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Simple back-of-the-envelope test takes the sample maximum and minimum and computes their z-score, or more properly t-statistic (number of sample standard deviations that a sample is above or below the sample mean), and compares it to the 68–95–99.7 rule: if one has a 3σ event (properly, a 3s event) and substantially fewer than 300 samples, or a 4s event and substantially fewer than 15,000 ...
The Shapiro–Wilk test tests the null hypothesis that a sample x 1, ..., x n came from a normally distributed population. The test statistic is = (= ()) = (¯), where with parentheses enclosing the subscript index i is the ith order statistic, i.e., the ith-smallest number in the sample (not to be confused with ).
The sample extrema can be used for a simple normality test, specifically of kurtosis: one computes the t-statistic of the sample maximum and minimum (subtracts sample mean and divides by the sample standard deviation), and if they are unusually large for the sample size (as per the three sigma rule and table therein, or more precisely a Student ...
Student's t-test assumes that the sample means being compared for two populations are normally distributed, and that the populations have equal variances. Welch's t-test is designed for unequal population variances, but the assumption of normality is maintained. [1] Welch's t-test is an approximate solution to the Behrens–Fisher problem.
Multivariate normality tests include the Cox–Small test [33] and Smith and Jain's adaptation [34] of the Friedman–Rafsky test created by Larry Rafsky and Jerome Friedman. [35] Mardia's test [36] is based on multivariate extensions of skewness and kurtosis measures. For a sample {x 1, ..., x n} of k-dimensional vectors we compute
Heckman's correction involves a normality assumption, provides a test for sample selection bias and formula for bias corrected model. Suppose that a researcher wants to estimate the determinants of wage offers, but has access to wage observations for only those who work.
As a reference, a straight line can be fit to the points. The further the points vary from this line, the greater the indication of departure from normality. If the sample has mean 0, standard deviation 1 then a line through 0 with slope 1 could be used. With more points, random deviations from a line will be less pronounced.
Lilliefors test is a normality test based on the Kolmogorov–Smirnov test.It is used to test the null hypothesis that data come from a normally distributed population, when the null hypothesis does not specify which normal distribution; i.e., it does not specify the expected value and variance of the distribution. [1]