Student's t-distribution

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This article is about the mathematics of Student's t-distribution. Student's t-distribution_sentence_0

For its uses in statistics, see Student's t-test. Student's t-distribution_sentence_1

Student's t-distribution_table_infobox_0

Student's tStudent's t-distribution_table_caption_0
ParametersStudent's t-distribution_header_cell_0_0_0 ν

> 0


{\displaystyle \nu >0}

degrees of freedom (real)Student's t-distribution_cell_0_0_1
SupportStudent's t-distribution_header_cell_0_1_0 x

∈ ( − ∞ , ∞ )


{\displaystyle x\in (-\infty ,\infty )}Student's t-distribution_cell_0_1_1

PDFStudent's t-distribution_header_cell_0_2_0 Γ

(


ν + 1

2


)



ν π


Γ

(


ν 2


)




(

1 +


x

2


ν


)



ν + 1

2





{\displaystyle \textstyle {\frac {\Gamma \left({\frac {\nu +1}{2}}\right)}{{\sqrt {\nu \pi }}\,\Gamma \left({\frac {\nu }{2}}\right)}}\left(1+{\frac {x^{2}}{\nu }}\right)^{-{\frac {\nu +1}{2}}}\!}Student's t-distribution_cell_0_2_1

CDFStudent's t-distribution_header_cell_0_3_0 1

2


+ x Γ

(


ν + 1

2


)

×






2


F

1


(


1 2


,


ν + 1

2



3 2



x

2


ν


)



π ν


Γ

(


ν 2


)






{\displaystyle {\begin{matrix}{\frac {1}{2}}+x\Gamma \left({\frac {\nu +1}{2}}\right)\times \\[0.5em]{\frac {\,_{2}F_{1}\left({\frac {1}{2}},{\frac {\nu +1}{2}};{\frac {3}{2}};-{\frac {x^{2}}{\nu }}\right)}{{\sqrt {\pi \nu }}\,\Gamma \left({\frac {\nu }{2}}\right)}}\end{matrix}}}



    where 2F1 is the hypergeometric functionStudent's t-distribution_cell_0_3_1
MeanStudent's t-distribution_header_cell_0_4_0 0 for


ν > 1


{\displaystyle \nu >1}

, otherwise undefinedStudent's t-distribution_cell_0_4_1

MedianStudent's t-distribution_header_cell_0_5_0 0Student's t-distribution_cell_0_5_1
ModeStudent's t-distribution_header_cell_0_6_0 0Student's t-distribution_cell_0_6_1
VarianceStudent's t-distribution_header_cell_0_7_0 ν

ν − 2




{\displaystyle \textstyle {\frac {\nu }{\nu -2}}}

for 


ν > 2


{\displaystyle \nu >2}

, ∞ for


1 < ν ≤ 2


{\displaystyle 1<\nu \leq 2}

, otherwise undefinedStudent's t-distribution_cell_0_7_1

SkewnessStudent's t-distribution_header_cell_0_8_0 0 for


ν > 3


{\displaystyle \nu >3}

, otherwise undefinedStudent's t-distribution_cell_0_8_1

Ex. kurtosisStudent's t-distribution_header_cell_0_9_0 6

ν − 4




{\displaystyle \textstyle {\frac {6}{\nu -4}}}

for 


ν > 4


{\displaystyle \nu >4}

, ∞ for


2 < ν ≤ 4


{\displaystyle 2<\nu \leq 4}

, otherwise undefinedStudent's t-distribution_cell_0_9_1

EntropyStudent's t-distribution_header_cell_0_10_0 ν

+ 1

2


[

ψ

(


1 + ν

2


)

− ψ

(


ν 2


)


]



+ ln ⁡


[


ν


B

(


ν 2


,


1 2


)


]




(nats)





{\displaystyle {\begin{matrix}{\frac {\nu +1}{2}}\left[\psi \left({\frac {1+\nu }{2}}\right)-\psi \left({\frac {\nu }{2}}\right)\right]\\[0.5em]+\ln {\left[{\sqrt {\nu }}B\left({\frac {\nu }{2}},{\frac {1}{2}}\right)\right]}\,{\scriptstyle {\text{(nats)}}}\end{matrix}}}Student's t-distribution_cell_0_10_1

MGFStudent's t-distribution_header_cell_0_11_0 undefinedStudent's t-distribution_cell_0_11_1
CFStudent's t-distribution_header_cell_0_12_0 K

ν

/

2


(


ν


t


)


(


ν


t


)


ν

/

2



Γ ( ν

/

2 )

2

ν

/

2 − 1





{\displaystyle \textstyle {\frac {K_{\nu /2}\left({\sqrt {\nu }}|t|\right)\cdot \left({\sqrt {\nu }}|t|\right)^{\nu /2}}{\Gamma (\nu /2)2^{\nu /2-1}}}}

for 


ν > 0


{\displaystyle \nu >0}Student's t-distribution_cell_0_12_1

In probability and statistics, Student's t-distribution (or simply the t-distribution) is any member of a family of continuous probability distributions that arise when estimating the mean of a normally-distributed population in situations where the sample size is small and the population's standard deviation is unknown. Student's t-distribution_sentence_2

It was developed by English statistician William Sealy Gosset under the pseudonym "Student". Student's t-distribution_sentence_3

The t-distribution plays a role in a number of widely used statistical analyses, including Student's t-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis. Student's t-distribution_sentence_4

The Student's t-distribution also arises in the Bayesian analysis of data from a normal family. Student's t-distribution_sentence_5

The t-distribution is symmetric and bell-shaped, like the normal distribution, but has heavier tails, meaning that it is more prone to producing values that fall far from its mean. Student's t-distribution_sentence_6

This makes it useful for understanding the statistical behavior of certain types of ratios of random quantities, in which variation in the denominator is amplified and may produce outlying values when the denominator of the ratio falls close to zero. Student's t-distribution_sentence_7

The Student's t-distribution is a special case of the generalised hyperbolic distribution. Student's t-distribution_sentence_8

