Student's t-Distribution

The setup — and why we need t at all

Suppose X₁, X₂, …, Xₙ are i.i.d. N(μ, σ²). The standardized sample mean is:

Z = (X̄ − μ) / (σ/√n)   ~ N(0, 1)

Beautiful — but only if you know σ. You almost never do. Replace σ with the sample standard deviation s, and the standardized statistic becomes:

T = (X̄ − μ) / (s/√n)

This is no longer Normal. The denominator s is itself a random variable. Its uncertainty thickens the tails of T. Gosset (1908) worked out exactly what distribution T follows: t(n − 1).

Construction from Normal and chi-squared

Let Z ~ N(0, 1) and V ~ χ²(ν) be independent. Then:

T = Z / √(V/ν)   ~ t(ν)

Why? In the sample-mean setup:

Numerator:   (X̄ − μ)/(σ/√n) ~ N(0, 1) = Z
Denominator: s/σ where (n−1)s²/σ² ~ χ²(n−1)

T = (X̄ − μ)/(s/√n) = Z / (s/σ) = Z / √(χ²(n−1)/(n−1)) = t(n−1).

So the t-distribution literally captures the joint behavior of "true Normal numerator" and "wobbly sample-variance denominator."

Density and moments

The pdf of t(ν):

f(t; ν) = Γ((ν+1)/2) / [√(νπ) · Γ(ν/2)] · (1 + t²/ν)^(-(ν+1)/2)

Quantity	Formula	ν=3	ν=10	ν=30
Mean	0 (for ν > 1)	0	0	0
Variance	ν/(ν − 2) (for ν > 2)	3	1.25	1.07
Skewness	0 (symmetric)	0	0	0
Excess kurtosis	6/(ν − 4)	—	1.0	0.23
97.5% critical (2-sided)	—	3.18	2.23	2.04

The 2.5% critical values for two-tailed tests show how t differs from Normal (1.96). At df = 30, you're paying only a few percent in interval width relative to assuming Normal.

t → Normal as ν → ∞

Plotting t(ν) for ν = 1, 2, 5, 10, 30, ∞ shows the curves stack from heavy-tailed Cauchy (ν=1) to standard Normal (ν=∞). Specific comparisons:

ν	Variance	97.5% critical	Heavier-tailed than Normal by
1 (Cauchy)	∞	12.71	6.5× wider
2	∞	4.30	2.2× wider
5	1.67	2.57	31% wider
10	1.25	2.23	14% wider
30	1.07	2.04	4% wider
100	1.02	1.98	1% wider
∞ (Normal)	1.00	1.96	—

For df ≤ 5 the difference is substantial; from df = 30 onward you might as well use Normal.

The t-test — by example

Suppose a battery manufacturer claims a mean lifetime of μ₀ = 1000 hours. You test n = 10 batteries and get sample mean X̄ = 950 hours, sample SD s = 50 hours. Is the claim plausible?

Null: μ = 1000
Test statistic: t = (X̄ − μ₀)/(s/√n) = (950 − 1000)/(50/√10) = −50/15.81 ≈ −3.16

Under null, t ~ t(9). Critical value at α = 0.05 two-tailed: ±2.26.

|t| = 3.16 > 2.26 → reject the null.

Conclusion: at α = 0.05, the data are inconsistent with mean lifetime 1000 hours.

Note: had we (mistakenly) used Normal's 1.96 critical value, we'd still reject — but with the wrong α level. The t-test gets the size of the test right.

t vs Normal vs Cauchy

Distribution	Tails	Mean	Variance	Use when
Normal N(0,1)	e^{−x²/2} (very light)	0	1	Large samples, known σ
t(30)	~Normal (slightly heavier)	0	30/28	n > 30, σ unknown
t(5)	polynomial decay, ν=5	0	5/3	n ≈ 6–10, σ unknown
t(2)	polynomial, infinite var	0	∞	Heavy-tailed test stat, robust inference
t(1) = Cauchy	1/(π(1+x²)), mean undefined	—	∞	Pathological / sub-CLT models
Laplace	e^{−|x|} (medium-heavy)	0	2	L1 regression, robust estimators

t(ν) is a continuous interpolation between Cauchy (ν=1, all-heavy) and Normal (ν=∞, all-light), with degrees of freedom literally tuning the tail weight.

Confidence intervals — wider for small n

95% confidence interval for μ with unknown σ:

X̄ ± t_{0.975}(n−1) · s/√n

For n = 10, the multiplier is 2.26 (vs Normal's 1.96 — interval is 15% wider). For n = 30, it's 2.05 (4.5% wider). For n = 100, it's 1.98 (1% wider). Small samples pay a real width penalty for not knowing σ.

Where the t-distribution shows up

• One-sample t-test. Test whether a sample mean differs from a hypothesized value when σ is unknown.
• Paired t-test. Test mean of paired differences (before/after, treatment/control with matched subjects).
• Two-sample t-test. Test difference of means in two groups; pooled (equal variances) or Welch (unequal).
• Regression coefficient inference. Each estimated β̂ⱼ has t-distributed standardized statistic; standard p-values in linear regression come from t-tests.
• Confidence intervals for means. Whenever the population variance is unknown — which is almost always in practice.
• Bayesian inference with vague priors. Posterior of a Normal mean under improper σ-prior is t-distributed — appears throughout objective Bayesian regression.
• Robust statistics. Heavy-tailed t-distributions (low ν) model outlier-prone data; t-distributed errors in regression resist outliers better than Gaussian errors.
• A/B testing. Welch's t-test is the standard significance test for mean differences between two groups.

Common pitfalls

• Using Z when you should use t. If σ is unknown (almost always), use t. The Normal approximation gets the type-I error rate wrong for small samples.
• Forgetting df = n − 1. One degree of freedom is lost estimating the mean. For paired t-tests, df = n − 1 where n is the number of pairs; for two-sample pooled t-tests, df = n₁ + n₂ − 2.
• Assuming normality of the underlying data. The t-test is robust to mild deviations from normality, especially for n ≥ 30. For severe non-normality, use Wilcoxon signed-rank or other non-parametric alternatives.
• Welch vs pooled t-test confusion. Use Welch's t-test (separate variances) when sample variances differ noticeably; pooled t-test assumes equal population variances and is biased otherwise.
• One-tailed vs two-tailed. Two-tailed is the default; use one-tailed only when you have a strong directional hypothesis pre-registered.
• Calling the result "significant" without effect size. A statistically significant t doesn't tell you whether the effect matters. Report the mean difference, confidence interval, and Cohen's d alongside the p-value.

The Guinness story

William Sealy Gosset (1876–1937) was head experimental brewer at Guinness in Dublin. Quality control on small batches of barley and hops required statistical inference with very few samples — n = 5 or 10, not n = 1000. Existing Normal-based methods overstated precision; Gosset noticed his confidence intervals were too narrow. Working with R. A. Fisher and Karl Pearson by correspondence, he derived the exact distribution of (X̄ − μ)/(s/√n) for small n. Guinness, fearing competitors would learn its statistical methods, forbade publishing under his real name. Gosset published as "Student" in Biometrika, 1908. The pseudonym stuck. The Guinness brewery's prohibition on Gosset's name is the reason an entire branch of inference is named after a nameless student.