Student's T Distribution

Let T represent the random variable of Student's T distribution. Let t represent a specific value of T.

What type of probability?

left, P(T<t)
right, P(T>t)
center, P(|T|<t)
two-tail, P(|T|>t)

Degree of freedom, df

df =

Boundary and Probability

Enter either the boundary or the probability to calculate the other.
t =
P =

Output: statement and visualization

P(T<0) = 0.5

Density visualization

The dotted line represents the standard normal distribution's density . The solid line represents the T distribution's density. Notice the T distribution has fatter tails: more density on the sides and less density in the middle. This additional spread occurs when the standard error is estimated (not known exactly) from the sample, and thus there is more possibility for the sample statistic to be far from the population parameter (because the standard error may be larger than estimated). Notice with larger samples, the estimated standard error is more certain, so the T distribution aligns with the standard normal distribution.