# Tukey's range test

Tukey methodTukey range testTukey's HSD
Tukey's range test, also known as the Tukey's test, Tukey method, Tukey's honest significance test, or Tukey's HSD (honestly significant difference) test, is a single-step multiple comparison procedure and statistical test.wikipedia
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### John Tukey

TukeyTukey, JohnJohn W. Tukey
Named after John Tukey, it compares all possible pairs of means, and is based on a studentized range distribution (q) (this distribution is similar to the distribution of t from the t-test.
The Tukey range test, the Tukey lambda distribution, the Tukey test of additivity, and the Teichmüller–Tukey lemma all bear his name.

### Multiple comparisons problem

multiple comparisonsmultiple testingmultiple comparison
Tukey's range test, also known as the Tukey's test, Tukey method, Tukey's honest significance test, or Tukey's HSD (honestly significant difference) test, is a single-step multiple comparison procedure and statistical test.
Methods which rely on an omnibus test before proceeding to multiple comparisons. Typically these methods require a significant ANOVA, MANOVA, or Tukey's range test. These methods generally provide only "weak" control of Type I error, except for certain numbers of hypotheses.

### Analysis of variance

ANOVAanalysis of variance (ANOVA)corrected the means
It can be used on raw data or in conjunction with an ANOVA (post-hoc analysis) to find means that are significantly different from each other.
Post hoc tests such as Tukey's range test most commonly compare every group mean with every other group mean and typically incorporate some method of controlling for Type I errors.

### Studentized range

studentized range statistics
The Tukey method uses the studentized range distribution.
q is the basic statistic for 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.

### Studentized range distribution

Named after John Tukey, it compares all possible pairs of means, and is based on a studentized range distribution (q) (this distribution is similar to the distribution of t from the t-test.
Critical values of the studentized range distribution are used in Tukey's range test.

### Newman–Keuls method

Newman-KeulsStudent-Newman-Keuls
Newman–Keuls method
The Newman–Keuls method is similar to Tukey's range test as both procedures use studentized range statistics.

### Statistical hypothesis testing

hypothesis testingstatistical teststatistical tests
Tukey's range test, also known as the Tukey's test, Tukey method, Tukey's honest significance test, or Tukey's HSD (honestly significant difference) test, is a single-step multiple comparison procedure and statistical test.

### Post hoc analysis

post-hoc analysispost hocpost-hoc
It can be used on raw data or in conjunction with an ANOVA (post-hoc analysis) to find means that are significantly different from each other.

### Sample mean and covariance

sample meansample covariancesample covariance matrix
Named after John Tukey, it compares all possible pairs of means, and is based on a studentized range distribution (q) (this distribution is similar to the distribution of t from the t-test.

### Student's t-test

t-testt''-testStudent's ''t''-test
Named after John Tukey, it compares all possible pairs of means, and is based on a studentized range distribution (q) (this distribution is similar to the distribution of t from the t-test.

### Bland–Altman plot

agreementBland-Altman bias plotsBland–Altman diagram
The Tukey HSD tests should not be confused with the Tukey Mean Difference tests (also known as the Bland–Altman diagram).

### Set (mathematics)

setsetsmathematical set
The confidence coefficient for the set, when all sample sizes are equal, is exactly 1 -

### Independence (probability theory)

independentstatistically independentindependence
1) The observations being tested are independent within and among the groups.

### Normal distribution

normally distributednormalGaussian
2) The groups associated with each mean in the test are normally distributed.

### Homoscedasticity

homoscedastichomoskedasticityhomogeneity of variance
3) There is equal within-group variance across the groups associated with each mean in the test (homogeneity of variance).

### Standard error

SEstandard errorsstandard error of the mean
and identifies any difference between two means that is greater than the expected standard error.

### Null hypothesis

nullnull hypotheseshypotheses
Since the null hypothesis for Tukey's test states that all means being compared are from the same population (i.e. μ 1 = μ 2 = μ 3 = ... = μ k ), the means should be normally distributed (according to the central limit theorem). 1) α (the Type I error rate, or the probability of rejecting a true null hypothesis)

### Central limit theorem

limit theoremsA proof of the central limit theoremcentral limit
Since the null hypothesis for Tukey's test states that all means being compared are from the same population (i.e. μ 1 = μ 2 = μ 3 = ... = μ k ), the means should be normally distributed (according to the central limit theorem).

### Student's t-distribution

Student's ''t''-distributiont''-distributiont-distribution
To understand which table it is, we can compute the result for k = 2 and compare it to the result of the Student's t-distribution with the same degrees of freedom and the same α.

### R (programming language)

RR programming languageCRAN
In addition, R offers a cumulative distribution function and a quantile function for q.

### Cumulative distribution function

distribution functionCDFcumulative probability distribution function
In addition, R offers a cumulative distribution function and a quantile function for q.

### Quantile function

quantileinverse distribution functionnormal quantile function
In addition, R offers a cumulative distribution function and a quantile function for q.

### Type I and type II errors

type I errorfalse positivefalse-positive
1) α (the Type I error rate, or the probability of rejecting a true null hypothesis)

### Confidence interval

confidence intervalsconfidence levelconfidence
The confidence coefficient for the set, when all sample sizes are equal, is exactly 1 -

### Family-wise error rate

familywise error ratefamily wise error rate(familywise error rate)
In fact, Tukey's test is essentially a t-test, except that it corrects for family-wise error rate.