History and etymology Student's t-distribution_section_0

In statistics, the t-distribution was first derived as a posterior distribution in 1876 by Helmert and Lüroth. Student's t-distribution_sentence_9

The t-distribution also appeared in a more general form as Pearson Type IV distribution in Karl Pearson's 1895 paper. Student's t-distribution_sentence_10

In the English-language literature the distribution takes its name from William Sealy Gosset's 1908 paper in Biometrika under the pseudonym "Student". Student's t-distribution_sentence_11

Gosset worked at the Guinness Brewery in Dublin, Ireland, and was interested in the problems of small samples – for example, the chemical properties of barley where sample sizes might be as few as 3. Student's t-distribution_sentence_12

One version of the origin of the pseudonym is that Gosset's employer preferred staff to use pen names when publishing scientific papers instead of their real name, so he used the name "Student" to hide his identity. Student's t-distribution_sentence_13

Another version is that Guinness did not want their competitors to know that they were using the t-test to determine the quality of raw material. Student's t-distribution_sentence_14

Gosset's paper refers to the distribution as the "frequency distribution of standard deviations of samples drawn from a normal population". Student's t-distribution_sentence_15

It became well known through the work of Ronald Fisher, who called the distribution "Student's distribution" and represented the test value with the letter t. Student's t-distribution_sentence_16

How Student's distribution arises from sampling Student's t-distribution_section_1

Let Student's t-distribution_sentence_17

be the sample mean and let Student's t-distribution_sentence_18

be the (Bessel-corrected) sample variance. Student's t-distribution_sentence_19

Then the random variable Student's t-distribution_sentence_20

has a standard normal distribution (i.e. normal with expected mean 0 and variance 1), and the random variable Student's t-distribution_sentence_21

Definition Student's t-distribution_section_2

Probability density function Student's t-distribution_section_3

Student's t-distribution has the probability density function given by Student's t-distribution_sentence_22

Cumulative distribution function Student's t-distribution_section_4

The cumulative distribution function can be written in terms of I, the regularized incomplete beta function. Student's t-distribution_sentence_23

For t > 0, Student's t-distribution_sentence_24

where Student's t-distribution_sentence_25

where 2F1 is a particular case of the hypergeometric function. Student's t-distribution_sentence_26

For information on its inverse cumulative distribution function, see quantile function § Student's t-distribution. Student's t-distribution_sentence_27

Special cases Student's t-distribution_section_5

Student's t-distribution_description_list_0

  • Distribution function:Student's t-distribution_item_0_0

Student's t-distribution_description_list_1

  • Density function:Student's t-distribution_item_1_1

Student's t-distribution_description_list_2

Student's t-distribution_description_list_3

  • Distribution function:Student's t-distribution_item_3_3

Student's t-distribution_description_list_4

  • Density function:Student's t-distribution_item_4_4

Student's t-distribution_description_list_5

  • Distribution function:Student's t-distribution_item_5_5

Student's t-distribution_description_list_6

  • Density function:Student's t-distribution_item_6_6

Student's t-distribution_description_list_7

  • Distribution function:Student's t-distribution_item_7_7

Student's t-distribution_description_list_8

  • Density function:Student's t-distribution_item_8_8

Student's t-distribution_description_list_9

  • Distribution function:Student's t-distribution_item_9_9

Student's t-distribution_description_list_10

  • Density function:Student's t-distribution_item_10_10

Student's t-distribution_description_list_11

  • Distribution function:Student's t-distribution_item_11_11

Student's t-distribution_description_list_12

Student's t-distribution_description_list_13

  • Density function:Student's t-distribution_item_13_13

Student's t-distribution_description_list_14

How the t-distribution arises Student's t-distribution_section_6

Sampling distribution Student's t-distribution_section_7

The resulting t-value is Student's t-distribution_sentence_28

Bayesian inference Student's t-distribution_section_8

Main article: Bayesian inference Student's t-distribution_sentence_29

In Bayesian statistics, a (scaled, shifted) t-distribution arises as the marginal distribution of the unknown mean of a normal distribution, when the dependence on an unknown variance has been marginalized out: Student's t-distribution_sentence_30

The marginalization integral thus becomes Student's t-distribution_sentence_31

so Student's t-distribution_sentence_32

But the z integral is now a standard Gamma integral, which evaluates to a constant, leaving Student's t-distribution_sentence_33

This is a form of the t-distribution with an explicit scaling and shifting that will be explored in more detail in a further section below. Student's t-distribution_sentence_34

It can be related to the standardized t-distribution by the substitution Student's t-distribution_sentence_35

Characterization Student's t-distribution_section_9

As the distribution of a test statistic Student's t-distribution_section_10

where Student's t-distribution_sentence_36

A different distribution is defined as that of the random variable defined, for a given constant μ, by Student's t-distribution_sentence_37

This random variable has a noncentral t-distribution with noncentrality parameter μ. Student's t-distribution_sentence_38

This distribution is important in studies of the power of Student's t-test. Student's t-distribution_sentence_39

Derivation Student's t-distribution_section_11

Suppose X1, ..., Xn are independent realizations of the normally-distributed, random variable X, which has an expected value μ and variance σ. Student's t-distribution_sentence_40

Let Student's t-distribution_sentence_41

be the sample mean, and Student's t-distribution_sentence_42

be an unbiased estimate of the variance from the sample. Student's t-distribution_sentence_43

It can be shown that the random variable Student's t-distribution_sentence_44

As a maximum entropy distribution Student's t-distribution_section_12

Properties Student's t-distribution_section_13

Moments Student's t-distribution_section_14

Monte Carlo sampling Student's t-distribution_section_15

There are various approaches to constructing random samples from the Student's t-distribution. Student's t-distribution_sentence_45

The matter depends on whether the samples are required on a stand-alone basis, or are to be constructed by application of a quantile function to uniform samples; e.g., in the multi-dimensional applications basis of copula-dependency. Student's t-distribution_sentence_46

