1417:
1348:, 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. Hence a second version of the etymology of the term Student is that Guinness did not want their competitors to know that they were using the t-test to determine the quality of raw material. Although it was William Gosset after whom the term "Student" is penned, it was actually through the work of
25:
350:
1109:
1336:) using his pseudonym "Student" because his employer preferred their staff to use pen names when publishing scientific papers instead of their real name, so he used the name "Student" to hide his identity. Gosset worked at the
1240:
707:
416:
863:
247:
926:
1544:
Schlyvitch, B. (October 1937). "Untersuchungen über den anastomotischen Kanal zwischen der
Arteria coeliaca und mesenterica superior und damit in Zusammenhang stehende Fragen".
788:– while defined in terms of the sample mean, its sampling distribution does not depend on the population parameters, and thus it can be used regardless of what these may be.
632:
537:
449:
223:
966:
1284:
1145:
998:
1371:: If the population parameters are known, then rather than computing the t-statistic, one can compute the z-score; analogously, rather than using a
1153:
1529:
1496:
1589:"Die Genauigkeit der Formel von Peters zur Berechnung des wahrscheinlichen Beobachtungsfehlers directer Beobachtungen gleicher Genauigkeit"
1438:
792:
1934:
1461:
1383:
108:
733:
1972:
1456:
770:
137:
46:
660:
369:
805:
89:
1977:
61:
42:
868:
to compute an estimate for the number of standard deviations a given sample is from the mean, as a sample version of a
1967:
1665:
Pfanzagl, J. (1996). "Studies in the history of probability and statistics XLIV. A forerunner of the t-distribution".
1304:-statistic" is abbreviated from "hypothesis test statistic". In statistics, the t-distribution was first derived as a
419:
133:
1446:
1325:
575:
1400:), and thus one must divide the different residuals by different estimates for the error, yielding what are called
164:-statistic is used when the sample size is small or the population standard deviation is unknown. For example, the
68:
35:
1396:, the standard errors of the estimators at different data points vary (compare the middle versus endpoints of a
1466:
1397:
1309:
345:{\displaystyle t_{\hat {\beta }}={\frac {{\hat {\beta }}-\beta _{0}}{\operatorname {s.e.} ({\hat {\beta }})}},}
75:
740:-statistic will asymptotically have one of the Dickey–Fuller distributions (depending on the test setting).
544:
188:
when running hypothesis tests where the p-value tells us what the odds are of the results to have happened.
1332:
who was first to publish the result in
English in his 1908 paper titled "The Probable Error of a Mean" (in
540:
889:
1305:
567:
173:
57:
1416:
1747:"X. Contributions to the mathematical theory of evolution.—II. Skew variation in homogeneous material"
1795:
1758:
1639:
1600:
1401:
1389:
1329:
1317:
643:
635:
608:
513:
425:
199:
1451:
1393:
1353:
1295:
989:
985:
973:
881:
774:
762:
548:
141:
1825:
1727:
1569:
1422:
796:
181:
724:-statistic is different from the normal distribution, even asymptotically. For example, when a
362:
is a non-random, known constant, which may or may not match the actual unknown parameter value
156:-test to determine whether to support or reject the null hypothesis. It is very similar to the
1940:
1930:
1874:
1776:
1719:
1561:
1525:
1492:
1486:
935:
234:
1844:"T Table | History of T Table, Etymology, one-tail T Table, two-tail T Table and T-statistic"
1256:
1117:
984:) that is a statistic (computed from observations). This allows one to compute a frequentist
1905:
1866:
1817:
1809:
1766:
1711:
1676:
1647:
1608:
1553:
1517:
1337:
785:
552:
1688:
1104:{\displaystyle {\frac {X_{n+1}-{\overline {X}}_{n}}{s_{n}{\sqrt {1+n^{-1}}}}}\sim T^{n-1}.}
132:
is the ratio of the difference in a number’s estimated value from its assumed value to its
1684:
714:
169:
1313:
476:-statistics are used to test the significance of corresponding regressor). However, when
82:
1762:
1643:
1604:
157:
1961:
1731:
1349:
1573:
1321:
1870:
1521:
1245:
from which one may compute predictive confidence intervals – given a probability
725:
177:
24:
1821:
1800:
1680:
1667:
1412:
1333:
122:
1944:
1780:
1723:
1651:
1612:
1565:
729:
226:
1878:
1813:
1771:
1746:
1512:
Szabó, István (2003), "Systeme aus einer endlichen Anzahl starrer Körper",
1316:. The t-distribution also appeared in a more general form as Pearson Type
1894:"Subgroup Analysis of Topical Tranexamic Acid in Total Knee Arthroplasty"
1352:
that the distribution became well known as "Student's distribution" and "
1235:{\displaystyle {\overline {X}}_{n}+s_{n}{\sqrt {1+n^{-1}}}\cdot T^{n-1},}
1910:
1893:
709:
correctly estimates the asymptotic variance of this estimator, then the
1829:
1715:
1557:
1365:
1345:
869:
185:
1843:
1430:
1376:
1341:
749:
1627:
1588:
1702:
Sheynin, Oscar (1995). "Helmert's work in the theory of errors".
