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Please help improve this article by adding citations to reliable sources. Unverifiable material may be challenged and removed. | In statistics, Spearman's rank correlation coefficient, named after Charles Spearman and often denoted by the Greek letter ρ (rho) or as rs, is a non-parametric measure of correlation – that is, it assesses how well an arbitrary monotonic function could describe the relationship between two variables, without making any assumptions about the frequency distribution of the variables. Image File history File links Question_book-3. ...
This article is about the field of statistics. ...
Charles Edward Spearman (September 10, 1863 - September 7, 1945) was an English psychologist known for work in statistics, as a pioneer of factor analysis, and for Spearmans rank correlation coefficient. ...
Rho (upper case Ρ, lower case Ï) is a letter of the Greek alphabet. ...
Non-Parametric statistics are statistics where it is not assumed that the population fits any parametrized distributions. ...
Positive linear correlations between 1000 pairs of numbers. ...
In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ...
In computer science and mathematics, a variable (pronounced ) (sometimes called an object or identifier in computer science) is a symbolic representation used to denote a quantity or expression. ...
In statistics, a frequency distribution is a list of the values that a variable takes in a sample. ...
In computer science and mathematics, a variable is a symbol denoting a quantity or symbolic representation. ...
Assumptions and alternatives
Unlike the Pearson product-moment correlation coefficient, Spearman's rank correlation coefficient does not require the assumption that the relationship between the variables is linear, nor does it require the variables to be measured on interval scales; it can be used for variables measured at the ordinal level. In statistics, the Pearson product-moment correlation coefficient (sometimes known as the PMCC) (r) is a measure of the correlation of two variables X and Y measured on the same object or organism, that is, a measure of the tendency of the variables to increase or decrease together. ...
Graph sample of linear equations A linear equation is an algebraic equation in which each term is either a constant or the product of a constant times the first power of a variable. ...
The level of measurement of a variable in mathematics and statistics is a classification that was proposed in order to describe the nature of information contained within numbers assigned to objects and, therefore, within the variable. ...
The level of measurement of a variable in mathematics and statistics is a classification that was proposed in order to describe the nature of information contained within numbers assigned to objects and, therefore, within the variable. ...
However, Spearman's rho does assume that subsequent ranks indicate equi-distant positions on the variable measured. For example, using Spearman's rho for Likert scales often used in psychology, sociology, biology and related disciplines assumes that the (psychologically) "felt distances" between scale points are the same for all betweens of the Likert scale used. A Likert scale (pronounced lick-urt) is a type of psychometric response scale often used in questionnaires, and is the most widely used scale in survey research. ...
Where equi-distance cannot be justified, correlation between ordinal-level variables can be calculated by using Kendall's  (tau). The Kendall tau rank correlation coefficient (or simply the Kendall tau coefficient, Kendalls Ï„ or Tau test(s)) is used to measure the degree of correspondence between two rankings and assessing the significance of this correspondence. ...
Calculation In principle, ρ is simply a special case of the Pearson product-moment coefficient in which the data are converted to rankings before calculating the coefficient.[1] In practice, however, a simpler procedure is normally used to calculate ρ. The raw scores are converted to ranks, and the differences d between the ranks of each observation on the two variables are calculated. This article needs to be cleaned up to conform to a higher standard of quality. ...
In statistics and data analysis, a raw score is an original datum that has not been transformed – for example, the original result obtained by a student on a test (i. ...
If there are no tied ranks, i.e. 
then ρ is given by:
 where: - di = the difference between each rank of corresponding values of x and y, and
- n = the number of pairs of values.
If tied ranks exist, classic Pearson's correlation coefficient between ranks has to be used instead of this formula.[1] You have to assign the same rank to each of the equal values. It is an average of their positions in the ascending order of the values: In probability theory and statistics, correlation, also called correlation coefficient, is a numeric measure of the strength of linear relationship between two random variables. ...
An Example of Averaging Ranks
| Variable | Position in the descending order | Rank | | 0.8 | 5 | 5 | | 1.2 | 4 |  | | 1.2 | 3 | | | 2.3 | 2 | 2 | | 18 | 1 | 1 | Spearman's rank correlation coefficient is equivalent to Pearson correlation on ranks. The formula above is a short-cut to its product-moment form, assuming no tie. The product-moment form can be used in both tied and untied cases.
A version of this correlation is called Spearman's rho. In this case ranks are calculated as above, but in the formula of Pearson's correlation a standard deviation is taken as there were no ties.
