Student's t-distribution
Probability density function | |||||||||
Cumulative distribution function | |||||||||
Parameters | degrees of freedom (real, almost always a positive integer) | ||||||||
---|---|---|---|---|---|---|---|---|---|
Support | |||||||||
CDF |
where is the hypergeometric function | ||||||||
Mean | for otherwise undefined | ||||||||
Median | |||||||||
Mode | |||||||||
Variance |
for ∞ for otherwise undefined | ||||||||
Skewness |
for otherwise Excess kurtosis |
for ∞ for otherwise Entropy |
| ||||||
MGF | undefined | ||||||||
CF |
for | ||||||||
Expected shortfall |
Where is the inverse standardized Student t CDF, and is the standardized Student t PDF.[2] |
In probability and statistics, Student's t distribution (or simply the t distribution) is a continuous probability distribution that generalizes the standard normal distribution. Like the latter, it is symmetric around zero and bell-shaped.
However, has heavier tails and the amount of probability mass in the tails is controlled by the parameter For the Student's t distribution becomes the standard Cauchy distribution, which has very "fat" tails; whereas for it becomes the standard normal distribution which has very "thin" tails.
The Student's 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.
In the form of the location-scale t distribution it generalizes the
History and etymology
In statistics, the t distribution was first derived as a
In the English-language literature, the distribution takes its name from William Sealy Gosset's 1908 paper in Biometrika under the pseudonym "Student".[10] 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. 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.[11][12]
Gosset worked at the
Definition
Probability density function
Student's t distribution has the probability density function (PDF) given by
where is the number of degrees of freedom and is the gamma function. This may also be written as
where is the Beta function. In particular for integer valued degrees of freedom we have:
For and even,
For and odd,
The probability density function is
The following images show the density of the t distribution for increasing values of The normal distribution is shown as a blue line for comparison. Note that the t distribution (red line) becomes closer to the normal distribution as increases.
Cumulative distribution function
The cumulative distribution function (CDF) can be written in terms of I, the regularized
where
Other values would be obtained by symmetry. An alternative formula, valid for is
where is a particular instance of the hypergeometric function.
For information on its inverse cumulative distribution function, see quantile function § Student's t-distribution.
Special cases
Certain values of give a simple form for Student's t-distribution.
CDF | notes | ||
---|---|---|---|
1 | See Cauchy distribution | ||
2 | |||
3 | |||
4 | |||
5 | |||
See Normal distribution, Error function |
Moments
For the
Moments of order or higher do not exist.[16]
The term for k even, may be simplified using the properties of the gamma function to
For a t distribution with degrees of freedom, the expected value is if and its variance is if The skewness is 0 if and the
Location-scale t distribution
Location-scale transformation
Student's t distribution generalizes to the three parameter location-scale t distribution by introducing a location parameter and a scale parameter With
and
we get
The resulting distribution is also called the non-standardized Student's t distribution.
Density and first two moments
The location-scale t distribution has a density defined by:[17]
Equivalently, the density can be written in terms of :
Other properties of this version of the distribution are:[17]
Special cases
- If follows a location-scale t distribution then for is normally distributed with mean and variance
- The location-scale t distribution with degree of freedom is equivalent to the Cauchy distribution
- The location-scale t distribution with and reduces to the Student's t distribution
How the t distribution arises (characterization)
Sampling distribution of t-statistic
The t distribution arises as the sampling distribution of the t statistic. Below the one-sample t statistic is discussed, for the corresponding two-sample t statistic see Student's t-test.
Unbiased variance estimate
Let be independent and identically distributed samples from a normal distribution with mean and variance The sample mean and unbiased
The resulting (one sample) t statistic is given by
and is distributed according to a Student's t distribution with degrees of freedom.
Thus for inference purposes the t statistic is a useful "pivotal quantity" in the case when the mean and variance are unknown population parameters, in the sense that the t statistic has then a probability distribution that depends on neither nor
ML variance estimate
Instead of the unbiased estimate we may also use the maximum likelihood estimate
yielding the statistic
This is distributed according to the location-scale t distribution:
Compound distribution of normal with inverse gamma distribution
The location-scale t distribution results from
Equivalently, this distribution results from compounding a Gaussian distribution with a
The reason for the usefulness of this characterization is that in Bayesian statistics the inverse gamma distribution is the conjugate prior distribution of the variance of a Gaussian distribution. As a result, the location-scale t distribution arises naturally in many Bayesian inference problems.[18]
Maximum entropy distribution
Student's t distribution is the maximum entropy probability distribution for a random variate X for which is fixed.