18:
1628:"Vergleichung von zwei Werthen des wahrscheinlichen Fehlers"
1927:
Probability & statistics for engineers & scientists
1751:
1324:'s 1895 paper. However, the T-Distribution, also known as
241:-statistic for this parameter is any quantity of the form
1929:. Myers, H. Raymond. (7th ed.). New Delhi: Pearson.
1892:
Tuttle, Md; Anazonwu, Bs, Walter; Rubin, Md, Lee (2014).
882:
Prediction interval § Unknown mean, unknown variance
480:-statistic is needed to test the hypothesis of the form
976:– a pivotal quantity (does not depend on the values of
602:
is the number of regressors (including the intercept).
702:{\displaystyle \operatorname {s.e.} ({\hat {\beta }})}
411:{\displaystyle \operatorname {s.e.} ({\hat {\beta }})}
1259:
1156:
1120:
1001:
938:
892:
858:{\displaystyle g(x,X)={\frac {x-{\overline {X}}}{s}}}
808:
663:
611:
555:
error terms), and if the true value of the parameter
516:
428:
372:
250:
202:
1546:
Zeitschrift für
Anatomie und Entwicklungsgeschichte
872:, the z-score requiring the population parameters.
49:. Unsourced material may be challenged and removed.
1278:
1234:
1139:
1103:
960:
920:
857:
701:
626:
531:
443:
410:
344:
217:
1516:, Springer Berlin Heidelberg, pp. 196–199,
8:
1491:. Academic Press. 29 May 2020. p. 397.
1857:Wendl, M. C. (2016). "Pseudonymous fame".
1798:) (1908). "The Probable Error of a Mean".
1909:
1770:
1264:
1258:
1249:, one may compute intervals such that 100
1217:
1199:
1187:
1181:
1168:
1158:
1155:
1125:
1119:
1086:
1065:
1053:
1047:
1035:
1025:
1009:
1002:
1000:
943:
937:
909:
891:
839:
830:
807:
685:
684:
664:
662:
613:
612:
610:
605:In the majority of models, the estimator
518:
517:
515:
430:
429:
427:
394:
393:
373:
371:
322:
321:
301:
293:
275:
274:
271:
256:
255:
249:
204:
203:
201:
109:Learn how and when to remove this message
713:-statistic will asymptotically have the
458:By default, statistical packages report
1478:
720:In some models the distribution of the
184:is unknown. It is also used along with
1704:Archive for History of Exact Sciences
1514:Einführung in die Technische Mechanik
992:), via the following t-distribution:
646:. If the true value of the parameter
168:-statistic is used in estimating the
7:
1488:The Microbiome in Health and Disease
1253:% of the time, the next observation
928:with unknown mean and variance, the
773:, and in the computation of certain
47:adding citations to reliable sources
1147:yields the prediction distribution
932:-statistic of a future observation
921:{\displaystyle N(\mu ,\sigma ^{2})}
598:is the number of observations, and
671:
665:
380:
374:
308:
302:
14:
1462:Folded-t and half-t distributions
1415:
23:
627:{\displaystyle {\hat {\beta }}}
532:{\displaystyle {\hat {\beta }}}
444:{\displaystyle {\hat {\beta }}}
218:{\displaystyle {\hat {\beta }}}
34:needs additional citations for
915:
896:
824:
812:
771:statistical hypothesis testing
696:
690:
681:
618:
523:
435:
405:
399:
390:
333:
327:
318:
280:
261:
209:
1:
1871:10.1126/science.351.6280.1406
160:but with the difference that
1522:10.1007/978-3-642-61925-0_16
1286:will fall in that interval.
1163:
1030:
886:Given a normal distribution
844:
734:augmented Dickey–Fuller test
1925:Walpole, Ronald E. (2006).