In the case of stand-alone sampling, an extension of the Box–Muller method and its polar form is easily deployed. Student's t-distribution_sentence_47

It has the merit that it applies equally well to all real positive degrees of freedom, ν, while many other candidate methods fail if ν is close to zero. Student's t-distribution_sentence_48

Integral of Student's probability density function and p-value Student's t-distribution_section_16

The function A(t | ν) is the integral of Student's probability density function, f(t) between −t and t, for t ≥ 0. Student's t-distribution_sentence_49

It thus gives the probability that a value of t less than that calculated from observed data would occur by chance. Student's t-distribution_sentence_50

Therefore, the function A(t | ν) can be used when testing whether the difference between the means of two sets of data is statistically significant, by calculating the corresponding value of t and the probability of its occurrence if the two sets of data were drawn from the same population. Student's t-distribution_sentence_51

This is used in a variety of situations, particularly in t-tests. Student's t-distribution_sentence_52

For the statistic t, with ν degrees of freedom, A(t | ν) is the probability that t would be less than the observed value if the two means were the same (provided that the smaller mean is subtracted from the larger, so that t ≥ 0). Student's t-distribution_sentence_53

It can be easily calculated from the cumulative distribution function Fν(t) of the t-distribution: Student's t-distribution_sentence_54

where Ix is the regularized incomplete beta function (a, b). Student's t-distribution_sentence_55

For statistical hypothesis testing this function is used to construct the p-value. Student's t-distribution_sentence_56

Generalized Student's t-distribution Student's t-distribution_section_17

or Student's t-distribution_sentence_57

The resulting non-standardized Student's t-distribution has a density defined by: Student's t-distribution_sentence_58

Other properties of this version of the distribution are: Student's t-distribution_sentence_59

In terms of inverse scaling parameter λ Student's t-distribution_section_18

Other properties of this version of the distribution are: Student's t-distribution_sentence_60

Related distributions Student's t-distribution_section_19

Student's t-distribution_unordered_list_15

  • One can generate Student-t samples by taking the ratio of variables from the normal distribution and the square-root of χ-distribution. If we use instead of the normal distribution, e.g., the Irwin–Hall distribution, we obtain over-all a symmetric 4-parameter distribution, which includes the normal, the uniform, the triangular, the Student-t and the Cauchy distribution. This is also more flexible than some other symmetric generalizations of the normal distribution.Student's t-distribution_item_15_15
  • t-distribution is an instance of ratio distributionsStudent's t-distribution_item_15_16

Uses Student's t-distribution_section_20

In frequentist statistical inference Student's t-distribution_section_21

Student's t-distribution arises in a variety of statistical estimation problems where the goal is to estimate an unknown parameter, such as a mean value, in a setting where the data are observed with additive errors. Student's t-distribution_sentence_61

If (as in nearly all practical statistical work) the population standard deviation of these errors is unknown and has to be estimated from the data, the t-distribution is often used to account for the extra uncertainty that results from this estimation. Student's t-distribution_sentence_62

In most such problems, if the standard deviation of the errors were known, a normal distribution would be used instead of the t-distribution. Student's t-distribution_sentence_63

Confidence intervals and hypothesis tests are two statistical procedures in which the quantiles of the sampling distribution of a particular statistic (e.g. the standard score) are required. Student's t-distribution_sentence_64

In any situation where this statistic is a linear function of the data, divided by the usual estimate of the standard deviation, the resulting quantity can be rescaled and centered to follow Student's t-distribution. Student's t-distribution_sentence_65

Statistical analyses involving means, weighted means, and regression coefficients all lead to statistics having this form. Student's t-distribution_sentence_66

Quite often, textbook problems will treat the population standard deviation as if it were known and thereby avoid the need to use the Student's t-distribution. Student's t-distribution_sentence_67

These problems are generally of two kinds: (1) those in which the sample size is so large that one may treat a data-based estimate of the variance as if it were certain, and (2) those that illustrate mathematical reasoning, in which the problem of estimating the standard deviation is temporarily ignored because that is not the point that the author or instructor is then explaining. Student's t-distribution_sentence_68

Hypothesis testing Student's t-distribution_section_22

A number of statistics can be shown to have t-distributions for samples of moderate size under null hypotheses that are of interest, so that the t-distribution forms the basis for significance tests. Student's t-distribution_sentence_69

For example, the distribution of Spearman's rank correlation coefficient ρ, in the null case (zero correlation) is well approximated by the t distribution for sample sizes above about 20. Student's t-distribution_sentence_70

Confidence intervals Student's t-distribution_section_23

Suppose the number A is so chosen that Student's t-distribution_sentence_71

when T has a t-distribution with n − 1 degrees of freedom. Student's t-distribution_sentence_72

By symmetry, this is the same as saying that A satisfies Student's t-distribution_sentence_73

and this is equivalent to Student's t-distribution_sentence_74

Therefore, the interval whose endpoints are Student's t-distribution_sentence_75

is a 90% confidence interval for μ. Student's t-distribution_sentence_76

Therefore, if we find the mean of a set of observations that we can reasonably expect to have a normal distribution, we can use the t-distribution to examine whether the confidence limits on that mean include some theoretically predicted value – such as the value predicted on a null hypothesis. Student's t-distribution_sentence_77

It is this result that is used in the Student's t-tests: since the difference between the means of samples from two normal distributions is itself distributed normally, the t-distribution can be used to examine whether that difference can reasonably be supposed to be zero. Student's t-distribution_sentence_78

If the data are normally distributed, the one-sided (1 − α)-upper confidence limit (UCL) of the mean, can be calculated using the following equation: Student's t-distribution_sentence_79

Prediction intervals Student's t-distribution_section_24

The t-distribution can be used to construct a prediction interval for an unobserved sample from a normal distribution with unknown mean and variance. Student's t-distribution_sentence_80

In Bayesian statistics Student's t-distribution_section_25

The Student's t-distribution, especially in its three-parameter (location-scale) version, arises frequently in Bayesian statistics as a result of its connection with the normal distribution. Student's t-distribution_sentence_81

