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Tests of Normality

This calculator tests normality using one of several methods. The calculator assumes that samples are drawn from a distribution specified by the user with the associated parameters.

Methods

The Omnibus method, described by D'Agostino and Pearson, combines the skew and kurtosis to test normality. If \(n < 20\), the result if not valid.

This method performs the Kolmogorov-Smirnov goodness of fit test against the normal distribution. This test works by comparing the empirical distribution function of the data to the CDF of the reference distrution (in this case, a normal distribution).

This methods tests the null hypothesis that the kurtosis is the same as that of a normal distribution. If \(n < 20\), the result if not valid.

This method performs the Shapiro-Wilk test of normality which compares the order statistics of the given distribution to that of a normal distribution.

This methods tests the null hypothesis that the skew is the same as that of a normal distribution. If \(n < 8\), the result if not valid.

Distribution

The beta distribution is a continuous distribution over the interval \([0, 1]\) parameterized by two shape parameters \(\alpha, \beta > 0\). The pdf of the beta distribution is given by:

\[ f(x; \alpha, \beta) = \frac{1}{\text{B}(\alpha, \beta)} x^{\alpha - 1} (1 - x)^{\beta - 1} \]

The Cauchy (or Lorentz) distribution is a symmetrical disitribution whose mean, variance and higher order moments are all undefined. The pdf of the Cauchy distribution is given by:

\[ f(x) = \frac{1}{\pi(1 + x^2)} \]

The chi-squared distribution arises from the sum of the squared values of \(\nu\) independant standard normal random variables. It is parameterized by \(\nu\), the degrees of freedom, and its pdf is given by:

\[ f(x; \nu) = \frac{x^{\nu/2 - 1}e^{-x/2}}{2^{\nu/2}\Gamma\left( \frac{k}{2} \right)} \]

for \(x > 0\)

The exponential distribution is a particular case of the gamma distribution where \(\alpha = 1\). It's pdf is given by:

\[ f(x; \lambda) = \lambda e^{-\lambda x} \]

for \(x, \lambda \gt 0\).

The f-distribution arises from the ratio of two scaled chi-squared distributions. It is parameterized by the degrees of freedom from the two chi-squared distributions, \(\nu_1\) and \(\nu_2\) and its pdf is given by:

\[ f(x; \nu_1, \nu_2) = \frac{1}{\text{B}\left( \frac{\nu_1}{2}, \frac{\nu_2}{2} \right)} \left( \frac{\nu_1}{\nu_2} \right)^{\nu_1/2} x^{\nu_1/2 - 1} \left( 1 + \frac{\nu_1}{\nu_2} x \right)^{-(\nu_1 + \nu_2) / 2} \]

for \(x \ge 0\).

The gamma distribution is parameterized by a shape, \(\alpha\), and rate, \(beta\). The pdf of the gamma distribution is given by:

\[ f(x; \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\beta x} \]

for \(x, \alpha, \beta > 0\). The gamme distribution can be alternatively parameterized by \(k = \alpha\) and \(\theta = 1/\beta\).

The right-skewed Gumbel distribition is a probability distribution commonly used when analyzing extreme values. The pdf is given by:

\[ f(x) = e^{-\left(x - e^{-x} \right)} \]

The left-skewed Gumbel distribition is a probability distribution commonly used when analyzing extreme values. The pdf is given by:

\[ f(x) = e^{\left(x - e^{x} \right)} \]

The Laplace distribution, also known as a double exponential distribution, is parameterized by a location, \(\mu\), and scale, \(b > 0\). Its pdf is given by:

\[ f(x; \mu, b) = \frac{1}{2b} \exp \left(-\frac{|x - \mu|}{b} \right) \]

A log-normal random variable is one whose logarithm is a normal random variable. The pdf of the log-normal distribution with mean, \(\mu\), and standard deviation, \(\sigma\) is given by:

\[ f(x; \mu, \sigma) = \frac{1}{x \sqrt{2 \sigma^2 \pi}} e^{-\frac{(\ln x - \mu)^2}{2\sigma^2}} \]

The \(t\)-distribution is parameterized by the degrees of freedom, \(\nu\), and has a pdf given by:

\[ f(x; \nu) = \frac{1}{\sqrt{\nu \pi} \text{ B} \left( \frac{1}{2}, \frac{\nu}{2} \right)} \left( 1 + \frac{x^2}{\nu} \right)^{-(\nu+1)/2} \]

The distributions can be shifted and scalled using a location and scale parameter. For many distributions, including the normal distribution, the location and scale parameter are their natural parameters.

Note:This calculator uses simulations to calculate sample size, power and alpha.

Note: For some tests and some distributions, as the sample size increases, the power decreases. This is because the distribution becomes more normal-like with respect to the aspects that the test tests. In that case, we will report the largest sample size with the appropriate power, or the minimium sample size for which the test is valid.

Power Calculation Parameters

To input multiple values, seperate them by a comma.

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Distributions's Parameters

Only single values are permited here.

Power Calculation Explanation

Solve For
The unknown you are interested in solving for.
N
The sample size used to test the hypothesis.
Alpha
The \(\alpha\) (Type I error rate) level of the hypothesis test.
Power
The power (1 - Type II error rate) of the hypothesis test.
Distribution
The distribution from which the sample is drawn
Show All Distribution
By default, we only show
Distribution Parameters
The values of the distribution's parameters described above
Location
The location parameter
Scale
The scale parameter

Note: For many of the methods, the power is invariant under location and/or scale changes

Calculation Results

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References

  • D'Agostino, R and Pearson, E.S. (1973), "Tests for Departure from Normality," Biometrica, 60, 613-622.
  • Shapiro, S.S. & Wilk, M.B. (1965), "An Analysis of Variance Test for Normality (Complete Samples)," Biometrica, 52, 591-611.