Dear Mohsen Ahmadkhani,

Pearson's correlation is a measure of the linear relationship between two continuous random variables. It does not assume normality although it does assume finite variances and finite covariance. When the variables are bivariate normal, Pearson's correlation provides a complete description of the association.

Spearman's correlation applies to ranks and so provides a measure of a monotonic relationship between two continuous random variables. It is also useful with ordinal data and is robust to outliers (unlike Pearson's correlation).

The distribution of either correlation coefficient will depend on the underlying distribution, although both are asymptotically normal because of the central limit theorem.

Don't forget Kendall's tau! Roger Newson has argued for the superiority of Kendall's τa over Spearman's correlation rS as a rank-based measure of correlation in a paper whose full text is now freely available online:

Newson R. Parameters behind "nonparametric" statistics: Kendall's tau,Somers' D and median differences. Stata Journal 2002; 2(1):45-64.

He references (on p47) Kendall & Gibbons (1990) as arguing that "...confidence intervals for Spearman’s rS are less reliable and less interpretable than confidence intervals for Kendall’s τ-parameters, but the sample Spearman’s rS is much more easily calculated without a computer" (which is no longer of much importance of course). Unfortunately I don't have easy access to a copy of their book:

Kendall, M. G. and J. D. Gibbons. 1990. Rank Correlation Methods. 5th ed. London: Griffin.

The issue is that non normal data will violate the tests of significance.  With cumulative data you will violate the assumption.  If you use non cumulative data you might be fine--depends on if each frequency is skewed. 

If your data are badly skewed, bimodal, or otherwise violate the assumptions of the general linear model, then you'd be better off using Spearman's rho (rank-order correlation) or Kendall's tau.

For Spearman's rho, the first step is to rank-order the results. Assign a value of '1' to the lowest value, "2" to the next, and so forth. (In other words, you're creating a new variable based on the scores.) If you're calculating the results by hand, you can find the equation in most statistics textbooks. But it works out the same as Pearson's r, so if you're using SPSS or some other stats program, just run the usual correlation procedure on the rank-ordered data.

Basically, you're exercising due caution so that 1 or 2 extreme values don't exert undue influence on your results. It's a wise choice.

Dear Mohsen

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Good Luck

In case of non-linear observations, rank correlation  would be appropriate not Pearson's  correlation method.

Certainly need to to use rank correlation for non-normally distributed data as it will keep you free from any unexpected correlations obtained from Pearson correlation analysis.

you need to use non parametric statistics.

Can someone give me a solution for finding correlation Coefficient where there are many repeated values while ranking. How can I avoid Distortion?

In some cases, continuous data may obey well with both methods. then Pearson r is more appropriate in such cases.

nominal or ordinals should test non parametric way. If assumptions are fulfilled best option is Pearson r than non parametric alternatives

Dineth Samarawickrama I am working on wind turbine's SCADA data. It has non-normal distribution but is continuous time-series data e.g. power generated over a year. I applied both Spearman and Pearson and their results are different. I have searched for relevant work which one people are using for this kind of data, but there is confusion as both have been used. What should we do in such cases?

Article On the Selection of Input Variables for a Wind Turbine Load ...

Article Improved power curve monitoring of wind turbines

Nadia Masood Khan, It is difficult to say which method perfect to your study. considering the method of calculation, spearman rank correlation conducted through ranking and then correlation coefficient calculated using assigned ranks. But the Pearson correlation coefficient calculated based on the raw data.

If normality assumption violates more, non-parametric method is prescribed.

However, my opinion is in most cases, Pearson correlation coefficient is good measure than Spearman. However, it depends on the type and nature of the data. we have to check nature of data and assumptions.

Nadia Masood Khan - What you have is an interesting finding in & of itself! If the results are radically different (e.g., Pearson's r = .12 but rank-ordered correlation = .30) with one clearly nonsignificant and the other clearly significant, then my guess is that the data are very badly skewed, so the rank-ordered correlation is a more accurate measure. But it would be fun to report both, emphasizing which one is to be trusted and raising awareness in your field of the fact that choice of statistical method really makes a difference!

One other thought, though: Remember that all correlation coefficients are meant to estimate the strength of linear relationships. If the relationship in question is not linear, then correlational analysis will be unrevealing. For instance, what if there is a threshold effect, such that on one side of the threshold there is really no relationship between the variables, but on the other there is a strong one? In your case, maybe the relationship only holds when wind speed is very high, so you have just a few measurements of winds fast enough to make the difference. (I don't know exactly what your study is about & I'm no meteorologist anyway, but I hope you get the general idea.)

Hey, Stephen Joy thank you so much for giving a detailed answer to my question. This certainly gave me a better idea of how to proceed. You are right that the difference must be reported.

Stephen Joy thank you, very nice answer with simply stated insight.

Nadia Masood Khan there are several ways to select best method. first check normality assumptions of data. if data obeys normality assumptions, then test with pearson method is the perfect way. if normality assumption fails then test with Spearman method or Kendall Tau.

Another way is if data falls under scale variable (continous or discrete - integers), some people prefer to do Pearson correlation than Spearman.

I would like to point assumptions to be considered prior to do Pearson correlation

1. Data should be continuous variables (any value within a interval)

2. Normality (data points close to mean)

3. Linearity (data follows linear relationship)

4. Homoscedascity (Two variables has equal variances)

5. Paired observations (data points must be in pairs)

6. No outliers in the data (data points not significantly skewed)

Scatter plots are one of best tool to check above assumptions. You can test Shapiro-wilk test, levene tests for obtaining clear picture of scenario.

My personal opinion is not to present Pearson and Spearman both and if more assumptions met, you can go to Pearson method. If not it satisfied best to go Spearman or Kendall Tau.

Best of luck

Pearson's correlation coefficient is very efficient for measuring strength of relationship between normally distributed data. For non-normal data, I would advise Spearman rank correlation method.

Have you got the answer？