543:estimator in the classical
1994:
1382:. This is rare outside of
1293:
879:
784:statistic is that it is a
747:
594:degrees of freedom, where
1632:Astronomische Nachrichten
1593:Astronomische Nachrichten
1652:10.1002/asna.18760871402
1613:10.1002/asna.18760880802
1467:Chi-squared distribution
1447:Student's T-Distribution
1398:simple linear regression
1369:-score (standardization)
1326:Student's T Distribution
961:{\displaystyle X_{n+1},}
780:The key property of the
152:-statistic is used in a
1681:10.1093/biomet/83.4.891
1279:{\displaystyle X_{n+1}}
1140:{\displaystyle X_{n+1}}
761:statistics are used in
644:asymptotically normally
545:linear regression model
192:Definition and features
1772:10.1098/rsta.1895.0010
1745:Pearson, Karl (1895).
1306:posterior distribution
1280:
1236:
1141:
1105:
962:
922:
859:
791:One can also divide a
703:
628:
541:ordinary least squares
533:
445:
412:
346:
219:
1973:Parametric statistics
1898:Reconstructive Review
1402:studentized residuals
1294:Further information:
1281:
1237:
1142:
1106:
963:
923:
880:Further information:
860:
704:
629:
568:sampling distribution
534:
446:
413:
347:
220:
174:sampling distribution
1814:10.1093/biomet/6.1.1
1796:William Sealy Gosset
1390:Studentized residual
1384:standardized testing
1330:William Sealy Gosset
1257:
1154:
1118:
999:
972:observations, is an
936:
890:
806:
775:confidence intervals
732:is regressed in the
661:
609:
549:normally distributed
514:
426:
370:
248:
200:
43:improve this article
1978:Normal distribution
1911:10.15438/rr.v4i2.72
1763:1895RSPTA.186..343P
1644:1876AN.....87..209L
1626:Lüroth, J. (1876).
1605:1876AN.....88..113H
1394:regression analysis
1328:gets its name from
990:confidence interval
986:prediction interval
974:ancillary statistic
968:after one has made
657:, and the quantity
642:and is distributed
16:Ratio in statistics
1968:Statistical ratios
1822:10338.dmlcz/143545
1716:10.1007/BF00374700
1558:10.1007/bf02118337
1457:Hypothesis testing
1423:Mathematics portal
1375:-test, one uses a
1276:
1232:
1137:
1101:
958:
918:
855:
797:standard deviation
699:
624:
574:-statistic is the
529:
500:, then a non-zero
441:
408:
342:
215:
182:standard deviation
180:if the population
138:hypothesis testing
1531:978-3-540-13293-6
1498:978-0-12-820001-8
1208:
1166:
1077:
1074:
1033:
853:
847:
757:Most frequently,
693:
621:
526:
438:
422:of the estimator
402:
337:
330:
283:
264:
235:statistical model
212:
119:
118:
111:
93:
1985:
1949:
1948:
1922:
1916:
1915:
1913:
1889:
1883:
1882:
1854:
1848:
1847:
1840:
1834:
1833:
1791:
1785:
1784:
1774:
1742:
1736:
1735:
1699:
1693:
1692:
1662:
1656:
1655:
1623:
1617:
1616:
1599:(8–9): 113–131.
1587:Helmert (1876).
1584:
1578:
1577:
1541:
1535:
1534:
1509:
1503:
1502:
1483:
1452:Student's t-test
1425:
1420:
1419:
1360:Related concepts
1354:Student's t-test
1338:Guinness Brewery
1320:distribution in
1296:Student's t-test
1285:
1283:
1282:
1277:
1275:
1274:
1241:
1239:
1238:
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1102:
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1078:
1076:
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1054:
1052:
1051:
1041:
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1039:
1034:
1026:
1020:
1019:
1003:
967:
965:
964:
959:
954:
953:
927:
925:
924:
919:
914:
913:
864:
862:
861:
856:
854:
849:
848:
840:
831:
786:pivotal quantity
708:
706:
705:
700:
695:
694:
686:
677:
633:
631:
630:
625:
623:
622:
614:
593:
538:
536:
535:
530:
528:
527:
519:
499:
471:
462:-statistic with
450:
448:
447:
442:
440:
439:
431:
417:
415:
414:
409:
404:
403:
395:
386:
351:
349:
348:
343:
338:
336:
332:
331:
323:
314:
299:
298:
297:
285:
284:
276:
272:
267:
266:
265:
257:
224:
222:
221:
216:
214:
213:
205:
136:. It is used in
114:
107:
103:
100:
94:
92:
51:
27:
19:
1993:
1992:
1988:
1987:
1986:
1984:
1983:
1982:
1958:
1957:
1955:
1953:
1952:
1937:
1924:
1923:
1919:
1891:
1890:
1886:
1856:
1855:
1851:
1842:
1841:
1837:
1793:
1792:
1788:
1744:
1743:
1739:
1701:
1700:
1696:
1664:
1663:
1659:
1638:(14): 209–220.
