T Distribution Calculator (Student)

Calculates the t distribution or the inverse t distribution.

The t distribution calculator computes the cumulative distribution (p) or the percentile (𝑥₁), inverse t distribution calculator

Choose 𝑥1 to calculate the cumulative t probability based on the t percentile, p(X ≤ 𝑥1) to calculate the t percentile based on the t cumulative probability, 𝑥1, 𝑥2 to calculate p(𝑥1 ≤ X ≤ 𝑥2) or p(X ≤ 𝑥1), p(X ≤ 𝑥2) to calculate x1, x2, p(𝑥1 ≤ X ≤ 𝑥2)

What is the T distribution?

The T-student distribution is an artificial distribution used for a normally distributed population when we don't know the population's standard deviation. Since we use the sample standard deviation, the is more uncertainty, and the T-distribution has heavier tails than the normal distribution (Leptokurtic kurtosis)

T distribution looks similar to the normal distribution but lower in the middle and with thicker tails. The shape depends on the degrees of freedom, number of independent observations, usually number of observations minus one (n-1). The higher the degree of freedom the more it resembles the normal distribution.

When to use the T distribution?

You should use the t distribution when the population's distribution is normal and you don't know the population's standard deviation.

When the sample size is big, as a rule of thumb is more than 30, the result of the normal distribution is similar to the t distribution.
In the old days when people used tables to calculate the probability, the z-table was more detailed, since it doesn't have the degrees of freedom dimension, hence it was more accurate to use the z-table instead of the t-table.
Today, when you estimate the standard deviation (S), you should always use the t distribution calculator.
Normal distribution vs T distribution

T distribution formula

Probability density function (PDF)

f(𝑥) =1(1+𝑥2)-(k+1)/2
B(1/2,k/2)√kk
Where:
k - degrees of freedom.
B - the Beta function.

Sample average distribution

The sample average's distribution is the t distribution.
t =x̄ - μ
S/√n