As described by Joshua Isaac Walters, "the assumption of homoskedasticity simplifies mathematical and computational treatment and usually leads to adequate estimation results (e.g. in data mining) even if the assumption is not true." Serious violations in homoskedasticity (assuming a distribution of data is homoskedastic when in actuality it is heteroskedastic) result in underemphasizing the Pearson coefficient.
In a scatterplot of data, homoskedasticity looks like an oval (most x values are concentrated around the mean of y, with fewer and fewer x values as y becomes more extreme in either direction). If a scatterplot looks like any geometric shape other than an oval, the rules of homoskedasticity may have been violated.
Residuals can be tested for homoskedasticity using the Breusch-Pagan test, which regresses square residuals to independent variables. The BP test is sensitive to normality so for general purpose the Koenkar-Basset or generalized Breusch-Pagan test statistic is used. For testing for groupwise heteroskedasticity, the Goldfeld-Quandt test is needed.
Two or more normal distributions, , are homoskedastic if they share a common covariance (or correlation) matrix, . Homoskedastic distributions are especially useful to derive statistical pattern recognition and machine learning algorithms. One popular example is Fisher's linear discriminant analysis.