Whenever the variance of a normally distributed random variable is unknown and a conjugate prior placed over it that follows an inverse gamma distribution, the resulting marginal distribution of the variable will follow a Student's t-distribution. Student's t-distribution_sentence_82

Equivalent constructions with the same results involve a conjugate scaled-inverse-chi-squared distribution over the variance, or a conjugate gamma distribution over the precision. Student's t-distribution_sentence_83

If an improper prior proportional to σ is placed over the variance, the t-distribution also arises. Student's t-distribution_sentence_84

This is the case regardless of whether the mean of the normally distributed variable is known, is unknown distributed according to a conjugate normally distributed prior, or is unknown distributed according to an improper constant prior. Student's t-distribution_sentence_85

Related situations that also produce a t-distribution are: Student's t-distribution_sentence_86

Student's t-distribution_unordered_list_16

Robust parametric modeling Student's t-distribution_section_26

The t-distribution is often used as an alternative to the normal distribution as a model for data, which often has heavier tails than the normal distribution allows for; see e.g. Lange et al. Student's t-distribution_sentence_87

The classical approach was to identify outliers (e.g., using Grubbs's test) and exclude or downweight them in some way. Student's t-distribution_sentence_88

However, it is not always easy to identify outliers (especially in high dimensions), and the t-distribution is a natural choice of model for such data and provides a parametric approach to robust statistics. Student's t-distribution_sentence_89

A Bayesian account can be found in Gelman et al. Student's t-distribution_sentence_90

The degrees of freedom parameter controls the kurtosis of the distribution and is correlated with the scale parameter. Student's t-distribution_sentence_91

The likelihood can have multiple local maxima and, as such, it is often necessary to fix the degrees of freedom at a fairly low value and estimate the other parameters taking this as given. Student's t-distribution_sentence_92

Some authors report that values between 3 and 9 are often good choices. Student's t-distribution_sentence_93

Venables and Ripley suggest that a value of 5 is often a good choice. Student's t-distribution_sentence_94

Student's t-process Student's t-distribution_section_27

Table of selected values Student's t-distribution_section_28

Note that the last row with infinite ν gives critical points for a normal distribution since a t-distribution with infinitely many degrees of freedom is a normal distribution. Student's t-distribution_sentence_95

