Student's t
Estimating SD adds uncertainty
A t score divides by a sample SD. In a small sample, that SD can underestimate the population spread and inflate the score. The heavier tails account for this extra uncertainty.
Compare five observations (4 degrees of freedom) with 101 observations (100 degrees of freedom). Watch the t curve approach the normal curve.
Standard normal & Student's t densities
Both curves describe possible standardised scores. At small degrees of freedom, the t curve assigns more probability to extreme scores. As the degrees of freedom increase, it approaches the standard normal curve.
Worked comparison
Follow the calculation
What the symbols mean
x̄ is the sample mean and μ₀ is the null mean. s is the sample standard deviation and n is the sample size. The one-sample t statistic has n − 1 degrees of freedom.
Assumptions and full calculation
Both curves are probability densities centred at zero. The t curve gives the distribution of the one-sample t statistic under the null for independent observations from a normal population. Both distributions extend beyond the plotted range. With one degree of freedom, t has no defined mean or finite variance. With two degrees of freedom, its variance is still not finite.