Example The raw data used in this example is shown below. | IQ | Hours of TV per week. | | 106 | 7 | | 86 | 0 | | 100 | 27 | | 101 | 50 | | 99 | 28 | | 103 | 29 | | 97 | 20 | | 113 | 12 | | 112 | 6 | | 110 | 17 | The first step is to sort this data by the first column. Next, two more columns are created. Both of these are for ranking the first two columns. Notice how the rank of values that are the same is the mean of what their ranks would otherwise be. Then a column "d" is created to hold the differences between the two rank columns. Finally another column "d2" should be created. This is just column d squared.
After doing this process with the example data you should end up with something like:
| IQ (i) | Hours of TV per week (t) | rank (i) | rank (t) | d | d2 | | 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 | The values in the d2 column can now be added to find  . The value of n is 10. So these values can now be substituted back into the equation,  which evaluates to ρ = − 0.175758. 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 The modern approach to testing whether an observed value of ρ is significantly different from zero (we will always have 1 ≥ ρ ≥ −1) is to calculate the probability that it would be greater than or equal to the observed ρ, given the null hypothesis, by using a permutation test. This approach is almost always superior to traditional methods, unless the data set is so large that computing power is not sufficient to generate permutations, or unless an algorithm for creating permutations that are logical under the null hypothesis is difficult to devise for the particular case (but usually these algorithms are straightforward). In statistics, a null hypothesis is a hypothesis set up to be nullified or refuted in order to support an alternative hypothesis. ...
In statistics, resampling is any of a variety of methods for doing one of the following: Estimating the precision of sample statistics (medians, variances, percentiles) by using subsets of available data (jackknife) or drawing randomly with replacement from a set of data points (bootstrapping) Exchanging labels on data points when...
A data set (or dataset) is a collection of data, usually presented in tabular form. ...
Although the permutation test is often trivial to perform for anyone with computing resources and programming experience, traditional methods for determining significance are still widely used. The most basic approach is to compare the observed ρ with published tables for various levels of significance. This is a simple solution if the significance only needs to be known within a certain range or less than a certain value, as long as tables are available that specify the desired ranges. A reference to such a table is given below. However, generating these tables is computationally intensive and complicated mathematical tricks have been used over the years to generate tables for larger and larger sample sizes, so it is not practical for most people to extend existing tables.
An alternative approach available for sufficiently large sample sizes is an approximation to the Student's t-distribution. For sample sizes above about 20, the variable In probability and statistics, the t-distribution or Students t-distribution is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. ...
  has a Student's t-distribution in the null case (zero correlation). In the non-null case (i.e. to test whether an observed ρ is significantly different from a theoretical value, or whether two observed ρs differ significantly) tests are much less powerful, though the t-distribution can again be used.
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 we predict 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 we predict 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 and is usually referred to as Page's trend test for ordered alternatives. In statistics, the Page test for multiple comparisons between ordered alternatives is a generalisation of the test of the statistical significance of a correlation performed using Spearmans rank correlation coefficient. ...
See also The Kendall tau rank correlation coefficient (or simply the Kendall tau coefficient, Kendalls Ï„ or Tau test(s)) is used to measure the degree of correspondence between two rankings and assessing the significance of this correspondence. ...
In statistics, rank correlation is the study of relationships between different rankings on the same set of items. ...
In mathematics, Chebyshevs sum inequality, named after Pafnuty Chebyshev, states that if and then Chebyshevs sum inequality follows from the rearrangement inequality. ...
Let be real numbers and be any permutation of . ...
In statistics, the Pearson product-moment correlation coefficient (sometimes known as the PMCC) (r) is a measure of the correlation of two variables X and Y measured on the same object or organism, that is, a measure of the tendency of the variables to increase or decrease together. ...
External links Microsoft Excel (full name Microsoft Office Excel) is a spreadsheet application written and distributed by Microsoft for Microsoft Windows and Mac OS. It features calculation and graphing tools which, along with aggressive marketing, have made Excel one of the most popular microcomputer applications to date. ...
References - ^ a b Myers, Jerome L.; Arnold D. Well (2003). Research Design and Statistical Analysis, second edition, Lawrence Erlbaum, p. 508. ISBN 0805840370.
- C. Spearman, "The proof and measurement of association between two things" Amer. J. Psychol. , 15 (1904) pp. 72–101
- M.G. Kendall, "Rank correlation methods" , Griffin (1962)
- M. Hollander, D.A. Wolfe, "Nonparametric statistical methods" , Wiley (1973)
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