(See Related distributions above). Student's t-distribution_sentence_96

Student's t-distribution_table_general_1

One-sidedStudent's t-distribution_header_cell_1_0_0 75%Student's t-distribution_header_cell_1_0_1 80%Student's t-distribution_header_cell_1_0_2 85%Student's t-distribution_header_cell_1_0_3 90%Student's t-distribution_header_cell_1_0_4 95%Student's t-distribution_header_cell_1_0_5 97.5%Student's t-distribution_header_cell_1_0_6 99%Student's t-distribution_header_cell_1_0_7 99.5%Student's t-distribution_header_cell_1_0_8 99.75%Student's t-distribution_header_cell_1_0_9 99.9%Student's t-distribution_header_cell_1_0_10 99.95%Student's t-distribution_header_cell_1_0_11
Two-sidedStudent's t-distribution_header_cell_1_1_0 50%Student's t-distribution_header_cell_1_1_1 60%Student's t-distribution_header_cell_1_1_2 70%Student's t-distribution_header_cell_1_1_3 80%Student's t-distribution_header_cell_1_1_4 90%Student's t-distribution_header_cell_1_1_5 95%Student's t-distribution_header_cell_1_1_6 98%Student's t-distribution_header_cell_1_1_7 99%Student's t-distribution_header_cell_1_1_8 99.5%Student's t-distribution_header_cell_1_1_9 99.8%Student's t-distribution_header_cell_1_1_10 99.9%Student's t-distribution_header_cell_1_1_11
1Student's t-distribution_header_cell_1_2_0 1.000Student's t-distribution_cell_1_2_1 1.376Student's t-distribution_cell_1_2_2 1.963Student's t-distribution_cell_1_2_3 3.078Student's t-distribution_cell_1_2_4 6.314Student's t-distribution_cell_1_2_5 12.71Student's t-distribution_cell_1_2_6 31.82Student's t-distribution_cell_1_2_7 63.66Student's t-distribution_cell_1_2_8 127.3Student's t-distribution_cell_1_2_9 318.3Student's t-distribution_cell_1_2_10 636.6Student's t-distribution_cell_1_2_11
2Student's t-distribution_header_cell_1_3_0 0.816Student's t-distribution_cell_1_3_1 1.080Student's t-distribution_cell_1_3_2 1.386Student's t-distribution_cell_1_3_3 1.886Student's t-distribution_cell_1_3_4 2.920Student's t-distribution_cell_1_3_5 4.303Student's t-distribution_cell_1_3_6 6.965Student's t-distribution_cell_1_3_7 9.925Student's t-distribution_cell_1_3_8 14.09Student's t-distribution_cell_1_3_9 22.33Student's t-distribution_cell_1_3_10 31.60Student's t-distribution_cell_1_3_11
3Student's t-distribution_header_cell_1_4_0 0.765Student's t-distribution_cell_1_4_1 0.978Student's t-distribution_cell_1_4_2 1.250Student's t-distribution_cell_1_4_3 1.638Student's t-distribution_cell_1_4_4 2.353Student's t-distribution_cell_1_4_5 3.182Student's t-distribution_cell_1_4_6 4.541Student's t-distribution_cell_1_4_7 5.841Student's t-distribution_cell_1_4_8 7.453Student's t-distribution_cell_1_4_9 10.21Student's t-distribution_cell_1_4_10 12.92Student's t-distribution_cell_1_4_11
4Student's t-distribution_header_cell_1_5_0 0.741Student's t-distribution_cell_1_5_1 0.941Student's t-distribution_cell_1_5_2 1.190Student's t-distribution_cell_1_5_3 1.533Student's t-distribution_cell_1_5_4 2.132Student's t-distribution_cell_1_5_5 2.776Student's t-distribution_cell_1_5_6 3.747Student's t-distribution_cell_1_5_7 4.604Student's t-distribution_cell_1_5_8 5.598Student's t-distribution_cell_1_5_9 7.173Student's t-distribution_cell_1_5_10 8.610Student's t-distribution_cell_1_5_11
5Student's t-distribution_header_cell_1_6_0 0.727Student's t-distribution_cell_1_6_1 0.920Student's t-distribution_cell_1_6_2 1.156Student's t-distribution_cell_1_6_3 1.476Student's t-distribution_cell_1_6_4 2.015Student's t-distribution_cell_1_6_5 2.571Student's t-distribution_cell_1_6_6 3.365Student's t-distribution_cell_1_6_7 4.032Student's t-distribution_cell_1_6_8 4.773Student's t-distribution_cell_1_6_9 5.893Student's t-distribution_cell_1_6_10 6.869Student's t-distribution_cell_1_6_11
6Student's t-distribution_header_cell_1_7_0 0.718Student's t-distribution_cell_1_7_1 0.906Student's t-distribution_cell_1_7_2 1.134Student's t-distribution_cell_1_7_3 1.440Student's t-distribution_cell_1_7_4 1.943Student's t-distribution_cell_1_7_5 2.447Student's t-distribution_cell_1_7_6 3.143Student's t-distribution_cell_1_7_7 3.707Student's t-distribution_cell_1_7_8 4.317Student's t-distribution_cell_1_7_9 5.208Student's t-distribution_cell_1_7_10 5.959Student's t-distribution_cell_1_7_11
7Student's t-distribution_header_cell_1_8_0 0.711Student's t-distribution_cell_1_8_1 0.896Student's t-distribution_cell_1_8_2 1.119Student's t-distribution_cell_1_8_3 1.415Student's t-distribution_cell_1_8_4 1.895Student's t-distribution_cell_1_8_5 2.365Student's t-distribution_cell_1_8_6 2.998Student's t-distribution_cell_1_8_7 3.499Student's t-distribution_cell_1_8_8 4.029Student's t-distribution_cell_1_8_9 4.785Student's t-distribution_cell_1_8_10 5.408Student's t-distribution_cell_1_8_11
8Student's t-distribution_header_cell_1_9_0 0.706Student's t-distribution_cell_1_9_1 0.889Student's t-distribution_cell_1_9_2 1.108Student's t-distribution_cell_1_9_3 1.397Student's t-distribution_cell_1_9_4 1.860Student's t-distribution_cell_1_9_5 2.306Student's t-distribution_cell_1_9_6 2.896Student's t-distribution_cell_1_9_7 3.355Student's t-distribution_cell_1_9_8 3.833Student's t-distribution_cell_1_9_9 4.501Student's t-distribution_cell_1_9_10 5.041Student's t-distribution_cell_1_9_11
9Student's t-distribution_header_cell_1_10_0 0.703Student's t-distribution_cell_1_10_1 0.883Student's t-distribution_cell_1_10_2 1.100Student's t-distribution_cell_1_10_3 1.383Student's t-distribution_cell_1_10_4 1.833Student's t-distribution_cell_1_10_5 2.262Student's t-distribution_cell_1_10_6 2.821Student's t-distribution_cell_1_10_7 3.250Student's t-distribution_cell_1_10_8 3.690Student's t-distribution_cell_1_10_9 4.297Student's t-distribution_cell_1_10_10 4.781Student's t-distribution_cell_1_10_11
10Student's t-distribution_header_cell_1_11_0 0.700Student's t-distribution_cell_1_11_1 0.879Student's t-distribution_cell_1_11_2 1.093Student's t-distribution_cell_1_11_3 1.372Student's t-distribution_cell_1_11_4 1.812Student's t-distribution_cell_1_11_5 2.228Student's t-distribution_cell_1_11_6 2.764Student's t-distribution_cell_1_11_7 3.169Student's t-distribution_cell_1_11_8 3.581Student's t-distribution_cell_1_11_9 4.144Student's t-distribution_cell_1_11_10 4.587Student's t-distribution_cell_1_11_11
11Student's t-distribution_header_cell_1_12_0 0.697Student's t-distribution_cell_1_12_1 0.876Student's t-distribution_cell_1_12_2 1.088Student's t-distribution_cell_1_12_3 1.363Student's t-distribution_cell_1_12_4 1.796Student's t-distribution_cell_1_12_5 2.201Student's t-distribution_cell_1_12_6 2.718Student's t-distribution_cell_1_12_7 3.106Student's t-distribution_cell_1_12_8 3.497Student's t-distribution_cell_1_12_9 4.025Student's t-distribution_cell_1_12_10 4.437Student's t-distribution_cell_1_12_11
12Student's t-distribution_header_cell_1_13_0 0.695Student's t-distribution_cell_1_13_1 0.873Student's t-distribution_cell_1_13_2 1.083Student's t-distribution_cell_1_13_3 1.356Student's t-distribution_cell_1_13_4 1.782Student's t-distribution_cell_1_13_5 2.179Student's t-distribution_cell_1_13_6 2.681Student's t-distribution_cell_1_13_7 3.055Student's t-distribution_cell_1_13_8 3.428Student's t-distribution_cell_1_13_9 3.930Student's t-distribution_cell_1_13_10 4.318Student's t-distribution_cell_1_13_11
