T distribution is a very important topic within statistics. This page is based on t distribution, first brief description is given on t distribution, and further the properties of t distribution are provided. Grab this learning and gain quality statistics help.
The student’s distribution table comes under the probability distribution tables. The tables that can be included in the probability distribution are cumulative distribution table, upper critical value of students t distribution table, upper critical value of F distribution, upper critical value for chi-square distribution, critical value for t distribution, and upper critical value for PPCC distribution table. In this section we will see about student’s t distribution table.
t-distribution is nothing but a continuous distribution which arises for a small distribution. If we want to estimate the normally distributed function for a small sample size then we will take the normal distribution. It is special case of distribution. Let us consider a small sample size n, drawn from a normal population with the mean µ and standard deviation s. If ‘barx’ and ‘sigma ‘s be the sample mean and standard deviation , then the t distribution statistics is defined as ,
t = ‘(barx – mu)/(sigma)’ ‘sqrt(n)’ or t = ‘(barx – mu)/(sigma)’ ‘sqrt(n – 1)’
where v = n – 1 denotes the distribution function of t.If we calculate t Distribution statistics for each sample, we obtain the sampling distribution for t. This distribution known as Student’s Distribution statistics, is given by
y = y0 / (1 + t2) / v)(v + 1) / 2
Student t distribution
Below are student t distribution properties for a complete t distribution learning:
The t-distribution curve is symmetrical about the line t = 0. it is like the normal curve, Since only even powers of t- distribution statistics appear in the above equation. But it is more peaked than the normal curve with the same distribution. The t-curve approaches the horizontal axis less rapidly than the normal curve. Also t- Distribution statistics curve attains its maximum value at t = 0, So that its mode coincides with the mean.
The limiting form of t-distribution statistics is when v ‘->’ ‘oo’ is given by yoe-1/2 t^2 which is a normal curve. This shows that t is normally distributed for large samples.
More t distribution properties
The property P that the value of t will exceed t is given by
P = ‘int_t^ooydx ‘
The values of t have been tabulated for various values of v from 1 to 30.
Moments about the mean
All the moments of add order about the mean are zero, due to its symmetry about the line t = 0
Even order moments about the mean are
µ2 = ‘(v)/(v-2)’ , µ4 = ‘(3v^2)/((v – 2)(v – 4). . . )’
The t Distribution statistics is often used in tests of hypothesis about the mean when the population standard deviation s is unknown.