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Studentized Range
In statistics, the studentized range, denoted ''q'', is the difference between the largest and smallest data in a sample normalized by the sample standard deviation. It is named after William Sealy Gosset (who wrote under the pseudonym "''Student''"), and was introduced by him in 1927. The concept was later discussed by Newman (1939), Keuls (1952), and John Tukey in some unpublished notes. Its statistical distribution is the ''studentized range distribution'', which is used for multiple comparison procedures, such as the single step procedure Tukey's range test, the Newman–Keuls method, and the Duncan's step down procedure, and establishing confidence intervals that are still valid after data snooping has occurred. Description The value of the studentized range, most often represented by the variable ''q'', can be defined based on a random sample ''x''1, ..., ''x''''n'' from the ''N''(0, 1) distribution of numbers, and another random variable ''s'' that is independen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") 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 statistical survey, surveys and experimental design, experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Null Hypothesis
In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is due to chance alone, and an underlying causative relationship does not exist, hence the term "null". In addition to the null hypothesis, an alternative hypothesis is also developed, which claims that a relationship does exist between two variables. Basic definitions The ''null hypothesis'' and the ''alternative hypothesis'' are types of conjectures used in statistical tests, which are formal methods of reaching conclusions or making decisions on the basis of data. The hypotheses are conjectures about a statistical model of the population, which are based on a sample of the population. The tests are core elements of statistical inference, heavily used in the interpretation of scientific experimental data, to separate scientific claims fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Summary Statistics
In descriptive statistics, summary statistics are used to summarize a set of observations, in order to communicate the largest amount of information as simply as possible. Statisticians commonly try to describe the observations in * a measure of location, or central tendency, such as the arithmetic mean * a measure of statistical dispersion like the standard deviation, standard mean absolute deviation * a measure of the shape of the distribution like skewness or kurtosis * if more than one variable is measured, a measure of correlation and dependence, statistical dependence such as a Pearson product-moment correlation coefficient, correlation coefficient A common collection of order statistics used as summary statistics are the five-number summary, sometimes extended to a seven-number summary, and the associated box plot. Entries in an analysis of variance table can also be regarded as summary statistics. Examples Location Common measures of location, or central tendency, are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Studentized Range Distribution
In probability and statistics, studentized range distribution is the continuous probability distribution of the studentized range of an i.i.d. sample from a normally distributed population. Suppose that we take a sample of size ''n'' from each of ''k'' populations with the same normal distribution ''N''(''μ'', ''σ''2) and suppose that \bar_ is the smallest of these sample means and \bar_ is the largest of these sample means, and suppose ''s''² is the pooled sample variance from these samples. Then the following statistic has a Studentized range distribution. :q = \frac Definition Probability density function Differentiating the cumulative distribution function with respect to ''q'' gives the probability density function. ::f_\text(q;k,\nu) = \frac\int_0^\infty s^\nu \, \varphi(\sqrt \,s)\,\left int_^\infty \varphi(z+q\,s)\,\varphi(z)\, \left[\Phi(z+q\,s)-\Phi(z)\right \, \mathrmz\right] \, \mathrms Note that in the outer part of the integral, the equation ::\varphi(\s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Studentized Residual
In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation. It is a form of a Student's ''t''-statistic, with the estimate of error varying between points. This is an important technique in the detection of outliers. It is among several named in honor of William Sealey Gosset, who wrote under the pseudonym ''Student''. Dividing a statistic by a sample standard deviation is called studentizing, in analogy with standardizing and normalizing. Motivation The key reason for studentizing is that, in regression analysis of a multivariate distribution, the variances of the ''residuals'' at different input variable values may differ, even if the variances of the ''errors'' at these different input variable values are equal. The issue is the difference between errors and residuals in statistics, particularly the behavior of residuals in regressions. Consider the simple linear regression model : Y = \a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