Further properties
Monte Carlo sampling
There are various approaches to constructing random samples from the Student's t distribution. The matter depends on whether the samples are required on a stand-alone basis, or are to be constructed by application of a
Integral of Student's probability density function and p-value
The function A(t | ν) is the integral of Student's probability density function, f(t) between -t and t, for t ≥ 0 . It thus gives the probability that a value of t less than that calculated from observed data would occur by chance. 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. This is used in a variety of situations, particularly in
where Ix(a, b) is the regularized incomplete beta function.
For statistical hypothesis testing this function is used to construct the p-value.
Related distributions
- The noncentral t distribution generalizes the t distribution to include a noncentrality parameter. Unlike the nonstandardized t distributions, the noncentral distributions are not symmetric (the median is not the same as the mode).
- The discrete Student's t distribution is defined by its probability mass function at r being proportional to:[21] Here a, b, and k are parameters. This distribution arises from the construction of a system of discrete distributions similar to that of the Pearson distributions for continuous distributions.[22]
- One can generate Student A(t | ν) samples by taking the ratio of variables from the normal distribution and the square-root of the χ² distribution. If we use instead of the normal distribution, e.g., the . This is also more flexible than some other symmetric generalizations of the normal distribution.
- t distribution is an instance of ratio distributions.
Uses
In frequentist statistical inference
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
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. 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.
Hypothesis testing
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. 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.[citation needed]
Confidence intervals
Suppose the number A is so chosen that
when T has a t distribution with n − 1 degrees of freedom. By symmetry, this is the same as saying that A satisfies
so A is the "95th percentile" of this probability distribution, or Then
and this is equivalent to
Therefore, the interval whose endpoints are
is a 90% confidence interval for μ. 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.
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.
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:
The resulting UCL will be the greatest average value that will occur for a given confidence interval and population size. In other words, being the mean of the set of observations, the probability that the mean of the distribution is inferior to UCL1 − α is equal to the confidence level 1 − α .
Prediction intervals
The t distribution can be used to construct a prediction interval for an unobserved sample from a normal distribution with unknown mean and variance.
In Bayesian statistics
The Student's t distribution, especially in its three-parameter (location-scale) version, arises frequently in
Related situations that also produce a t distribution are:
- The posterior distributionof the unknown mean of a normally distributed variable, with unknown prior mean and variance following the above model.
- The independent identically distributednormally distributed data points have been observed, with prior mean and variance as in the above model.
Robust parametric modeling
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.
A Bayesian account can be found in Gelman et al.[24] The degrees of freedom parameter controls the kurtosis of the distribution and is correlated with the scale parameter. 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. Some authors[citation needed] report that values between 3 and 9 are often good choices. Venables and Ripley[citation needed] suggest that a value of 5 is often a good choice.
Student's t process
For practical regression and prediction needs, Student's t processes were introduced, that are generalisations of the Student t distributions for functions. A Student's t process is constructed from the Student t distributions like a Gaussian process is constructed from the Gaussian distributions. For a Gaussian process, all sets of values have a multidimensional Gaussian distribution. Analogously, is a Student t process on an interval if the correspondent values of the process () have a joint multivariate Student t distribution.[25] These processes are used for regression, prediction, Bayesian optimization and related problems. For multivariate regression and multi-output prediction, the multivariate Student t processes are introduced and used.[26]
Table of selected values
The following table lists values for t distributions with ν degrees of freedom for a range of one-sided or two-sided critical regions. The first column is ν, the percentages along the top are confidence levels and the numbers in the body of the table are the factors described in the section on confidence intervals.
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. (See Related distributions above).
