# Student's t-distribution

Student's t Probability density function Cumulative distribution function Parameters deg. of freedom (real) Support pdf cdf Mean Median Mode Variance for Skewness for Kurtosis for Entropy : digamma function, : beta function mgf see text for raw/central moments Char. func. In probability and statistics, the t-distribution or Student's t-distribution 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 t-tests 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 t-distribution is a special case of the generalised hyperbolic distribution. Although the t-distribution is attributed to William Sealy Gosset, it was actually first derived as a posterior distribution in 1876 by Helmert and Luroth. Before William Gosset published the derivation of t-distribution, it also appeared in a more general form as Pearson Type IV in Karl Pearson's 1895 paper. In English literature, the derivation of the t-distribution was first published in 1908 by William Sealy Gosset, while he worked at a Guinness brewery in Dublin. He published it in the paper Biometrika and as he was not allowed to publish under his own name, he penned it under the pseudonym Student. The t-test and the associated theory became well-known through the work of R.A. Pescador, 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 data-based estimate of the variance as if it were certain, y (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. Contenido 1 Occurrence and specification of Student's t-distribution 2 Confidence intervals derived from Student's t-distribution 3 Further theory 4 Table of selected values 5 Special cases 5.1 5.2 6 Related distributions 7 Ver también 8 Referencias 9 External links Occurrence and specification of Student's t-distribution Suppose X1, ..., Xn are independent random variables that are normally distributed with expected value μ and variance σ2. 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. Gosset studied a related quantity, and showed that T has the probability density function with equal to n − 1. The distribution of T is now called the t-distribution. The parameter is conventionally called the number of degrees of freedom. The distribution depends on , but not or the lack of dependence on and is what makes the t-distribution important in both theory and practice. is the Gamma function. Confidence intervals derived from Student's t-distribution Suppose the number A is so chosen that when T has a t-distribution with n − 1 degrees of freedom (this is the same as so A is the "95th percentage point" of this probability distribution). Then and this is equivalent to Therefore the interval whose endpoints are is a 90-percent confidence interval for μ. Por lo tanto, 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 is normally distributed, the one-sided (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. En otras palabras., being the mean of the set of observations, the probability that the mean of the distribution is inferior to is equal to the confidence level A number of other 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 in other situations as well as when examining the differences between means. Por ejemplo, the distribution of Spearman's rank correlation coefficient, rho, 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. Further theory Gosset's result can be stated more generally. (Ver, por ejemplo, Hogg and Craig, Sections 4.4 y 4.8.) Let Z have a normal distribution with mean 0 and variance 1. Let V have a chi-square distribution with ν degrees of freedom. Further suppose that Z and V are independent (see Cochran's theorem). Then the ratio has a t-distribution with ν degrees of freedom. For a t-distribution with ν degrees of freedom, the expected value is 0, and its variance is ν/(ν − 2) if ν > 2. The skewness is 0 and the kurtosis is 6/(ν − 4) if ν > 4. The cumulative distribution function is given by an incomplete beta function, with The t-distribution is related to the F-distribution as follows: the square of a value of t with ν degrees of freedom is distributed as F with 1 and ν degrees of freedom. The overall shape of the probability density function of the t-distribution 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 t-distribution approaches the normal distribution with mean 0 and variance 1. The following images show the density of the t-distribution for increasing values of ν. 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 ν increases. For ν = 30 the t-distribution is almost the same as the normal distribution. Density of the t-distribution (red and green) para 1, 2, 3, 5, 10, y 30 df compared to normal distribution (azul) Table of selected values The following table lists a few selected values for distributions with r degrees of freedom for the 90%, 95%, 97.5%, y 99.5% confidence intervals. These are "one-sided", es decir,, where we see "90%", "4 degrees of freedom", y "1.533", it means Pr(T < 1.533) = 0.9; it does not mean Pr(−1.533 < T < 1.533) = 0.9. Consequently, by the symmetry of this distribution, we have Pr(−1.533 < T) = 0.9, and consequently Pr(−1.533 < T < 1.533) = 0.8. r 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 1.37218 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 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: And, still at 90% confidence, we have a true mean lying over: So that at 80% confidence, we have a true mean lying between Special cases Certain values of give an especially simple form. Distribution function Distribution function Density function Related distributions is a F-distribution if and is a Student's t-distribution. is a normal distribution as where. is a Cauchy distribution if. See also Gamma function Hotelling's T-square distribution Noncentral t-distribution References "Student" (W.S. Gosset) (1908) The probable error of a mean. Biometrika 6(1):1--25. M. Abramowitz and I. A. Stegun, eds. (1972) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover. (See Section 26.7.) R.V. Hogg and A.T. Craig (1978) Introduction to Mathematical Statistics. New York: Macmillan. Helmert FR (1875). "Über die Berechnung des wahrscheinlichen Fehlers aus einer endlichen Anzahl wahrer Beobachtungsfehler". Z. Math. U. Physik. 20: 300–3. Lüroth J (1876). "Vergleichung von zwei Werten des wahrscheinlichen Fehlers". Astron. Nachr. 87 (14): 209–20. Bibcode:1876AN.....87..209L. doi:10.1002/asna.18760871402. T Table | History of T Table, Etymology, one-tail T Table, two-tail T Table and T-statistic Pearson, K. (1895-01-01). "Contributions to the Mathematical Theory of Evolution. II. Skew Variation in Homogeneous Material". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 186: 343–414 (374). doi:10.1098/rsta.1895.0010. ISSN 1364-503X. Wendl MC (2016). "Pseudonymous fame". Science. 351 (6280): 1406. doi:10.1126/science.351.6280.1406. PMID 27013722. External links VassarStats Density plot, critical values, etc., calculated for a user-specified number of d.f. Earliest Known Uses of Some of the Words of Mathematics (S) (Remarks on the history of the term "Student's distribution") Distribution Calculator Calculates probabilities and critical values for normal, t-, chi2- and F-distribution de:Students t-Verteilung su:Sebaran-t student nl:Student-verdeling fr: Loi de Student This page uses Creative Commons Licensed content from Wikipedia (view authors).

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