The square of a random variable having standard normal distribution is distributed as chi-square with 1 degree of freedom. The sum of squares of 'n' independently distributed standard normal variables has a Chi-Square distribution with 'n' degrees of freedom. The distribution is typically used to compare multiple-sample count data in contingency tables to expected values under a null hypothesis.

Factorial ANOVA:

Factorial ANOVA (factorial analysis of variance ) is aimed at assessing the relative importance of various combinations of independent variables. Factorial ANOVA is used when there are at least two independent variables.

A statistical technique which helps in making inference whether three or more samples might come from populations having the same mean; specifically, whether the differences among the samples might be caused by chance variation

An ordinal scale is a measurement scale that assigns values to objects based on their ranking with respect to one another. For example, a doctor might use a scale of 0-10 to indicate degree of improvement in some condition, from 0 (no improvement) to 10 (disappearance of the condition). While you know that a 4 is better than a 2, there is no implication that a 4 is twice as good as a 2. Nor is the improvement from 2 to 4 necessarily the same "amount" of improvement as the improvement from 6 to 8. All we know is that there are 11 categories, with 1 being better than 0, 2 being better than 1, etc.

Descriptive Statistics

Descriptive statistics refers to statistical techniques used to summarize and describe a data set, and also to the statistics (measures) used in such summaries. Measures of central tendency (e.g. mean, median) and variation (e.g. range, standard deviation) are the main descriptive statistics. Displays of data such as histograms and box-plots are also considered techniques of descriptive statistics.

A ratio scale is a measurement scale in which a certain distance along the scale means the same thing no matter where on the scale you are, and where "0" on the scale represents the absence of the thing being measured. Thus a "4" on such a scale implies twice as much of the thing being measured as a "2."

Interval Scale:

An interval scale is a measurement scale in which a certain distance along the scale means the same thing no matter where on the scale you are, but where "0" on the scale does not represent the absence of the thing being measured. Fahrenheit and Celsius temperature scales are examples

Nominal Scale:

A nominal scale is really a list of categories to which objects can be classified. For example, people who receive a mail order offer might be classified as "no response," "purchase and pay," "purchase but return the product," and "purchase and neither pay nor return." The data so classified are termed categorical data.

The mean score statistic is one of the statistics used in the generalized Cochran-Mantel-Haenszel tests . It is applicable when the response levels (columns) are measured at an ordinal scale .

If the two variables are independent of each other in all strata, the asymptotic distribution of the mean score statistic is the chi-square distribution with (R-1) degrees of freedom , where R is the number of the treatment groups (rows).

The mean score statistic tends to take on higher values if the mean scores of the response vary across the treatment groups in at least one strata. This statistic is also called "nonparametric ANOVA statistic".

Hypothesis testing (also called "significance testing") is a statistical procedure for discriminating between two statistical hypotheses - the null hypothesis (H₀) and the alternative hypothesis ( H_a, often denoted as H₁). Hypothesis testing rests on the presumption of validity of the null hypothesis - that is, the null hypothesis is accepted unless the data at hand testify strongly enough against it.

The philosophical basis for hypothesis testing lies in the fact that random variation pervades all aspects of life, and in the desire to avoid being fooled by what might be chance variation. The alternative hypothesis typically describes some change or effect that you expect or hope to see confirmed by data. For example, new drug A works better than standard drug B. Or the accuracy of a new weapon targeting system is better than historical standards. The null hypothesis embodies the presumption that nothing has changed, or that there is no difference.

Hypothesis testing comes into play if the observed data do, in fact, suggest that the alternative hypothesis is true (the new drug produces better survival times than the old one in an experiment, for example). We ask the question "is it possible that chance variation might have produced this result?"

As noted, the null hypothesis stands ("is accepted") unless the data at hand provide strong enough evidence against it. "Strong enough" means that probability that you would obtain a result as extreme as the observed result, given that the null hypothesis is true, is small enough (usually < 0.05) given the null hypothesis is true.

A statistical test is a procedure for statistical hypothesis testing . The outcome of a statistical test is a decision to reject or accept the null hypothesis for given probability of type I error .

The outcome is frequently reported as p-value - the minimal level of significance for which the null hypothesis is still being rejected. The smaller the p-value, the stronger the evidence against the null hypothesis.

In a test of significance, Type II error is the error of accepting the null hypothesis when it is false -- of failing to declare a real difference as statistically significant. Obviously, the bigger your samples, the more likely your test is to detect any difference that exists. The probability of detecting a real difference of specified size (i.e. of not committing a Type II error) is called the power of the test

In statistical inference procedures (hypothesis tests and confidence intervals), nonparametric procedures are those that are relatively free of assumptions about population parameters. For an example of a nonparametric test, see sign test.

In hypothesis testing, the null hypothesis is the one you are hoping can be disproven by the observed data. Typically, it asserts that chance variation is responsible for an effect seen in observed data (for example, a difference between treatment and placebo, an apparent correlation between one variable and another, a divergence between a sample measure and some benchmark, etc.)

Post-hoc tests (or post-hoc comparison tests) are used at the second stage of the analysis of variance (ANOVA) or multiple analysis of variance (MANOVA) if the null hypothesis is rejected. The question of interest at this stage is which groups significantly differ from others in respect to the mean or, in case of MANOVA, in respect to centroid s.

The Bonferroni adjustment and the Dunn test are examples of post-hoc testing procedures.