Normal Distribution - Normality Tests

Normality Tests

Normality tests assess the likelihood that the given data set {x₁, …, x_n} comes from a normal distribution. Typically the null hypothesis H₀ is that the observations are distributed normally with unspecified mean μ and variance σ2, versus the alternative H_a that the distribution is arbitrary. A great number of tests (over 40) have been devised for this problem, the more prominent of them are outlined below:

• "Visual" tests are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.
  • Q-Q plot — is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it's a plot of point of the form (Φ−1(p_k), x_(k)), where plotting points p_k are equal to p_k = (k − α)/(n + 1 − 2α) and α is an adjustment constant which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
  • P-P plot — similar to the Q-Q plot, but used much less frequently. This method consists of plotting the points (Φ(z_(k)), p_k), where . For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).
  • Wilk–Shapiro test employs the fact that the line in the Q-Q plot has the slope of σ. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
  • Normal probability plot (rankit plot)

• Moment tests:
  • D'Agostino's K-squared test
  • Jarque–Bera test

• Empirical distribution function tests:
  • Lilliefors test (an adaptation of the Kolmogorov–Smirnov test)
  • Anderson–Darling test