services CV publications contact Value-at-Risk statistical navigator SK
Two numerical variables are described using:
b) Scatter Plot
Three-dimensional histograms are used to visualize crosstabulations of values in two variables. They can be considered a conjunction of two simple (i.e., univariate) histograms, combined such that the frequencies of co-occurrences of values on the two analyzed variables can be examined.
The scatterplot visualizes a relation (correlation) between two variables X and Y (e.g., weight and height). Individual data points are represented in two-dimensional space (see below), where axes represent the variables (X on the horizontal axis and Y on the vertical axis). The x-y scatter plot is one of the most powerful tools for analyzing data.
Correlation coefficient is a measure of association between two numerical variables. Pearson's product moment correlation coefficient (1896) is a measure of linear association between the two variables:
Numerator is called covariance and measures how values of two variables vary together. Positive value means that they vary in the same direction, negative value means that they vary in the opposite direction and value near zero mean that they vary independently. Pearson coefficent is standardized covariance (divided by standard deviations). The correlations ranges between plus and minus one. 0 is no correlation, 1 is perfect positive correlation, −1 is perfect negative.R-squared (r 2) is called coefficient of determination. It is the proportion of the variation in Y that is accounted by the variation in X. R2 varies between zero (no linear relationship) and one (perfect linear relationship).
Pearson correlation is influenced by outliers, unequal variances, nonnormality, and nonlinearities. One of the chief competitors of the Pearson correlation coefficient is the Spearman-rank correlation coefficient. This latter correlation is calculated by applying the Pearson correlation formulas to the ranks of the data rather than to the actual data values themselves. In so doing, many of the distortions that infect the Pearson correlation are reduced considerably.
Because of its construction Spearman's rank correlation is difficult to interpret. Kendall's rank correlation (1948) improves upon this by reflecting the strength of the dependence between the variables being compared. Consider two samples, x and y, each of size n. The total number of possible pairings of x with y observations is n(n−1)/2. Now consider ordering the pairs by the x values and then by the y values. If x3 > y3 when ordered on both x and y then the third pair is concordant, otherwise the third pair is discordant. S is the difference between the number of concordant (ordered in the same way, nc ) and discordant (ordered differently, nd) pairs.
The gamma coefficient is given as a measure of association that is highly resistant to tied data (Goodman and Kruskal, 1963):
Gamma is also a probability; specifically, it is computed as the difference between the probability that the rank ordering of the two variables agree minus the probability that they disagree, divided by 1 minus the probability of ties. Thus, Gamma is basically equivalent to Kendall tau, except that ties are explicitly taken into account.