T-Distribution | What It Is and How To Use It (With Examples)
The t-distribution, also known as Student’s t-distribution, is a way of describing data that follow a bell curve when plotted on a graph, with the greatest number of observations close to the mean and fewer observations in the tails.
In statistics, the t-distribution is most often used to:
- Find the critical values for a confidence interval when the data is approximately normally distributed.
- Find the corresponding p-value from a statistical test that uses the t-distribution (t-tests, regression analysis).
What is a t-distribution?
The t-distribution is a type of normal distribution that is used for smaller sample sizes. Normally-distributed data form a bell shape when plotted on a graph, with more observations near the mean and fewer observations in the tails.
The t-distribution is used when data are approximately normally distributed, which means the data follow a bell shape but the population variance is unknown. The variance in a t-distribution is estimated based on the degrees of freedom of the data set (total number of observations minus 1).
It is a more conservative form of the standard normal distribution, also known as the z-distribution. This means that it gives a lower probability to the center and a higher probability to the tails than the standard normal distribution.
T-distribution and the standard normal distribution
As the degrees of freedom (total number of observations minus 1) increases, the t-distribution will get closer and closer to matching the standard normal distribution, a.k.a. the z-distribution, until they are almost identical.
Above 30 degrees of freedom, the t-distribution roughly matches the z-distribution. Therefore, the z-distribution can be used in place of the t-distribution with large sample sizes.
The z-distribution is preferable over the t-distribution when it comes to making statistical estimates because it has a known variance. It can make more precise estimates than the t-distribution, whose variance is approximated using the degrees of freedom of the data.
T-distribution and t-scores
In statistics, t-scores are primarily used to find two things:
- The upper and lower bounds of a confidence interval when the data are approximately normally distributed.
- The p-value of the test statistic for t-tests and regression tests.
T-scores and confidence intervals
Confidence intervals use t-scores to calculate the upper and lower bounds of the prediction interval. The t-score used to generate the upper and lower bounds is also known as the critical value of t, or t*.
T-scores and p-values
Statistical tests generate a test statistic showing how far from the null hypothesis of the statistical test your data is. They then calculate a p-value that describes the likelihood of your data occurring if the null hypothesis were true.
The test statistic for t-tests and regression tests is the t-score. While most statistical programs will automatically calculate the corresponding p-value for the t-score, you can also look up the values in a t-table, using your degrees of freedom and t-score to find the p-value.
The t-score which generates a p-value below your threshold for statistical significance is known as the critical value of t, or t*.
Frequently asked questions
- What is a t-distribution?
The t-distribution is a way of describing a set of observations where most observations fall close to the mean, and the rest of the observations make up the tails on either side. It is a type of normal distribution used for smaller sample sizes, where the variance in the data is unknown.
The t-distribution forms a bell curve when plotted on a graph. It can be described mathematically using the mean and the standard deviation.
- What is the difference between the t-distribution and the standard normal distribution?
In this way, the t-distribution is more conservative than the standard normal distribution: to reach the same level of confidence or statistical significance, you will need to include a wider range of the data.
- What is a t-score?
A t-score (a.k.a. a t-value) is equivalent to the number of standard deviations away from the mean of the t-distribution.
