Normal vs Student’s t

Two symmetric bell-shaped curves can assign very different probabilities to an extreme result. Compare the event you care about, alongside the shape of the curves.

This experiment matches location and scale. The normal standard deviation equals its scale; Student’s t generally has a larger standard deviation, and for low degrees of freedom no finite variance.

Same location 0 and scale 1. Normal is solid; Student t is dashed. The shaded tails belong to t.

Probability outside [-2.000, 2.000], including tails beyond the drawing
Normal	Student t	t / normal
0.0455003	0.139326	3.062

Normal variance: 1.0000. Student t variance: 3.0000. Equal scale does not mean equal standard deviation.

Degrees of freedom 3.00 Distance from center, in scale units 2.00 Shared location 0.00 Shared scale 1.00

Both distributions extend over the whole real line. The displayed window covers only part of that support; the probability table uses the full distributions. Distances are measured in scale units, which are standard deviations only for the normal here.

Match the quantities you mean to match

Feature	Normal	Student’s t
Support	Whole real line	Whole real line
Center	Location equals mean and median	Location equals median; mean exists for degrees of freedom greater than 1
Spread	Variance is scale squared	Variance is scale squared times ν/(ν − 2), when ν > 2
Tails	Gaussian decay	Polynomial decay at any fixed finite ν
Shape control	Location and scale	Location, scale, and degrees of freedom ν

\operatorname{Var}(\mu+sT_\nu)=s^2\frac{\nu}{\nu-2},\qquad \nu>2

At ν = 3 and scale 1, the t variance is 3, compared with 1 for the normal. The larger tail area is therefore not a comparison at equal variance. Matching variances would require shrinking the t scale by the square root of (ν − 2)/ν, which only works when ν > 2.

Compare a tail event

At the default settings, the event is a value more than two scale units from the center. The normal probability is about 0.0455; for t with 3 degrees of freedom it is about 0.1393. The two sides of each distribution are both included.

Increase degrees of freedom to 30 and then 200. The t distribution approaches the normal. Then increase the tail distance: a close-looking center does not imply a small relative error in a rare-event probability.

Make a prediction

If you double the shared scale while keeping the distance at 2 scale units, do the displayed tail probabilities change?

Explore the answer

No. The cutoffs move twice as far from the center, so the standardized event stays the same. A fixed cutoff in physical units would answer a different question.

Why t appears in inference

For independent normal observations, subtracting the population mean from the sample mean and dividing by the estimated standard error gives a t statistic with n − 1 degrees of freedom:

\frac{\bar X-\mu}{S/\sqrt n}\sim t_{n-1}

Estimating the denominator adds uncertainty. Using a t critical value accounts for that uncertainty under this sampling model. It does not make a small sample from an arbitrary population obey the t distribution exactly. Explore confidence intervals to see the difference between a nominal level and repeated-sampling coverage.

When a model change is justified

A Student t observation model can accommodate more extreme values than a normal model at the same scale. That is a modelling choice to check against the measurement process and data. It does not follow solely from having a small sample. Conversely, the normal approximation is not automatically adequate once ν passes one universal threshold: the event and acceptable error matter.

Read the individual normal and Student t lessons, or use the distribution chooser.

Reference

NIST’s t distribution reference defines the location/scale transformation, moments, and normal limit. The comparison computes full two-sided tail probabilities numerically.

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