Student's t-Distribution¶

Mathematical Definition¶

If Z ~ N(0,1) and V ~ χ²(ν) are independent, then:

\[ t = \frac{Z}{\sqrt{V/\nu}} \]

follows a t-distribution with ν degrees of freedom (df).

The probability density function is:

\[ f(t) = \frac{\Gamma(\frac{\nu+1}{2})}{\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{t^2}{\nu}\right)^{-\frac{\nu+1}{2}} \]

Key Properties¶

Symmetric around 0, like the normal distribution
Heavier tails than normal (more prone to extreme values)
Mean = 0 (for ν > 1)
Variance = ν/(ν-2) for ν > 2 (higher than N(0,1) variance of 1)
As ν → ∞, t-distribution → N(0,1)

When to Use It¶

Use the t-distribution when: - You have a small sample (n < 30) - You know the sample mean but NOT the population standard deviation - You want to estimate confidence intervals or perform hypothesis tests

For large samples (n ≥ 30), the t-distribution converges to the standard normal, so the difference is negligible.

Derivation¶

The Problem¶

You have n samples from a normal population: X₁, X₂, ..., Xₙ. You want to know if the sample mean X̄ is consistent with a hypothesized population mean μ.

The sample mean is normally distributed:

\[ \bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right) \]

Standardization with Unknown σ¶

If we knew σ, we would use the Z-score:

\[ Z = \frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \sim N(0,1) \]

But in practice we don't know σ. We use the sample standard deviation s:

\[ s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(X_i - \bar{X})^2} \]

Substituting s for σ gives:

\[ t = \frac{\bar{X} - \mu}{s/\sqrt{n}} \]

This statistic does NOT follow N(0,1) because s introduces additional uncertainty.

Decomposing the t-Statistic¶

Multiply top and bottom by σ:

\[ t = \frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \cdot \frac{\sigma}{s} = \frac{Z}{s/\sigma} \]

Where Z ~ N(0,1).

Relationship to Chi-Squared¶

The scaled sample variance follows a chi-squared distribution:

\[ V = \frac{(n-1)s^2}{\sigma^2} \sim \chi^2(n-1) \]

Therefore:

\[ \frac{s}{\sigma} = \sqrt{\frac{V}{n-1}} \]

Final Result¶

Substituting:

\[ t = \frac{Z}{\sqrt{V/(n-1)}} \]

This is the t-statistic: a standard normal divided by the square root of a chi-squared (scaled by df).

Z and V are independent (property of the normal distribution), so t follows Student's t-distribution with ν = n-1 degrees of freedom.

Why Heavier Tails?¶

When n is small, s² is a noisy estimate of σ². Sometimes s² is too small, making t larger than Z. This creates occasional extreme values, producing heavier tails.

As n grows, s² → σ² (law of large numbers), so t → Z.

Common Uses¶

Confidence intervals for means when σ is unknown
t-tests: one-sample, two-sample, paired
Linear regression: testing coefficient significance

Brief History¶

Developed by William Sealy Gosset in 1908 while working at Guinness. He published under the pseudonym "Student" because the company banned employees from publishing.

Gosset had a practical problem: he needed to test beer quality with small samples (4-6 bottles per batch). Formulas based on the normal distribution gave unreliable results. He derived this distribution through manual simulations and solved the small-sample problem.

Ronald Fisher later generalized and rigorously proved the theory, making the t-distribution a fundamental tool in modern statistics.

Comparison with Normal¶

df = 1: Very heavy tails (almost like a Cauchy)
df = 3: Noticeably heavier tails than N(0,1)
df = 10: Minor differences
df = 30: Practically identical to N(0,1)
df = ∞: Exactly N(0,1)