Erlbaum. p. 508.

^ Maritz, J. S. (1981). DistributionFree Statistical Methods. Chapman & Hall. p. 217.

^ Yule, G. U.; Kendall, M. G. (1968). An Introduction to the Theory of Statistics (14th ed.). Charles Griffin & Co. p. 268.

^ Piantadosi, J.; Howlett, P.; Boland, J. (2007). "Matching the grade correlation coefficient using a copula with maximum disorder". Journal of Industrial and Management Optimization 3 (2): 305–312.

^ Choi, S. C. (1977). "Tests of Equality of Dependent Correlation Coefficients".

^ Fieller, E. C.; Hartley, H. O.; Pearson, E. S. (1957). "Tests for rank correlation coefficients. I". Biometrika 44: 470–481.

^ Press; Vettering; Teukolsky; Flannery (1992). Numerical Recipes in C: The Art of Scientific Computing (2nd ed.). p. 640.

^ Kendall, M. G.; Stuart, A. (1973). The Advanced Theory of Statistics, Volume 2: Inference and Relationship. Griffin. (Sections 31.19, 31.21)

^ Amerise, I.L.; Marozzi, M.; Tarsitano, A. "R package pvrank".

^ Page, E. B. (1963). "Ordered hypotheses for multiple treatments: A significance test for linear ranks". Journal of the American Statistical Association 58 (301): 216–230.

^ Kowalczyk, T.; Pleszczyńska E. , Ruland F. (eds.) (2004). Grade Models and Methods for Data Analysis with Applications for the Analysis of Data Populations. Studies in Fuzziness and Soft Computing 151. Berlin Heidelberg New York: Springer Verlag.
Further reading

Corder, G.W. & Foreman, D.I. (2014). Nonparametric Statistics: A StepbyStep Approach, Wiley. ISBN 9781118840313.

Daniel, Wayne W. (1990). "Spearman rank correlation coefficient". Applied Nonparametric Statistics (2nd ed.). Boston: PWSKent. pp. 358–365.

Spearman C (1904). "The proof and measurement of association between two things". Amer. J. Psychol. 15: 72–101.

Bonett DG, Wright, TA (2000). "Sample size requirements for Pearson, Kendall, and Spearman correlations". Psychometrika. 65: 23–28.

Kendall MG (1970). Rank correlation methods (4th ed.). London: Griffin.

Hollander M, Wolfe DA (1973). Nonparametric statistical methods. New York: Wiley.

Caruso JC, Cliff N (1997). "Empirical size, coverage, and power of confidence intervals for Spearman's Rho". Ed. and Psy. Meas. 57: 637–654.
External links

"Understanding Correlation vs. Copulas in Excel" by Eric Torkia, Technology Partnerz 2011

Table of critical values of ρ for significance with small samples

Spearman's rank online calculator

Spearman correlation calculator with humanreadable explanation

Chapter 3 part 1 shows the formula to be used when there are ties

An example of how to calculate Spearman's Rho along with basic R code.

Spearman's rank correlation: Simple notes for students with an example of usage by biologists and a spreadsheet for Microsoft Excel for calculating it (a part of materials for a Research Methods in Biology course).
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