t-Distribution — When σ is Unknown

Foundations of Statistics

The Real-World Workhorse for Means

The t-distribution accounts for the extra uncertainty when estimating σ with s, making it the standard for real-world mean comparisons. Its heavier tails provide more conservative inference than the normal distribution.

Quality Control — Comparing process means when population variance is unknown
Clinical Research — Testing treatment effects with small sample sizes
Business Analytics — A/B testing with limited data to make faster decisions

When σ is unknown, the t-distribution is your trusted companion.

Core Concepts

The t-distribution arises when we estimate the population standard deviation $\sigma$ with the sample standard deviation $s$ . It has heavier tails than the normal, reflecting additional uncertainty from estimating $\sigma$ .

Dft-Distribution

Let $Z \sim N(0,1)$ and $V \sim \chi^2_\nu$ be independent. Then $T = \frac{Z}{\sqrt{V/\nu}}$ follows a t-distribution with $\nu$ degrees of freedom, written $T \sim t_\nu$ .

PDF of t-Distribution

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

Here,

$\nu$ =Degrees of freedom
$\Gamma$ =Gamma function

Heavy Tails

The t-distribution has heavier tails than the normal, meaning more probability in the extremes. This reflects the additional uncertainty from estimating $\sigma$ . As $\nu \to \infty$ , the t-distribution approaches $N(0,1)$ .

Interactive Visualization

t-Distribution — Interactive Explorer

Mean (μ) = 0.0000Var = 1.6667σ = 1.2910

t-Distribution vs Normal — Heavy Tails

Mean (μ) = 0.0000Var = 1.6667σ = 1.2910

Derivation: Why the t-Distribution Appears

ThOrigin of the t-Statistic

If $X_1, \ldots, X_n \overset{\text{i.i.d.}}{\sim} N(\mu, \sigma^2)$ , then:

T = \frac{\bar{X} - \mu}{s/\sqrt{n}} \sim t_{n-1}

where $s^2 = \frac{1}{n-1}\sum(X_i - \bar{X})^2$ .

Proof Sketch

Step 1. Define $Z = \frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \sim N(0,1)$ by properties of the normal.

Step 2. By Fisher's lemma, $\bar{X}$ and $s^2$ are independent for normal samples. Moreover, $\frac{(n-1)s^2}{\sigma^2} \sim \chi^2_{n-1}$ .

Step 3. Therefore $T = \frac{Z}{\sqrt{\chi^2_{n-1}/(n-1)}}$ , which is the definition of $t_{n-1}$ .

The independence of $\bar{X}$ and $s^2$ is specific to the normal distribution — it fails for other distributions, which is why the t-test is not robust to non-normality for small $n$ .

Degrees of Freedom and Tail Behavior

t-Statistic

t = \frac{\bar{X} - \mu}{s/\sqrt{n}}, \quad \nu = n - 1

Here,

$\bar{X}$ =Sample mean
$\mu$ =Hypothesized population mean
$s$ =Sample standard deviation
$n$ =Sample size
$\nu = n-1$ =Degrees of freedom

Why Degrees of Freedom Matter

With $\nu$ degrees of freedom, the estimator $s^2$ uses $n-1$ independent pieces of information (one is lost estimating $\mu$ ). Fewer degrees of freedom means more uncertainty about $\sigma$ , hence heavier tails. The variance of $t_\nu$ is $\frac{\nu}{\nu-2}$ for $\nu > 2$ , which exceeds 1 (the normal variance) and decreases to 1 as $\nu \to \infty$ .

Critical Values

Common t-Critical Values

$\nu$	$t_{0.025}$ (95%)	$t_{0.005}$ (99%)	$z_{0.025}$ (normal)
5	2.571	4.032	1.960
10	2.228	3.169	1.960
29	2.045	2.756	1.960
100	1.984	2.626	1.960
$\infty$	1.960	2.576	1.960

As $\nu$ increases, t-critical values converge to z-critical values. The difference is substantial for small $\nu$ .

Worked Example

A biochemist measures enzyme reaction rates (in μmol/min) for $n = 16$ samples: $\bar{x} = 42.3$ , $s = 5.8$ . Test $H_0: \mu = 40$ vs $H_a: \mu \neq 40$ at $\alpha = 0.05$ .

Step 1. Compute the t-statistic:

t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} = \frac{42.3 - 40}{5.8/\sqrt{16}} = \frac{2.3}{1.45} = 1.586

Step 2. With $\nu = 15$ degrees of freedom, the critical values are $t_{0.025, 15} = 2.131$ .

Step 3. Since $|t| = 1.586 < 2.131$ , we fail to reject $H_0$ . The observed difference is not statistically significant at the 5% level.

Step 4. For comparison, if we had used the normal approximation: $z = 1.586$ with critical value $1.960$ . We would still fail to reject, but the normal approximation underestimates the tail probability. The exact p-value from $t_{15}$ is 0.133, while the normal approximation gives 0.113.

Small Sample Consequence

With $n = 16$ , the t-distribution is substantially wider than the normal. Using the normal approximation would underestimate the p-value by about 15% in this case. Always use the t-distribution when $\sigma$ is unknown and $n$ is small.

Convergence to Normal

ThAsymptotic Normality of t

As $\nu \to \infty$ , $t_\nu \xrightarrow{d} N(0,1)$ . More precisely, by Slutsky's theorem:

\frac{\bar{X} - \mu}{s/\sqrt{n}} = \frac{(\bar{X} - \mu)/(\sigma/\sqrt{n})}{s/\sigma} \xrightarrow{d} \frac{Z}{1} = Z

since $s \xrightarrow{p} \sigma$ by the law of large numbers.

Key Takeaways

Summary: t-Distribution

Used when $\sigma$ is unknown and estimated by $s$
$t = (\bar{X} - \mu)/(s/\sqrt{n}) \sim t_{n-1}$ for normal populations
Heavier tails than normal (more uncertainty); approaches normal as $\nu \to \infty$
Degrees of freedom: $\nu = n - 1$ (one lost estimating $\mu$ )
The variance of $t_\nu$ is $\nu/(\nu-2)$ for $\nu > 2$ , always $> 1$
Foundation for t-tests and t-intervals for the mean
Derived from the independence of $\bar{X}$ and $s^2$ under normality

t-Distribution — When σ is Unknown

t-Distribution — When σ is Unknown

The Real-World Workhorse for Means

Core Concepts

Dft-Distribution

PDF of t-Distribution

Interactive Visualization

Derivation: Why the t-Distribution Appears

ThOrigin of the t-Statistic

Degrees of Freedom and Tail Behavior

t-Statistic

Critical Values

Worked Example

Convergence to Normal

ThAsymptotic Normality of t

Key Takeaways

Summary: t-Distribution

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