One-sided | 75% | 80% | 85% | 90% | 95% | 97.5% | 99% | 99.5% | 99.75% | 99.9% | 99.95% |
---|---|---|---|---|---|---|---|---|---|---|---|
Two-sided | 50% | 60% | 70% | 80% | 90% | 95% | 98% | 99% | 99.5% | 99.8% | 99.9% |
1 | 1.000 | 1.376 | 1.963 | 3.078 | 6.314 | 12.706 | 31.821 | 63.657 | 127.321 | 318.309 | 636.619 |
2 | 0.816 | 1.061 | 1.386 | 1.886 | 2.920 | 4.303 | 6.965 | 9.925 | 14.089 | 22.327 | 31.599 |
3 | 0.765 | 0.978 | 1.250 | 1.638 | 2.353 | 3.182 | 4.541 | 5.841 | 7.453 | 10.215 | 12.924 |
4 | 0.741 | 0.941 | 1.190 | 1.533 | 2.132 | 2.776 | 3.747 | 4.604 | 5.598 | 7.173 | 8.610 |
5 | 0.727 | 0.920 | 1.156 | 1.476 | 2.015 | 2.571 | 3.365 | 4.032 | 4.773 | 5.893 | 6.869 |
6 | 0.718 | 0.906 | 1.134 | 1.440 | 1.943 | 2.447 | 3.143 | 3.707 | 4.317 | 5.208 | 5.959 |
7 | 0.711 | 0.896 | 1.119 | 1.415 | 1.895 | 2.365 | 2.998 | 3.499 | 4.029 | 4.785 | 5.408 |
8 | 0.706 | 0.889 | 1.108 | 1.397 | 1.860 | 2.306 | 2.896 | 3.355 | 3.833 | 4.501 | 5.041 |
9 | 0.703 | 0.883 | 1.100 | 1.383 | 1.833 | 2.262 | 2.821 | 3.250 | 3.690 | 4.297 | 4.781 |
10 | 0.700 | 0.879 | 1.093 | 1.372 | 1.812 | 2.228 | 2.764 | 3.169 | 3.581 | 4.144 | 4.587 |
11 | 0.697 | 0.876 | 1.088 | 1.363 | 1.796 | 2.201 | 2.718 | 3.106 | 3.497 | 4.025 | 4.437 |
12 | 0.695 | 0.873 | 1.083 | 1.356 | 1.782 | 2.179 | 2.681 | 3.055 | 3.428 | 3.930 | 4.318 |
13 | 0.694 | 0.870 | 1.079 | 1.350 | 1.771 | 2.160 | 2.650 | 3.012 | 3.372 | 3.852 | 4.221 |
14 | 0.692 | 0.868 | 1.076 | 1.345 | 1.761 | 2.145 | 2.624 | 2.977 | 3.326 | 3.787 | 4.140 |
15 | 0.691 | 0.866 | 1.074 | 1.341 | 1.753 | 2.131 | 2.602 | 2.947 | 3.286 | 3.733 | 4.073 |
16 | 0.690 | 0.865 | 1.071 | 1.337 | 1.746 | 2.120 | 2.583 | 2.921 | 3.252 | 3.686 | 4.015 |
17 | 0.689 | 0.863 | 1.069 | 1.333 | 1.740 | 2.110 | 2.567 | 2.898 | 3.222 | 3.646 | 3.965 |
18 | 0.688 | 0.862 | 1.067 | 1.330 | 1.734 | 2.101 | 2.552 | 2.878 | 3.197 | 3.610 | 3.922 |
19 | 0.688 | 0.861 | 1.066 | 1.328 | 1.729 | 2.093 | 2.539 | 2.861 | 3.174 | 3.579 | 3.883 |
20 | 0.687 | 0.860 | 1.064 | 1.325 | 1.725 | 2.086 | 2.528 | 2.845 | 3.153 | 3.552 | 3.850 |
21 | 0.686 | 0.859 | 1.063 | 1.323 | 1.721 | 2.080 | 2.518 | 2.831 | 3.135 | 3.527 | 3.819 |
22 | 0.686 | 0.858 | 1.061 | 1.321 | 1.717 | 2.074 | 2.508 | 2.819 | 3.119 | 3.505 | 3.792 |
23 | 0.685 | 0.858 | 1.060 | 1.319 | 1.714 | 2.069 | 2.500 | 2.807 | 3.104 | 3.485 | 3.767 |
24 | 0.685 | 0.857 | 1.059 | 1.318 | 1.711 | 2.064 | 2.492 | 2.797 | 3.091 | 3.467 | 3.745 |
25 | 0.684 | 0.856 | 1.058 | 1.316 | 1.708 | 2.060 | 2.485 | 2.787 | 3.078 | 3.450 | 3.725 |
26 | 0.684 | 0.856 | 1.058 | 1.315 | 1.706 | 2.056 | 2.479 | 2.779 | 3.067 | 3.435 | 3.707 |
27 | 0.684 | 0.855 | 1.057 | 1.314 | 1.703 | 2.052 | 2.473 | 2.771 | 3.057 | 3.421 | 3.690 |
28 | 0.683 | 0.855 | 1.056 | 1.313 | 1.701 | 2.048 | 2.467 | 2.763 | 3.047 | 3.408 | 3.674 |
29 | 0.683 | 0.854 | 1.055 | 1.311 | 1.699 | 2.045 | 2.462 | 2.756 | 3.038 | 3.396 | 3.659 |
30 | 0.683 | 0.854 | 1.055 | 1.310 | 1.697 | 2.042 | 2.457 | 2.750 | 3.030 | 3.385 | 3.646 |
40 | 0.681 | 0.851 | 1.050 | 1.303 | 1.684 | 2.021 | 2.423 | 2.704 | 2.971 | 3.307 | 3.551 |
50 | 0.679 | 0.849 | 1.047 | 1.299 | 1.676 | 2.009 | 2.403 | 2.678 | 2.937 | 3.261 | 3.496 |
60 | 0.679 | 0.848 | 1.045 | 1.296 | 1.671 | 2.000 | 2.390 | 2.660 | 2.915 | 3.232 | 3.460 |
80 | 0.678 | 0.846 | 1.043 | 1.292 | 1.664 | 1.990 | 2.374 | 2.639 | 2.887 | 3.195 | 3.416 |
100 | 0.677 | 0.845 | 1.042 | 1.290 | 1.660 | 1.984 | 2.364 | 2.626 | 2.871 | 3.174 | 3.390 |