1625:
1624:
1620:
1586:
1585:
1581:
1543:
1542:
1538:
1532:
1511:
1510:
1506:
1499:
1485:
1484:
1480:
1475:
1421:
1414:
1411:
1362:
1298:
1292:
1260:
1255:
1254:
1213:
1195:
1177:
1157:
1152:
1151:
1121:
1116:
1115:
1082:
1061:
1043:
1042:
1024:
1005:
1004:
997:
996:
939:
934:
933:
905:
888:
887:
884:
878:
832:
804:
803:
755:
746:
715:standard normal
659:
658:
656:
607:
606:
583:
565:
547:(that is, with
512:
511:
506:
498:
487:
481:
469:
463:
424:
423:
368:
367:
361:
300:
289:
273:
251:
246:
245:
198:
197:
194:
170:population mean
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
1991:
1989:
1981:
1980:
1975:
1970:
1960:
1959:
1951:
1950:
1935:
1917:
1884:
1865:(6280): 1406.
1849:
1835:
1786:
1737:
1694:
1675:(4): 891–898.
1657:
1618:
1579:
1552:(6): 709–737.
1536:
1530:
1504:
1497:
1477:
1476:
1474:
1471:
1470:
1469:
1464:
1459:
1454:
1449:
1444:
1436:
1427:
1426:
1410:
1407:
1406:
1405:
1387:
1361:
1358:
1291:
1288:
1273:
1270:
1267:
1263:
1243:
1242:
1231:
1226:
1223:
1220:
1216:
1212:
1205:
1202:
1198:
1194:
1191:
1184:
1180:
1176:
1171:
1165:
1162:
1134:
1131:
1128:
1124:
1112:
1111:
1100:
1095:
1092:
1089:
1085:
1081:
1071:
1068:
1064:
1060:
1057:
1050:
1046:
1038:
1032:
1029:
1023:
1018:
1015:
1012:
1008:
988:(a predictive
957:
952:
949:
946:
942:
917:
912:
908:
904:
901:
898:
895:
877:
874:
866:
865:
852:
846:
843:
838:
835:
829:
826:
823:
820:
817:
814:
811:
795:by the sample
748:Main article:
745:
742:
717:distribution.
698:
692:
689:
683:
680:
676:
673:
670:
667:
654:
620:
617:
563:
525:
522:
504:
496:
485:
467:
437:
434:
420:standard error
407:
401:
398:
392:
389:
385:
382:
379:
376:
359:
353:
352:
341:
335:
329:
326:
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317:
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307:
304:
296:
292:
288:
282:
279:
270:
263:
260:
254:
211:
208:
193:
190:
134:standard error
117:
116:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
1990:
1979:
1976:
1974:
1971:
1969:
1966:
1965:
1963:
1956:
1946:
1942:
1938:
1936:81-7758-404-9
1932:
1928:
1921:
1918:
1912:
1907:
1903:
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1888:
1885:
1880:
1876:
1872:
1868:
1864:
1860:
1853:
1850:
1845:
1839:
1836:
1831:
1827:
1823:
1819:
1815:
1811:
1807:
1803:
1802:
1797:
1790:
1787:
1782:
1778:
1773:
1768:
1764:
1760:
1756:
1752:
1748:
1741:
1738:
1733:
1729:
1725:
1721:
1717:
1713:
1710:(1): 73–104.
1709:
1705:
1698:
1695:
1690:
1686:
1682:
1678:
1674:
1670:
1669:
1661:
1658:
1653:
1649:
1645:
1641:
1637:
1634:(in German).
1633:
1629:
1622:
1619:
1614:
1610:
1606:
1602:
1598:
1595:(in German).
