What is the correlation coefficient and how do you calculate it?

A "coefficient of correlation" measures the strength and direction of the association between two variables. Different types of correlation coefficients relate to different types of associations. Pearson's "r" only relates to linear associations and is indeed closely related to the slope of the simple linear regression line that "links" the two variables. Spearman's "rho" relates to monotonic associations overall, where variables simply vary together in the same monotonic direction: either both up / both down or one up / one down. Kendall's "tau" relates to concordant associations overall, where the change in one variable matches or not the change in the other variable. No correlation coefficient implies and/or represents direct causation.

A dimensionless measure of the linear association between two variables, usually lying in the interval from −1 to +1, with zero indicating the absence of correlation (but not necessarily the independence of the two variables)

I would define using the word "Co" & "Relation" , so a relation between two variables. But, pay attention to the nature of the variables (you do not face always numerical variables, you may also have categorical variables). So, correlation is not always linear and is not always calculated based on numeric variables.

Mudelsee on Correlation, Part 1 A correlation coefficient measures how strong a coupling (association, relation, etc.) is between two or among several stochastic processes. The most common setting is bivariate, where you have two processes. From the bivariate measures, the most commonly employed one is Pearson's correlation coefficient, r (note the correct spelling!). It estimates the degree of the *linear* relationship. The big success of Pearson'r in applied research, since over a century ago, is that it is a single, dimensionless number between -1 and +1: easy to report and understand. There are, however, a number of issues to be considered, see below.

The correlation coefficient is a normalized measurement of how two variables in a data set are linearly related. It is calculated by dividing the covariance of the variables by the product of their standard deviations. .

Mudelsee on Correlation, Part 2 This is now mathematics, and hence we give the formulas. I use a version with possible serial dependence since this is useful in many applied fields, including mine (climatology). Let the discrete-time stochastic process X(i) consist just of a mean value plus some noise: X(i) = mu_X + S_X * X_noise(i), where i is the time counter, mu_X is the mean, S_X > 0 is the variability (assumed constant over time) and X_noise(i) is a zero-mean, unit-standard-deviation process. The noise has a certain distributional shape (e.g., Gaussian), and it follows a certain serial dependence model (e.g., AR(1)). Assume analogously for the second process, Y(i): Y(i) = mu_Y + S_Y * Y_noise(i). (To be continued.)

Correlation is a relative measure used to find tight association between two variables with the values ranging from -1 to +1. In the portfolio context, we would like the assets to be as close to the former in order to reduce the portfolio risk. Correlation can also be derived from Covariance as shown below: Correlation = [Covariance/(Sd-1 x Sd-2)] where Sd are the standard deviation of the two Securities' returns. Say, Sd of Microsoft and GM stock returns.

Mathematically, Pearson's "r" is the "scaled standardised" covariance between X and Y: Cov(X,Y) / sqrt[Var(X) * Var(Y)], implying a "symmetric" role of both variables in explaining their covariance. This is important in light of the relationship between Pearson's "r" and the slope "beta" of the simple linear regression (SLR) line linking Y to X, i.e. their covariance scaled *only* to the variance of the variable chosen to lie on the x-axis, the "predictor" or "explanatory" variable: Cov(X,Y) / Var(X). This implies an "asymmetric" role in "explaining" the covariance, which in SLR is described only through the variability in the predictor X, as a sort of "residual variability" in the response variable Y per unit variance of the predictor X.

Sure, think of the correlation coefficient as a numerical "wingman" for two variables on a date, trying to see if they click. It's a number from -1 to 1 that tells us just how much these two are dancing in sync. A score of 1 means they’re doing a perfect tango, moving together step by step. A score of -1 means they’re in a dance-off, doing completely opposite moves. And a zero? That's like two people stepping on each other's feet – no rhythm, no connection, just awkwardness. Calculating it is like scoring their dance routine, considering how often they move in harmony, and adjusting for any solo improvisations. Ways to calculate in python: 1. NumPy: numpy.corrcoef() 2. SciPy: scipy.stats.pearsonr() 3. Pandas: pandas.DataFrame.corr()

As it can be seen from the formula it is always bounded to give values between -1 and 1, as it is the ratio of covariances in a normalized fashion.

The correct values... n = 10 Σx = 1825 Σy = 725 Σxy = 134375 Σx^2 = 335125 Σy^2 = 54625 r comes out to be 1.0 There's a perfect positive linear relationship between Height and Weight.

The correlation coefficient is a statistical measure that quantifies the degree to which two variables are linearly related. It assesses the strength and direction of a linear relationship between two sets of data. The most commonly used correlation coefficient is the Pearson correlation coefficient, often denoted by the symbol "r." r=1 indicates a perfect positive linear relationship. r=−1 indicates a perfect negative linear relationship. A positive correlation suggests that as one variable increases, the other tends to increase as well. A negative correlation indicates that as one variable increases, the other tends to decrease. Correlation only measures the strength and direction of a linear relationship between two variables.

Mudelsee on Correlation, Part 3 Then the correlation coefficient, ρ, is *on the process level* defined as ρ = E[{X(i) - mu_X} * {Y(i) - mu_Y}] / (S_X * S_Y), where E[] is the expectation operator. ρ measures the degree of the linear relationship between X and Y. ρ is between -1 and +1 (inclusive of the endpoints). That is, as a single dimensionless number, ρ has a lot to say about the first order of the relation between two processes. This is the reason for the success of correlation analyses in applied research, this makes ρ the most often employed statistical parameter ever. Note that still we speak about theoretical processes. The next contribution (Part 4) is on the estimation of ρ, on the *sample level*. (To be continued.)