120 | 0.677 | 0.845 | 1.041 | 1.289 | 1.658 | 1.980 | 2.358 | 2.617 | 2.860 | 3.160 | 3.373 |
∞ | 0.674 | 0.842 | 1.036 | 1.282 | 1.645 | 1.960 | 2.326 | 2.576 | 2.807 | 3.090 | 3.291 |
One-sided | 75% | 80% | 85% | 90% | 95% | 97.5% | 99% | 99.5% | 99.75% | 99.9% | 99.95% |
Two-sided | 50% | 60% | 70% | 80% | 90% | 95% | 98% | 99% | 99.5% | 99.8% | 99.9% |
- Calculating the confidence interval
Let's say we have a sample with size 11, sample mean 10, and sample variance 2. For 90% confidence with 10 degrees of freedom, the one-sided t value from the table is 1.372 . Then with confidence interval calculated from
we determine that with 90% confidence we have a true mean lying below
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.
And with 90% confidence we have a true mean lying above
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.
So that at 80% confidence (calculated from 100% − 2 × (1 − 90%) = 80%), we have a true mean lying within the interval
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
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.
See also
- F-distribution
- Folded t and half t distributions
- Hotelling's T² distribution
- Multivariate Student distribution
- Standard normal table (Z-distribution table)
- t statistic
- internally studentized residuals
- Wilks' lambda distribution
- Wishart distribution
- Modified half-normal distribution[27] with the pdf on is given as where denotes the Fox–Wright Psi function.
Notes
- ^ Hurst, Simon. "The characteristic function of the Student t distribution". Financial Mathematics Research Report. Statistics Research Report No. SRR044-95. Archived from the original on February 18, 2010.
- S2CID 254231768. Retrieved 2023-02-27.
- ^ Helmert FR (1875). "Über die Berechnung des wahrscheinlichen Fehlers aus einer endlichen Anzahl wahrer Beobachtungsfehler". Zeitschrift für Angewandte Mathematik und Physik (in German). 20: 300–303.
- ^ Helmert FR (1876). "Über die Wahrscheinlichkeit der Potenzsummen der Beobachtungsfehler und uber einige damit in Zusammenhang stehende Fragen". Zeitschrift für Angewandte Mathematik und Physik (in German). 21: 192–218.
- .
- .
- MR 1766040.
- S2CID 121241599.
- ISSN 1364-503X.
- JSTOR 2331554.)
{{cite journal}}
: CS1 maint: numeric names: authors list (link - PMID 27013722.
- OCLC 156200058.
- ^ Fisher RA (1925). "Applications of 'Student's' distribution" (PDF). Metron. 5: 90–104. Archived from the original (PDF) on 5 March 2016.
- OCLC 818811849.
- OCLC 959632184.
- ISBN 9780534119584.
- ^ ISBN 9780470011546.
- ISBN 9780412039911.
- .
- ^ S2CID 120459654.
- ISBN 9780852641378.
- ISBN 9780852641378.
- JSTOR 2290063.
- ISBN 9781439898208.
- arXiv:1402.4306.
- .
- S2CID 237919587.
References
- Senn, S.; Richardson, W. (1994). "The first t test". PMID 8047737.
- ASIN B010WFO0SA.
- Venables, W. N.; Ripley, B. D. (2002). Modern Applied Statistics with S (Fourth ed.). Springer.
- Gelman, Andrew; John B. Carlin; Hal S. Stern; Donald B. Rubin (2003). Bayesian Data Analysis (Second ed.). CRC/Chapman & Hall. ISBN 1-58488-388-X.
External links
- "Student distribution", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- Earliest Known Uses of Some of the Words of Mathematics (S) (Remarks on the history of the term "Student's distribution")
- Rouaud, M. (2013), Probability, Statistics and Estimation (PDF) (short ed.) First Students on page 112.
- Student's t-Distribution, Archived 2021-04-10 at the Wayback Machine