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T-Ugawaji: Nini ni na Jinsi ya Kutumia (Kwa Mifano) T-Ugawaji ni njia ya kuelezea data ambayo hufuata mviringo wa kengele wakati wa kuchorwa kwenye grafu, na idadi kubwa ya uchunguzi karibu na wastani na uchunguzi mdogo katika mikia. Katika takwimu, t-usambazaji ni mara nyingi kutumika kwa: - Kupata maadili muhimu kwa ajili ya muda wa uaminifu wakati data ni takriban kawaida kusambazwa. - Kupata p-thamani inayolingana kutoka mtihani wa takwimu ambayo hutumia usambazaji wa t (mtihani wa t, uchambuzi wa kurudi nyuma). Ugawaji wa t ni nini? T-usambazaji ni aina ya usambazaji wa kawaida ambayo hutumiwa kwa ukubwa mdogo wa sampuli. Takwimu za kawaida kusambazwa fomu umbo la kengele wakati plotted juu ya grafu, na uchunguzi zaidi karibu na wastani na uchunguzi wachache katika mikia. Ugawaji wa t hutumiwa wakati data ni karibu kawaida kusambazwa, ambayo ina maana data kufuata umbo la kengele lakini idadi ya watu kutofautiana haijulikani. Utaratibu wa t-distribution ni tofauti na kiwango cha uhuru wa data (jumla ya idadi ya uchunguzi minus 1). Ni aina ya kihafidhina zaidi ya usambazaji wa kawaida wa kawaida, pia inajulikana kama usambazaji wa z. Hii ina maana kwamba inatoa uwezekano mdogo kwa kituo na uwezekano mkubwa kwa mikia kuliko usambazaji wa kawaida wa kawaida. T-Ugawanyiko wa kawaida (T-Ugawanyiko wa kawaida) ni kiwango cha kawaida cha kawaida ambacho kinahusiana na kiwango cha kawaida cha kawaida (T-Ugawanyiko wa kawaida). Z-distribution, mpaka wao ni karibu sawa. Zaidi ya digrii 30 za uhuru, t-usambazaji hufanana na z-usambazaji. Kwa hiyo, z-usambazaji inaweza kutumika badala ya t-usambazaji na ukubwa mkubwa wa sampuli. Z-usambazaji ni bora juu ya t-usambazaji linapokuja kufanya makadirio ya takwimu kwa sababu ina kutofautiana inayojulikana. Inaweza kufanya makadirio sahihi zaidi kuliko usambazaji wa t, ambao kutofautiana ni takriban kwa kutumia digrii ya uhuru wa data. T-score ni kipimo cha kiwango cha juu cha t-score, ambacho hutumiwa kwa sababu ya upendeleo wa data katika upatikanaji wa data ya kawaida. - p-thamani ya takwimu ya mtihani kwa t-mtihani na vipimo vya kurudi nyuma. T-scores na vipindi vya uaminifu Vipindi vya uaminifu hutumia t-scores kuhesabu mipaka ya juu na ya chini ya vipindi vya utabiri. T-score kutumika kuzalisha mipaka ya juu na ya chini pia inajulikana kama thamani muhimu ya t, au t*. T-scores na p-thamani: Uchunguzi wa takwimu hutokeza takwimu za mtihani zinazoonyesha jinsi data yako ilivyo mbali na nadharia ya null ya mtihani wa takwimu. Kisha wao kuhesabu p-thamani ambayo inaelezea uwezekano wa data yako kutokea kama nadharia ya sifuri walikuwa kweli. Takwimu ya mtihani kwa t-mtihani na vipimo vya kurudi nyuma ni t-score. Ingawa programu nyingi za takwimu zitahesabu moja kwa moja thamani ya p inayolingana kwa alama ya t, unaweza pia kuangalia maadili katika meza ya t, kwa kutumia digrii zako za uhuru na alama ya t kupata thamani ya p. T-score ambayo inazalisha p-thamani chini ya kizingiti chako kwa umuhimu wa takwimu inajulikana kama thamani muhimu ya t, au t*. Maswali yanayoulizwa mara kwa mara - Ugawaji wa t ni nini? T-Ugawanyiko ni njia ya kuelezea seti ya uchunguzi ambapo uchunguzi wengi kuanguka karibu na wastani, na mapumziko ya uchunguzi kufanya mkia pande zote mbili. Ni aina ya usambazaji wa kawaida kutumika kwa ukubwa mdogo sampuli, ambapo kutofautiana katika data haijulikani. usambazaji t fomu bell curve wakati plotted juu ya grafu. Inaweza kuelezwa kwa hisabati kwa kutumia wastani na kupotoka kwa kawaida. - Ni nini tofauti kati ya t-usambazaji na kawaida kawaida usambazaji? Kwa njia hii, t-usambazaji ni zaidi ya kihafidhina kuliko kawaida kawaida usambazaji: kufikia ngazi sawa ya uaminifu au umuhimu wa takwimu, unahitaji ni pamoja na mbalimbali pana ya data. - T-score ni nini? T-score (a.k.a. T-thamani ni sawa na idadi ya kupotoka kwa kawaida mbali na wastani wa usambazaji wa t. Quote hii Scribbr makala Kama unataka kunukuu chanzo hiki, unaweza kunakili na kuweka kunakili au bonyeza <unk>Cite hii Scribbr makala<unk> kifungo kwa moja kwa moja kuongeza kunakili kwa bure yetu Reference Generator.