13Student's t-distribution_header_cell_1_14_0 0.694Student's t-distribution_cell_1_14_1 0.870Student's t-distribution_cell_1_14_2 1.079Student's t-distribution_cell_1_14_3 1.350Student's t-distribution_cell_1_14_4 1.771Student's t-distribution_cell_1_14_5 2.160Student's t-distribution_cell_1_14_6 2.650Student's t-distribution_cell_1_14_7 3.012Student's t-distribution_cell_1_14_8 3.372Student's t-distribution_cell_1_14_9 3.852Student's t-distribution_cell_1_14_10 4.221Student's t-distribution_cell_1_14_11
14Student's t-distribution_header_cell_1_15_0 0.692Student's t-distribution_cell_1_15_1 0.868Student's t-distribution_cell_1_15_2 1.076Student's t-distribution_cell_1_15_3 1.345Student's t-distribution_cell_1_15_4 1.761Student's t-distribution_cell_1_15_5 2.145Student's t-distribution_cell_1_15_6 2.624Student's t-distribution_cell_1_15_7 2.977Student's t-distribution_cell_1_15_8 3.326Student's t-distribution_cell_1_15_9 3.787Student's t-distribution_cell_1_15_10 4.140Student's t-distribution_cell_1_15_11
15Student's t-distribution_header_cell_1_16_0 0.691Student's t-distribution_cell_1_16_1 0.866Student's t-distribution_cell_1_16_2 1.074Student's t-distribution_cell_1_16_3 1.341Student's t-distribution_cell_1_16_4 1.753Student's t-distribution_cell_1_16_5 2.131Student's t-distribution_cell_1_16_6 2.602Student's t-distribution_cell_1_16_7 2.947Student's t-distribution_cell_1_16_8 3.286Student's t-distribution_cell_1_16_9 3.733Student's t-distribution_cell_1_16_10 4.073Student's t-distribution_cell_1_16_11
16Student's t-distribution_header_cell_1_17_0 0.690Student's t-distribution_cell_1_17_1 0.865Student's t-distribution_cell_1_17_2 1.071Student's t-distribution_cell_1_17_3 1.337Student's t-distribution_cell_1_17_4 1.746Student's t-distribution_cell_1_17_5 2.120Student's t-distribution_cell_1_17_6 2.583Student's t-distribution_cell_1_17_7 2.921Student's t-distribution_cell_1_17_8 3.252Student's t-distribution_cell_1_17_9 3.686Student's t-distribution_cell_1_17_10 4.015Student's t-distribution_cell_1_17_11
17Student's t-distribution_header_cell_1_18_0 0.689Student's t-distribution_cell_1_18_1 0.863Student's t-distribution_cell_1_18_2 1.069Student's t-distribution_cell_1_18_3 1.333Student's t-distribution_cell_1_18_4 1.740Student's t-distribution_cell_1_18_5 2.110Student's t-distribution_cell_1_18_6 2.567Student's t-distribution_cell_1_18_7 2.898Student's t-distribution_cell_1_18_8 3.222Student's t-distribution_cell_1_18_9 3.646Student's t-distribution_cell_1_18_10 3.965Student's t-distribution_cell_1_18_11
18Student's t-distribution_header_cell_1_19_0 0.688Student's t-distribution_cell_1_19_1 0.862Student's t-distribution_cell_1_19_2 1.067Student's t-distribution_cell_1_19_3 1.330Student's t-distribution_cell_1_19_4 1.734Student's t-distribution_cell_1_19_5 2.101Student's t-distribution_cell_1_19_6 2.552Student's t-distribution_cell_1_19_7 2.878Student's t-distribution_cell_1_19_8 3.197Student's t-distribution_cell_1_19_9 3.610Student's t-distribution_cell_1_19_10 3.922Student's t-distribution_cell_1_19_11
19Student's t-distribution_header_cell_1_20_0 0.688Student's t-distribution_cell_1_20_1 0.861Student's t-distribution_cell_1_20_2 1.066Student's t-distribution_cell_1_20_3 1.328Student's t-distribution_cell_1_20_4 1.729Student's t-distribution_cell_1_20_5 2.093Student's t-distribution_cell_1_20_6 2.539Student's t-distribution_cell_1_20_7 2.861Student's t-distribution_cell_1_20_8 3.174Student's t-distribution_cell_1_20_9 3.579Student's t-distribution_cell_1_20_10 3.883Student's t-distribution_cell_1_20_11
20Student's t-distribution_header_cell_1_21_0 0.687Student's t-distribution_cell_1_21_1 0.860Student's t-distribution_cell_1_21_2 1.064Student's t-distribution_cell_1_21_3 1.325Student's t-distribution_cell_1_21_4 1.725Student's t-distribution_cell_1_21_5 2.086Student's t-distribution_cell_1_21_6 2.528Student's t-distribution_cell_1_21_7 2.845Student's t-distribution_cell_1_21_8 3.153Student's t-distribution_cell_1_21_9 3.552Student's t-distribution_cell_1_21_10 3.850Student's t-distribution_cell_1_21_11
21Student's t-distribution_header_cell_1_22_0 0.686Student's t-distribution_cell_1_22_1 0.859Student's t-distribution_cell_1_22_2 1.063Student's t-distribution_cell_1_22_3 1.323Student's t-distribution_cell_1_22_4 1.721Student's t-distribution_cell_1_22_5 2.080Student's t-distribution_cell_1_22_6 2.518Student's t-distribution_cell_1_22_7 2.831Student's t-distribution_cell_1_22_8 3.135Student's t-distribution_cell_1_22_9 3.527Student's t-distribution_cell_1_22_10 3.819Student's t-distribution_cell_1_22_11
22Student's t-distribution_header_cell_1_23_0 0.686Student's t-distribution_cell_1_23_1 0.858Student's t-distribution_cell_1_23_2 1.061Student's t-distribution_cell_1_23_3 1.321Student's t-distribution_cell_1_23_4 1.717Student's t-distribution_cell_1_23_5 2.074Student's t-distribution_cell_1_23_6 2.508Student's t-distribution_cell_1_23_7 2.819Student's t-distribution_cell_1_23_8 3.119Student's t-distribution_cell_1_23_9 3.505Student's t-distribution_cell_1_23_10 3.792Student's t-distribution_cell_1_23_11
23Student's t-distribution_header_cell_1_24_0 0.685Student's t-distribution_cell_1_24_1 0.858Student's t-distribution_cell_1_24_2 1.060Student's t-distribution_cell_1_24_3 1.319Student's t-distribution_cell_1_24_4 1.714Student's t-distribution_cell_1_24_5 2.069Student's t-distribution_cell_1_24_6 2.500Student's t-distribution_cell_1_24_7 2.807Student's t-distribution_cell_1_24_8 3.104Student's t-distribution_cell_1_24_9 3.485Student's t-distribution_cell_1_24_10 3.767Student's t-distribution_cell_1_24_11
24Student's t-distribution_header_cell_1_25_0 0.685Student's t-distribution_cell_1_25_1 0.857Student's t-distribution_cell_1_25_2 1.059Student's t-distribution_cell_1_25_3 1.318Student's t-distribution_cell_1_25_4 1.711Student's t-distribution_cell_1_25_5 2.064Student's t-distribution_cell_1_25_6 2.492Student's t-distribution_cell_1_25_7 2.797Student's t-distribution_cell_1_25_8 3.091Student's t-distribution_cell_1_25_9 3.467Student's t-distribution_cell_1_25_10 3.745Student's t-distribution_cell_1_25_11
25Student's t-distribution_header_cell_1_26_0 0.684Student's t-distribution_cell_1_26_1 0.856Student's t-distribution_cell_1_26_2 1.058Student's t-distribution_cell_1_26_3 1.316Student's t-distribution_cell_1_26_4 1.708Student's t-distribution_cell_1_26_5 2.060Student's t-distribution_cell_1_26_6 2.485Student's t-distribution_cell_1_26_7 2.787Student's t-distribution_cell_1_26_8 3.078Student's t-distribution_cell_1_26_9 3.450Student's t-distribution_cell_1_26_10 3.725Student's t-distribution_cell_1_26_11
26Student's t-distribution_header_cell_1_27_0 0.684Student's t-distribution_cell_1_27_1 0.856Student's t-distribution_cell_1_27_2 1.058Student's t-distribution_cell_1_27_3 1.315Student's t-distribution_cell_1_27_4 1.706Student's t-distribution_cell_1_27_5 2.056Student's t-distribution_cell_1_27_6 2.479Student's t-distribution_cell_1_27_7 2.779Student's t-distribution_cell_1_27_8 3.067Student's t-distribution_cell_1_27_9 3.435Student's t-distribution_cell_1_27_10 3.707Student's t-distribution_cell_1_27_11
27Student's t-distribution_header_cell_1_28_0 0.684Student's t-distribution_cell_1_28_1 0.855Student's t-distribution_cell_1_28_2 1.057Student's t-distribution_cell_1_28_3 1.314Student's t-distribution_cell_1_28_4 1.703Student's t-distribution_cell_1_28_5 2.052Student's t-distribution_cell_1_28_6 2.473Student's t-distribution_cell_1_28_7 2.771Student's t-distribution_cell_1_28_8 3.057Student's t-distribution_cell_1_28_9 3.421Student's t-distribution_cell_1_28_10 3.690Student's t-distribution_cell_1_28_11
28Student's t-distribution_header_cell_1_29_0 0.683Student's t-distribution_cell_1_29_1 0.855Student's t-distribution_cell_1_29_2 1.056Student's t-distribution_cell_1_29_3 1.313Student's t-distribution_cell_1_29_4 1.701Student's t-distribution_cell_1_29_5 2.048Student's t-distribution_cell_1_29_6 2.467Student's t-distribution_cell_1_29_7 2.763Student's t-distribution_cell_1_29_8 3.047Student's t-distribution_cell_1_29_9 3.408Student's t-distribution_cell_1_29_10 3.674Student's t-distribution_cell_1_29_11
29Student's t-distribution_header_cell_1_30_0 0.683Student's t-distribution_cell_1_30_1 0.854Student's t-distribution_cell_1_30_2 1.055Student's t-distribution_cell_1_30_3 1.311Student's t-distribution_cell_1_30_4 1.699Student's t-distribution_cell_1_30_5 2.045Student's t-distribution_cell_1_30_6 2.462Student's t-distribution_cell_1_30_7 2.756Student's t-distribution_cell_1_30_8 3.038Student's t-distribution_cell_1_30_9 3.396Student's t-distribution_cell_1_30_10 3.659Student's t-distribution_cell_1_30_11
30Student's t-distribution_header_cell_1_31_0 0.683Student's t-distribution_cell_1_31_1 0.854Student's t-distribution_cell_1_31_2 1.055Student's t-distribution_cell_1_31_3 1.310Student's t-distribution_cell_1_31_4 1.697Student's t-distribution_cell_1_31_5 2.042Student's t-distribution_cell_1_31_6 2.457Student's t-distribution_cell_1_31_7 2.750Student's t-distribution_cell_1_31_8 3.030Student's t-distribution_cell_1_31_9 3.385Student's t-distribution_cell_1_31_10 3.646Student's t-distribution_cell_1_31_11
40Student's t-distribution_header_cell_1_32_0 0.681Student's t-distribution_cell_1_32_1 0.851Student's t-distribution_cell_1_32_2 1.050Student's t-distribution_cell_1_32_3 1.303Student's t-distribution_cell_1_32_4 1.684Student's t-distribution_cell_1_32_5 2.021Student's t-distribution_cell_1_32_6 2.423Student's t-distribution_cell_1_32_7 2.704Student's t-distribution_cell_1_32_8 2.971Student's t-distribution_cell_1_32_9 3.307Student's t-distribution_cell_1_32_10 3.551Student's t-distribution_cell_1_32_11
50Student's t-distribution_header_cell_1_33_0 0.679Student's t-distribution_cell_1_33_1 0.849Student's t-distribution_cell_1_33_2 1.047Student's t-distribution_cell_1_33_3 1.299Student's t-distribution_cell_1_33_4 1.676Student's t-distribution_cell_1_33_5 2.009Student's t-distribution_cell_1_33_6 2.403Student's t-distribution_cell_1_33_7 2.678Student's t-distribution_cell_1_33_8 2.937Student's t-distribution_cell_1_33_9 3.261Student's t-distribution_cell_1_33_10 3.496Student's t-distribution_cell_1_33_11
60Student's t-distribution_header_cell_1_34_0 0.679Student's t-distribution_cell_1_34_1 0.848Student's t-distribution_cell_1_34_2 1.045Student's t-distribution_cell_1_34_3 1.296Student's t-distribution_cell_1_34_4 1.671Student's t-distribution_cell_1_34_5 2.000Student's t-distribution_cell_1_34_6 2.390Student's t-distribution_cell_1_34_7 2.660Student's t-distribution_cell_1_34_8 2.915Student's t-distribution_cell_1_34_9 3.232Student's t-distribution_cell_1_34_10 3.460Student's t-distribution_cell_1_34_11
80Student's t-distribution_header_cell_1_35_0 0.678Student's t-distribution_cell_1_35_1 0.846Student's t-distribution_cell_1_35_2 1.043Student's t-distribution_cell_1_35_3 1.292Student's t-distribution_cell_1_35_4 1.664Student's t-distribution_cell_1_35_5 1.990Student's t-distribution_cell_1_35_6 2.374Student's t-distribution_cell_1_35_7 2.639Student's t-distribution_cell_1_35_8 2.887Student's t-distribution_cell_1_35_9 3.195Student's t-distribution_cell_1_35_10 3.416Student's t-distribution_cell_1_35_11
100Student's t-distribution_header_cell_1_36_0 0.677Student's t-distribution_cell_1_36_1 0.845Student's t-distribution_cell_1_36_2 1.042Student's t-distribution_cell_1_36_3 1.290Student's t-distribution_cell_1_36_4 1.660Student's t-distribution_cell_1_36_5 1.984Student's t-distribution_cell_1_36_6 2.364Student's t-distribution_cell_1_36_7 2.626Student's t-distribution_cell_1_36_8 2.871Student's t-distribution_cell_1_36_9 3.174Student's t-distribution_cell_1_36_10 3.390Student's t-distribution_cell_1_36_11
120Student's t-distribution_header_cell_1_37_0 0.677Student's t-distribution_cell_1_37_1 0.845Student's t-distribution_cell_1_37_2 1.041Student's t-distribution_cell_1_37_3 1.289Student's t-distribution_cell_1_37_4 1.658Student's t-distribution_cell_1_37_5 1.980Student's t-distribution_cell_1_37_6 2.358Student's t-distribution_cell_1_37_7 2.617Student's t-distribution_cell_1_37_8 2.860Student's t-distribution_cell_1_37_9 3.160Student's t-distribution_cell_1_37_10 3.373Student's t-distribution_cell_1_37_11
Student's t-distribution_header_cell_1_38_0 0.674Student's t-distribution_cell_1_38_1 0.842Student's t-distribution_cell_1_38_2 1.036Student's t-distribution_cell_1_38_3 1.282Student's t-distribution_cell_1_38_4 1.645Student's t-distribution_cell_1_38_5 1.960Student's t-distribution_cell_1_38_6 2.326Student's t-distribution_cell_1_38_7 2.576Student's t-distribution_cell_1_38_8 2.807Student's t-distribution_cell_1_38_9 3.090Student's t-distribution_cell_1_38_10 3.291Student's t-distribution_cell_1_38_11
One-sidedStudent's t-distribution_header_cell_1_39_0 75%Student's t-distribution_header_cell_1_39_1 80%Student's t-distribution_header_cell_1_39_2 85%Student's t-distribution_header_cell_1_39_3 90%Student's t-distribution_header_cell_1_39_4 95%Student's t-distribution_header_cell_1_39_5 97.5%Student's t-distribution_header_cell_1_39_6 99%Student's t-distribution_header_cell_1_39_7 99.5%Student's t-distribution_header_cell_1_39_8 99.75%Student's t-distribution_header_cell_1_39_9 99.9%Student's t-distribution_header_cell_1_39_10 99.95%Student's t-distribution_header_cell_1_39_11
Two-sidedStudent's t-distribution_header_cell_1_40_0 50%Student's t-distribution_header_cell_1_40_1 60%Student's t-distribution_header_cell_1_40_2 70%Student's t-distribution_header_cell_1_40_3 80%Student's t-distribution_header_cell_1_40_4 90%Student's t-distribution_header_cell_1_40_5 95%Student's t-distribution_header_cell_1_40_6 98%Student's t-distribution_header_cell_1_40_7 99%Student's t-distribution_header_cell_1_40_8 99.5%Student's t-distribution_header_cell_1_40_9 99.8%Student's t-distribution_header_cell_1_40_10 99.9%Student's t-distribution_header_cell_1_40_11

