Glossary

The significance level of a statistical test.

Distribution function of the standard normal distribution: .

The -quantile of the -distribution with degrees of freedom (Table B.3 and Table B.4).

The -quantile of the t-distribution with degrees of freedom (Table B.2).

The -quantile of the standard normal distribution (Table B.1): .

The absolute value of .

The sample mean of a sample : .

Continuity correction

A continuity correction is often applied when approximating the cumulative probability function of a discrete random variable by the standard normal distribution function. Usually a correction factor of is used such that .

Empirical distribution function (EDF)

Let be a descending ordered sample, then the EDF is defined as:

Distribution function of the F-distribution with and degrees of freedom.

The -quantile of the F-distribution with and degrees of freedom (Tables B.5–B.7).

The null hypothesis of a test problem.

The alternative hypothesis of a test problem.

The sample size of a sample .

Ranks

Let be a sample. The ordered sample (from the lowest to the highest value) is . Then of is the rank of the corresponding value . For example, let be a sample of size 5, then the ordered sample is: . The rank of the sample value 2 is 1, the rank of sample value 3 is 2, and the rank of sample value 9 is 5.

Run

Let observations of random variable and observations of random variable be given. Assume that both samples are combined and (if at least ordinal) are arranged in increasing or time of occurrence order. A run is a group of successive observations generated from the same random variable. The same idea can be applied if the observations are coming from a binary random variable. For example, a coin is tossed 10 times; the result of these tosses are either (H)eads or (T)ails. The observed sequence is: HH T HH TTT H. This sequence has five runs, namely , , , , .

Mid ranks

This is a way of dealing with tied values, which are identical values in an ordered sequence. The same rank is assigned to these values, namely the mean of their ranks. For example, let be a sample. It is unclear if the observations 1, 3, or 4 will get the ranks 2, 3, or 4. The arithmetic mean of the ranks of the tied values is , so each value 4 will get the mid rank 3. The rank vector is while the sum of ranks is still 15.

p-value

The probability of observing a sample as discrepant with the null hypothesis as the observed sample under the null hypothesis.

Ties

If one or more observations in a sample have the same value they are called tied values.

Characteristic function: