See F. Levy, Dollars and Dreams: The Changing American Income Distribution (1988); K. M. Perkins, Production, Distribution, and Growth in Transitional Economies (1988).
In statistics, a frequency distribution in the shape of the classic bell curve. It accurately represents most variations in such attributes as height and weight. Any random variable with a normal distribution has a mean (see mean, median, and mode) and a standard deviation that indicates how much the data as a whole deviate from the mean. The standard deviation is smaller for data clustered closely around the mean value and larger for more dispersed data sets.
Learn more about normal distribution with a free trial on Britannica.com.
Crime of making an unauthorized imitation of a genuine article, typically money, with the intent to deceive or defraud. Because of the value conferred on money and the high level of technical skill required to imitate it, counterfeiting is singled out from other acts of forgery. It is generally punished as a felony (see felony and misdemeanour). The international police organization Interpol was established primarily to organize lawenforcement efforts against counterfeiting. Software, credit cards, designer clothing, and watches are among nonmoney items commonly counterfeited.
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{{Probability distribution 
name =Student's t
type =density
pdf_image =
cdf_image =
parameters =$nu\; >\; 0$ degrees of freedom (real)
support =$x\; in\; (infty;\; +infty)!$
pdf =$frac\{Gamma(frac\{nu+1\}\{2\})\}\; \{sqrt\{nupi\},Gamma(frac\{nu\}\{2\})\}\; left(1+frac\{x^2\}\{nu\}\; right)^\{(frac\{nu+1\}\{2\})\}!$
cdf =$begin\{matrix\}\; frac\{1\}\{2\}\; +\; x\; Gamma\; left(frac\{nu+1\}\{2\}\; right)\; cdot[0.5em]\; frac\{,\_2F\_1\; left\; (frac\{1\}\{2\},frac\{nu+1\}\{2\};frac\{3\}\{2\};\; frac\{x^2\}\{nu\}\; right)\}\; \{sqrt\{pinu\},Gamma\; (frac\{nu\}\{2\})\}\; end\{matrix\}$ median =$0$
mode =$0$
variance =$frac\{nu\}\{nu2\}text\{\; for\; \}nu>2!$, otherwise undefined
skewness =$0text\{\; for\; \}nu>3$
kurtosis =$frac\{6\}\{nu4\}text\{\; for\; \}nu>4!$ 
entropy =$begin\{matrix\}\; frac\{nu+1\}\{2\}left[\; psi(frac\{1+nu\}\{2\})\; \; psi(frac\{nu\}\{2\})$right] [0.5em]+ log{left[sqrt{nu}B(frac{nu}{2},frac{1}{2})right]} end{matrix}

mgf =(Not defined)
char =}} In probability and statistics, Student's tdistribution (or simply the tdistribution) is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. It is the basis of the popular Student's ttests for the statistical significance of the difference between two sample means, and for confidence intervals for the difference between two population means. The Student's tdistribution is a special case of the generalised hyperbolic distribution.
The derivation of the tdistribution was first published in 1908 by William Sealy Gosset, while he worked at a Guinness Brewery in Dublin. He was prohibited from publishing under his own name, so the paper was written under the pseudonym Student. The ttest and the associated theory became wellknown through the work of R.A. Fisher, who called the distribution "Student's distribution".
Student's distribution arises when (as in nearly all practical statistical work) the population standard deviation is unknown and has to be estimated from the data. Textbook problems treating the standard deviation as if it were known are of two kinds: (1) those in which the sample size is so large that one may treat a databased 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 tdistribution is the probability distribution of the ratio
where
where $nu$ is the number of degrees of freedom and $Gamma$ is the Gamma function.
The overall shape of the probability density function of the tdistribution resembles the bell shape of a normally distributed variable with mean 0 and variance 1, except that it is a bit lower and wider. As the number of degrees of freedom grows, the tdistribution approaches the normal distribution with mean 0 and variance 1.
The following images show the density of the tdistribution for increasing values of $nu$. The normal distribution is shown as a blue line for comparison.; Note that the tdistribution (red line) becomes closer to the normal distribution as $nu$ increases. For $nu$ = 30 the tdistribution is almost the same as the normal distribution.
Suppose X_{1}, ..., X_{n} are independent random variables that are normally distributed with expected value μ and variance σ^{2}. Let
be the sample mean, and
be the sample variance. It is readily shown that the quantity
is normally distributed with mean 0 and variance 1, since the sample mean $scriptstyle\; overline\{X\}\_n$ is normally distributed with mean $mu$ and standard error $scriptstylesigma/sqrt\{n\}$.
Gosset studied a related pivotal quantity,
which differs from Z in that the exact standard deviation $scriptstyle\; sigma$ is replaced by the random variable $scriptstyle\; S\_n$. Technically, $scriptstyle(n1)S\_n^2/sigma^2$ has a $scriptstylechi\_\{n1\}^2$ distribution by Cochran's theorem. Gosset's work showed that T has the probability density function
This may also be written as
The distribution of T is now called the tdistribution. The parameter $nu$ is called the number of degrees of freedom. The distribution depends on $nu$, but not μ or σ; the lack of dependence on μ and σ is what makes the tdistribution important in both theory and practice.