Calculating the confidence interval Student's t-distribution_sentence_97

Let's say we have a sample with size 11, sample mean 10, and sample variance 2. Student's t-distribution_sentence_98

For 90% confidence with 10 degrees of freedom, the one-sided t-value from the table is 1.372. Student's t-distribution_sentence_99

Then with confidence interval calculated from Student's t-distribution_sentence_100

we determine that with 90% confidence we have a true mean lying below Student's t-distribution_sentence_101

In other words, 90% of the times that an upper threshold is calculated by this method from particular samples, this upper threshold exceeds the true mean. Student's t-distribution_sentence_102

And with 90% confidence we have a true mean lying above Student's t-distribution_sentence_103

In other words, 90% of the times that a lower threshold is calculated by this method from particular samples, this lower threshold lies below the true mean. Student's t-distribution_sentence_104

So that at 80% confidence (calculated from 100% − 2 × (1 − 90%) = 80%), we have a true mean lying within the interval Student's t-distribution_sentence_105

Saying that 80% of the times that upper and lower thresholds are calculated by this method from a given sample, the true mean is both below the upper threshold and above the lower threshold is not the same as saying that there is an 80% probability that the true mean lies between a particular pair of upper and lower thresholds that have been calculated by this method; see confidence interval and prosecutor's fallacy. Student's t-distribution_sentence_106

Nowadays, statistical software, such as the R programming language, and functions available in many spreadsheet programs compute values of the t-distribution and its inverse without tables. Student's t-distribution_sentence_107

See also Student's t-distribution_section_29

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Student's t-distribution.