Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures used to analyze the differences among means. ANOVA was developed by the statistician Ronald Fisher. ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the t-test beyond two means. In other words, the ANOVA is used to test the difference between two or more means.
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments.
A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population parameter, describing a sample, or evaluating a hypothesis. The average of sample values is a statistic. The term statistic is used both for the function and for the value of the function on a given sample. When a statistic is being used for a specific purpose, it may be referred to by a name indicating its purpose.
A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis.
Nonparametric statistics is the branch of statistics that is not based solely on parametrized families of probability distributions. Nonparametric statistics is based on either being distribution-free or having a specified distribution but with the distribution's parameters unspecified. Nonparametric statistics includes both descriptive statistics and statistical inference. Nonparametric tests are often used when the assumptions of parametric tests are violated.
In inferential statistics, the null hypothesis is that two possibilities are the same. The null hypothesis is that the observed difference is due to chance alone. Using statistical tests, it is possible to calculate the likelihood that the null hypothesis is true.
A chi-squared test is a statistical hypothesis test that is valid to perform when the test statistic is chi-squared distributed under the null hypothesis, specifically Pearson's chi-squared test and variants thereof. Pearson's chi-squared test is used to determine whether there is a statistically significant difference between the expected frequencies and the observed frequencies in one or more categories of a contingency table.
The statistical power of a binary hypothesis test is the probability that the test correctly rejects the null hypothesis when a specific alternative hypothesis is true. It is commonly denoted by , and represents the chances of a "true positive" detection conditional on the actual existence of an effect to detect. Statistical power ranges from 0 to 1, and as the power of a test increases, the probability of making a type II error by wrongly failing to reject the null hypothesis decreases.
An F-test is any statistical test in which the test statistic has an F-distribution under the null hypothesis. It is most often used when comparing statistical models that have been fitted to a data set, in order to identify the model that best fits the population from which the data were sampled. Exact "F-tests" mainly arise when the models have been fitted to the data using least squares. The name was coined by George W. Snedecor, in honour of Sir Ronald A. Fisher. Fisher initially developed the statistic as the variance ratio in the 1920s.
A Z-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. Z-tests test the mean of a distribution. For each significance level in the confidence interval, the Z-test has a single critical value which makes it more convenient than the Student's t-test whose critical values are defined by the sample size. Both the Z test and Student's t-test have similarities in that they both help determine the significance of a set of data. However, the z-test is rarely used in practice because the population deviation is difficult to determine.
In statistics, the Mann–Whitney U test is a nonparametric test of the null hypothesis that, for randomly selected values X and Y from two populations, the probability of X being greater than Y is equal to the probability of Y being greater than X.
The t-test is any statistical hypothesis test in which the test statistic follows a Student's t-distribution under the null hypothesis.
In statistical significance testing, a one-tailed test and a two-tailed test are alternative ways of computing the statistical significance of a parameter inferred from a data set, in terms of a test statistic. A two-tailed test is appropriate if the estimated value is greater or less than a certain range of values, for example, whether a test taker may score above or below a specific range of scores. This method is used for null hypothesis testing and if the estimated value exists in the critical areas, the alternative hypothesis is accepted over the null hypothesis. A one-tailed test is appropriate if the estimated value may depart from the reference value in only one direction, left or right, but not both. An example can be whether a machine produces more than one-percent defective products. In this situation, if the estimated value exists in one of the one-sided critical areas, depending on the direction of interest, the alternative hypothesis is accepted over the null hypothesis. Alternative names are one-sided and two-sided tests; the terminology "tail" is used because the extreme portions of distributions, where observations lead to rejection of the null hypothesis, are small and often "tail off" toward zero as in the normal distribution, colored in yellow, or "bell curve", pictured on the right and colored in green.
A test statistic is a statistic used in statistical hypothesis testing. A hypothesis test is typically specified in terms of a test statistic, considered as a numerical summary of a data-set that reduces the data to one value that can be used to perform the hypothesis test. In general, a test statistic is selected or defined in such a way as to quantify, within observed data, behaviours that would distinguish the null from the alternative hypothesis, where such an alternative is prescribed, or that would characterize the null hypothesis if there is no explicitly stated alternative hypothesis.
The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test used either to test the location of a population based on a sample of data, or to compare the locations of two populations using two matched samples. The one-sample version serves a purpose similar to that of the one-sample Student's t-test. For two matched samples, it is a paired difference test like the paired Student's t-test. The Wilcoxon test can be a good alternative to the t-test when population means are not of interest; for example, when one wishes to test whether a population's median is nonzero, or whether there is a better than 50% chance that a sample from one population is greater than a sample from another population.
This glossary of statistics and probability is a list of definitions of terms and concepts used in the mathematical sciences of statistics and probability, their sub-disciplines, and related fields. For additional related terms, see Glossary of mathematics.
A permutation test is an exact statistical hypothesis test making use of the proof by contradiction in which the distribution of the test statistic under the null hypothesis is obtained by calculating all possible values of the test statistic under possible rearrangements of the observed data. Permutation tests are, therefore, a form of resampling.
The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measures can be used in statistical hypothesis testing, e.g. to test for normality of residuals, to test whether two samples are drawn from identical distributions, or whether outcome frequencies follow a specified distribution. In the analysis of variance, one of the components into which the variance is partitioned may be a lack-of-fit sum of squares.
The sign test is a statistical method to test for consistent differences between pairs of observations, such as the weight of subjects before and after treatment. Given pairs of observations for each subject, the sign test determines if one member of the pair tends to be greater than the other member of the pair.
In statistics, resampling is any of a variety of methods for doing one of the following:
- Estimating the precision of sample statistics by using subsets of available data (jackknifing) or drawing randomly with replacement from a set of data points (bootstrapping)
- Permutation tests are exact tests: Exchanging labels on data points when performing significance tests
- Validating models by using random subsets