Gosset's result can be stated more generally. (See, for example, Hogg and Craig, Sections 4.4 and 4.8.) Let Z have a normal distribution with mean 0 and variance 1. Let V have a chisquare distribution with $nu$ degrees of freedom. Further suppose that Z and V are independent (see Cochran's theorem). Then the ratio
has a tdistribution with $nu$ degrees of freedom.
with
It should be noted that the term for 0 < k < $nu$, k even, may be simplified using the properties of the Gamma function to
For a tdistribution with $nu$ degrees of freedom, the expected value is 0, and its variance is $nu$/($nu$ − 2) if $nu$ > 2. The skewness is 0 if $nu$ > 3 and the kurtosis is 6/($nu$ − 4) if $nu$ > 4.
Suppose the number A is so chosen that
when T has a tdistribution 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 $A=t\_\{(0.05,n1)\}$. Then
and this is equivalent to
is a 90percent 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 tdistribution 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 ttests: since the difference between the means of samples from two normal distributions is itself distributed normally, the tdistribution can be used to examine whether that difference can reasonably be supposed to be zero.
If the data are normally distributed, the onesided (1 − a)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, $overline\{X\}\_n$ being the mean of the set of observations, the probability that the mean of the distribution is inferior to UCL_{1−a}> is equal to the confidence level 1 − a.
A number of other statistics can be shown to have tdistributions for samples of moderate size under null hypotheses that are of interest, so that the tdistribution forms the basis for significance tests in other situations as well as when examining the differences between means. 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.
See prediction interval for another example of the use of this distribution.
The function $scriptstyle\; A(tnu)$ is the integral of Student's probability density function, ƒ(t) between −t and t. It thus gives the probability that a value of t less than that calculated from observed data would occur by chance. Therefore, the function $scriptstyle\; A(tnu)$ 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 ttests. For the statistic t, with $scriptstylenu$ degrees of freedom, $scriptstyle\; A(tnu)$ 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). It is defined for real t by the following formula:
where B is the Beta function. For t > 0, there is a relation to the regularized incomplete beta function I_{x}(a, b) as follows:
The probability that a value of the t statistic greater than or equal to that observed would happen by chance, if the two sets of data were drawn from the same population, is given by
Distribution function:
Density function:
Distribution function:
Density function:
Lange et al explored the use of the tdistribution for robust modelling of heavy tailed data in a variety of contexts. A Bayesian account can be found in Gelman et al. 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 report that values between 3 and 9 are often good choices. Venables and Ripley suggest that a value of 5 is often a good choice.
The following table lists a few selected values for tdistributions with $nu$ degrees of freedom for a range of onesided critical regions. For an example of how to read this table, take the fourth row, which begins with 4; that means $nu$, the number of degrees of freedom, is 4 (and if we are dealing, as above, with n values with a fixed sum, n = 5). Take the fifth entry, in the column headed 95%. The value of that entry is "2.132". Then the probability that T is less than 2.132 is 95% or Pr(−∞ < T < 2.132) = 0.95; the entry does not mean (as it might with other distributions) that Pr(−2.132 < T < 2.132) = 0.95.
In fact, by the symmetry of the distribution,
and so
Note that the last row also gives critical points: a tdistribution with infinitelymany degrees of freedom is a normal distribution. (See above: Related distributions).
$nu$  75%  80%  85%  90%  95%  97.5%  99%  99.5%  99.75%  99.9%  99.95% 

1  1.000  1.376  1.963  3.078  6.314  12.71  31.82  63.66  127.3  318.3  636.6 
2  0.816  1.061  1.386  1.886  2.920  4.303  6.965  9.925  14.09  22.33  31.60 
3  0.765  0.978  1.250  1.638  2.353  3.182  4.541  5.841  7.453  10.21  12.92 
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 
$infty$  0.674  0.842  1.036  1.282  1.645  1.960  2.326  2.576  2.807  3.090  3.291 
The number at the beginning of each row in the table above is $nu$ which has been defined above as n − 1. The percentage along the top is 100%(1 − α). The numbers in the main body of the table are t_{α,$nu$}. If a quantity T is distributed as a Student's t distribution with $nu$ degrees of freedom, then there is a probability 1 − α that T will be less than t_{α,$nu$}.(Calculated as for a onetailed or onesided test as opposed to a twotailed test.)
For example, given a sample with a sample variance 2 and sample mean of 10, taken from a sample set of 11 (10 degrees of freedom), using the formula
We can determine that at 90% confidence, we have a true mean lying below
(In other words, on average, 90% of the times that an upper threshold is calculated by this method, the true mean lies below this upper threshold.) And, still at 90% confidence, we have a true mean lying over
(In other words, on average, 90% of the times that a lower threshold is calculated by this method, the true mean lies above this lower threshold.) So that at 90% confidence, we have a true mean lying between
(In other words, on average, 90% of the times that upper and lower thresholds are calculated by this method, the true mean is both below the upper threshold and above the lower threshold. This is not the same thing as saying that there is an 90% 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.)
For information on the inverse cumulative distribution function see Quantile function.