There seems to be some error in the calculation, because the data now implies perfect correlation (i.e. r=1). The values of the sums shown here are mostly incorrect, except for the sum of x.

The formula is correct and the implementation of the formula is correct; however, the values are inaccurate. Regardless, our data is clearly linear so we would expect a line to be a decently accurate model, and thus would expect a high R-squared value. It also makes contextual sense that height and weight will be positively and (relatively) linearly related in a population.

Si dos variables son independientes, su coeficiente de correlación lineal tomará un valor cercano a 0, indicando no correlación lineal. Pero si el coeficiente de correlación lineal es 0, no podemos estar seguros de que las dos variables son independientes, ya que podría existir otro tipo de relación que no sea lineal, como por ejemplo, una relación cuadrática.

Mudelsee on Correlation, Part 4 Now we come to estimation. Let a bivariate sample of size n be given: {X(i), Y(i)}_i=1^i=n. The expression "_i=1^i=n" means that the counter, i, runs from 1 to n. Pearson (1896) introduced the correlation coefficient, r, as follows: r = (1/n) * Σ_i=1^i=n * { [X(i) -X_bar]/S_n,X } * { [Y(i) -Y_bar]/S_n,Y }, where X_bar = Σ_i=1^i=n [ X(i) / n ] and Y_bar = Σ_i=1^i=n [ Y(i) / n ] are the sample means and S_n,X = SQRT{ Σ_i=1^i=n [X(i) -X_bar]**2 /n} and S_n,Y = SQRT{ Σ_i=1^i=n [Y(i) -Y_bar]**2 /n} are the sample standard deviations calculated with denominator n. Now, important: r is an estimator of rho. (To be continued.)

Stress must be placed on the "linear" connotation of Pearson's "r". A Pearson's "r" close to 0 implies the absence of a *linear* relationship, and not the absence of any relationship (i.e. complete independence). Pearson's "r" with value 0 means complete independence *only if* the dataset has a bivariate joint Normal distribution.

Lack of correlation doesn't mean independence. It means the variables are not linearly related. For instance, correlation will be 0 for y = x^2, for x = -2,-1,1,2. However, they are clearly not independent as there is a quadratic relation between them.

For a simple linear set of data this is absolutely right! Because of this, we can reasonably assume that if we plug in a new parameter, the line will give us a reasonably accurate prediction.

Data scientists frequently evaluate variable relationships by using the Pearson correlation coefficient. Unfortunately, Pearson correlation can only be utilized for linear relationships, hence limiting its usefulness. ppscore is a Python library that overcomes this problem, by utilizing the Predictive Power Score. PPS is an asymmetric score that detects both linear and non-linear relationships, therefore being an excellent alternative to Pearson correlation.

One caveat also is to look at the significance of the correlation coefficient to test if the correlation is significantly different from 0

Pearson correlation just measures linear tendencies thus it is not the best option if there is no inherent linear relationship between two variables. It is the default option of most of the libraries, thus it makes it a little bit dangerous if you are not aware what are you measuring.

Pearson's "r" relates to *linear* associations and is closely related to the slope of the simple linear regression line that "links" the two variables (coinciding numerically with it when regressing standardised variables). Pearson's "r" has the same limitations that affect linear regression, the main ones being non-normal data distributions and presence of outliers or high-leverage points. If the relationship between variables is truly linear and data are at least close to a bivariate normal distribution, Kendall's, Spearman's and Pearson's will provide similar results, at least in sign, but in this specific case Pearson's will hold more statistical power, as it's specifically geared towards assessing this type of relationships.

The correlation coefficient is the measure of the magnitude/strength and direction of association between two continuous variables. The most common measure of association for two continuous variables is the Pearson’s Correlation Coefficient which explains about linear associations between two continuous variables. There are other know correlation coefficients such as Spearman’s, and Kendal’s Tau which differs in what they measures based on how the two variables vary. Correlation whether linear or not shouldn’t be confused with Causation which it doesn’t explain. The direction of association can also be explained by the sign of a slope coefficient in the regression model.

It can easily be demonstrated that: beta * StDev(X) = r * StDev(Y), where "beta" is the simple linear regression slope linking Y to X. A consequence of this is that if using standardised variables X_st and Y_st for linear regression, the value of the regression slope will match exactly the value of the Pearson's correlation coefficient between the two standardised variables. Caution should, however, be used when interpreting this: the correspondence is only purely numerical, because the correlation coefficient still considers both variables equally in terms of explaining their covariance, whereas the regression slope only explains it by considering the variability in the predictor X_st alone.

It is never too much to say that correlation does not imply causation. Other tools can assess causation, for instance, structural equation modeling (SEM) which is a well-known tool for epidemiological studies. Also, linear correlations can fit well as a tool in the arsenal of exploratory data analysis (EDA), especially considering the biological, physical, or chemical mechanism that theoretically explains the association between variables, whenever possible.

Another possible alternative is the mixture of Pearson’s and Spearman’s correlation coefficients. It gives better results than either Pearson’s or Spearman’s alone.

Correlation and are two concepts commonly confused. Two variables may be correlated, however be independent or not. If two variables are correlated this doesn't mean that if one of the moves in a way the other will move in the same expected direction. What it could happen is that if one variable is dependent on another, then if the first moves, then the second would move as well. Pearson's coefficient would no show they are correlated if the correlation between the two is not linear.