What is the t-distribution

The t-distribution is a probability distribution that the sample mean \bar{X} standardized by the sample standard deviation follows.

If the population mean is \mu and the standard deviation is \sigma, the distribution of the sample mean becomes a normal distribution with mean \mu and variance \frac{\sigma^2}{n} from central limit theorem. The z-value, which is the standardized value of the sample mean, is as follows:

Hypothesis testing is performed based on the standard normal distribution, which is the distribution that the above equation z follows. However, the standard deviation \sigma of the population in the above equation is usually an unknown value. Therefore, we use the sample standard deviation s instead of \sigma. Therefore, we will consider the following distribution that t follows.

The s^2 is also called unbiased variance. The distribution that the above equation t follows is the t-distribution. In other words, the t-distribution is an alternative distribution to the standard normal distribution.

The probability density function of the t-distribution is expressed by the following equation:

The probability density function of a t-distribution is determined by the shape of the distribution with only degrees of freedom (n). The graph of a t-distribution with 1 to 10 degrees of freedom looks like this.

As n degrees of freedom increase, the distribution approaches the standard normal distribution.

Thus, if n is sufficiently large, the t-distribution can be used as a proxy for the standard normal distribution in hypothesis testing where the population variance is unknown. This test is called the t-test.

Expected value and variance of the t-distribution

The expected value and variance of the t-distribution are respectively:

t-distribution table (upper side)

Since the t-distribution has only n parameters, the probabilities of the t-distribution can be summarized in a table called the t-distribution table. Below is a t-distribution table summarizing the degrees of freedom of the t-distribution for which the upper side probabilities \alpha are equal to 0.1, 0.05, 0.025, 0.01 and 0.005, respectively.

For example, if you want to find the upper 5% point of a t-distribution with 20 degrees of freedom, look for the value at the intersection of n=20 and \alpha=0.05. Thus, the upper 5% point you are looking for is 1.725.

Python Code

The following is the Python code used to draw the t-distribution.

import numpy as np
from scipy import stats
import matplotlib.pyplot as plt

plt.style.use('ggplot')
fig, ax = plt.subplots(facecolor="w", figsize=(10, 5))

x = np.linspace(-4, 4, 100)
z = stats.norm.pdf(x, loc=0, scale=1)

for df in range(1, 11):
    t = stats.t.pdf(x, df)
    plt.plot(x, t, label=f"t dist(df={df})")
plt.legend()

import numpy as np
from scipy import stats
import matplotlib.pyplot as plt

plt.style.use('ggplot')
fig, ax = plt.subplots(facecolor="w", figsize=(10, 5))

x = np.linspace(-4, 4, 100)
z = stats.norm.pdf(x, loc=0, scale=1)

for df in [1, 5, 10]:
    t = stats.t.pdf(x, df)
    plt.plot(x, t, label=f"t dist(df={df})")
plt.plot(x, z, label='Std norm dist', linewidth=4)
plt.legend()