1594:
1590:
1583:
1580:
1575:
1571:
1567:
1563:
1559:
1555:
1551:
1547:
1540:
1537:
1533:
1527:
1523:
1519:
1515:
1508:
1505:
1500:
1494:
1490:
1489:
1482:
1479:
1472:
1468:
1465:
1463:
1460:
1458:
1455:
1453:
1450:
1448:
1445:
1443:
1441:
1437:
1435:
1433:
1429:
1428:
1424:
1418:
1413:
1408:
1403:
1399:
1395:
1391:
1388:
1385:
1381:
1379:
1374:
1370:
1368:
1364:
1363:
1359:
1357:
1355:
1351:
1350:Ronald Fisher
1347:
1343:
1339:
1335:
1331:
1327:
1323:
1319:
1315:
1311:
1307:
1303:
1297:
1289:
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1268:
1265:
1261:
1252:
1248:
1229:
1224:
1221:
1218:
1214:
1210:
1203:
1200:
1196:
1192:
1189:
1182:
1178:
1174:
1169:
1160:
1150:
1149:
1148:
1132:
1129:
1126:
1122:
1098:
1093:
1090:
1087:
1083:
1079:
1069:
1066:
1062:
1058:
1055:
1048:
1044:
1036:
1027:
1021:
1016:
1013:
1010:
1006:
995:
994:
993:
991:
987:
983:
979:
975:
971:
955:
950:
947:
944:
940:
931:
910:
906:
902:
899:
893:
883:
875:
873:
871:
850:
841:
836:
833:
827:
821:
818:
815:
809:
802:
801:
800:
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789:
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768:
766:
760:
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735:
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727:
723:
718:
716:
712:
687:
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645:
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603:
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597:
591:
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581:
580:-distribution
579:
573:
569:
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558:
554:
553:homoscedastic
550:
546:
542:
520:
508:
507:may be used.
503:
495:
491:
484:
479:
475:
466:
461:
456:
454:
432:
421:
396:
387:
383:
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365:
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311:
305:
294:
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286:
277:
268:
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252:
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242:
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232:
229:of parameter
228:
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187:
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147:
145:
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131:
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67:
63:
60: –
59:
58:"T-statistic"
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
1954:
1926:
1920:
1904:(2): 37–41.
1901:
1897:
1887:
1862:
1858:
1852:
1838:
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1799:
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1754:
1750:
1740:
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1672:
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1439:
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1322:Karl Pearson
1301:
1299:
1250:
1246:
1244:
1114:Solving for
1113:
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977:
969:
929:
885:
867:
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781:
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769:, a form of
764:
758:
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710:
651:
650:is equal to
647:
639:
604:
599:
595:
589:
585:
577:
571:
560:
559:is equal to
556:
509:
501:
493:
489:
482:
477:
473:
464:
459:
457:
452:
363:
356:
354:
238:
230:
195:
178:sample means
165:
161:
153:
149:
143:
127:
126:
120:
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96:
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79:
72:
65:
53:
41:Please help
36:verification
33:
1808:(1): 1–25.
1794:"Student" (
1757:: 343–414.
1308:in 1876 by
736:, the test
726:time series
566:, then the
1962:Categories
1801:Biometrika
1668:Biometrika
1473:References
1442:-statistic
1334:Biometrika
1300:The term "
876:Prediction
763:Student's
636:consistent
576:Student's
142:Student's
130:-statistic
123:statistics
99:April 2020
69:newspapers
1945:818811849
1781:1364-503X
1732:121241599
1724:0003-9519
1566:0340-2061
1222:−
1211:⋅
1201:−
1164:¯
1091:−
1080:∼
1067:−
1031:¯
1022:−
907:σ
900:μ
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837:−
730:unit root
691:^
688:β
679:
619:^
616:β
524:^
521:β
436:^
433:β
400:^
397:β
388:
328:^
325:β
316:
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287:−
281:^
278:β
262:^
259:β
237:. Then a
227:estimator
210:^
207:β
1879:27013722
1574:27311567
1409:See also
793:residual
233:in some
1859:Science
1830:2331554
1759:Bibcode
1689:1766040
1640:Bibcode
1601:Bibcode
1346:Ireland
1310:Helmert
1290:History
870:z-score
728:with a
570:of the
472:(these
418:is the
186:p-value
172:from a
158:z-score
83:scholar
1943:
1933:
1877:
1828:
1779:
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1342:Dublin
1314:Lüroth
767:-tests
539:is an
366:, and
355:where
225:be an
148:. The
125:, the
85:
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71:
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1826:JSTOR
1728:S2CID
1570:S2CID
1434:-test
1392:: In
1380:-test
753:-test
582:with
146:-test
90:JSTOR
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1941:OCLC
1931:ISBN
1875:PMID
1777:ISSN
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638:for
551:and
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196:Let
140:via
62:news
1906:doi
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1755:186
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