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Student's t-Distribution — Definition, Formula & Examples

Student's t-distribution is a bell-shaped probability distribution used in place of the normal distribution when the sample size is small and the population standard deviation is unknown. It has heavier tails than the normal distribution, meaning extreme values are more likely.

If ZZ is a standard normal random variable and VV is an independent chi-squared random variable with ν\nu degrees of freedom, then the random variable T=ZV/νT = \frac{Z}{\sqrt{V/\nu}} follows a t-distribution with ν\nu degrees of freedom. As ν→∞\nu \to \infty, the t-distribution converges to the standard normal distribution N(0,1)N(0,1).

Key Formula

t=xˉ−μ0s / nt = \frac{\bar{x} - \mu_0}{s \,/\, \sqrt{n}}
Where:
  • xˉ\bar{x} = Sample mean
  • μ0\mu_0 = Hypothesized population mean
  • ss = Sample standard deviation
  • nn = Sample size

How It Works

You use the t-distribution whenever you estimate a population mean from a sample but do not know the population standard deviation σ\sigma. Instead of σ\sigma, you plug in the sample standard deviation ss, which introduces extra uncertainty — the t-distribution accounts for that extra uncertainty through its heavier tails. The distribution's exact shape depends on a single parameter called degrees of freedom, typically ν=n−1\nu = n - 1 for a one-sample test where nn is the sample size. With fewer degrees of freedom the tails are fatter, so critical values are larger and confidence intervals are wider. Once nn exceeds about 30, the t-distribution is nearly identical to the standard normal.

Worked Example

Problem: A sample of 16 light bulbs has a mean lifetime of 1200 hours with a sample standard deviation of 80 hours. Test whether the population mean differs from 1150 hours at the 0.05 significance level.
Step 1: Set up hypotheses: Null hypothesis H0H_0: μ=1150\mu = 1150. Alternative hypothesis HaH_a: μ≠1150\mu \neq 1150 (two-tailed test).
Step 2: Compute the t-statistic: Substitute the given values into the formula.
t=1200−115080/16=5020=2.50t = \frac{1200 - 1150}{80 / \sqrt{16}} = \frac{50}{20} = 2.50
Step 3: Find the critical value: Degrees of freedom ν=16−1=15\nu = 16 - 1 = 15. For a two-tailed test at α=0.05\alpha = 0.05, the critical values from a t-table are ± 2.131\pm\,2.131.
Step 4: Make a decision: Since ∣t∣=2.50>2.131|t| = 2.50 > 2.131, the test statistic falls in the rejection region. Reject H0H_0.
Answer: At the 0.05 significance level, there is sufficient evidence to conclude that the population mean lifetime differs from 1150 hours (t=2.50t = 2.50, df=15df = 15).

Another Example

Problem: Construct a 95% confidence interval for the population mean if a sample of 9 measurements gives xˉ=50\bar{x} = 50 and s=6s = 6.
Step 1: Identify degrees of freedom and critical value: With n=9n = 9, degrees of freedom ν=8\nu = 8. The two-tailed t∗t^* value for 95% confidence is 2.3062.306.
Step 2: Compute the margin of error: Margin of error =t∗⋅sn= t^* \cdot \dfrac{s}{\sqrt{n}}.
E=2.306×69=2.306×2=4.612E = 2.306 \times \frac{6}{\sqrt{9}} = 2.306 \times 2 = 4.612
Step 3: Build the interval: Add and subtract the margin of error from the sample mean.
50±4.612  ⇒  (45.39,  54.61)50 \pm 4.612 \;\Rightarrow\; (45.39,\; 54.61)
Answer: The 95% confidence interval for the population mean is approximately (45.39,  54.61)(45.39,\; 54.61).

Visualization

Why It Matters

The t-distribution is central to introductory statistics courses (AP Statistics, college-level intro stats) because real-world data rarely comes with a known population standard deviation. It underpins one-sample and two-sample t-tests, paired t-tests, and the construction of confidence intervals — tools used daily in fields like clinical trials, quality control, and social science research.

Common Mistakes

Mistake: Using the standard normal (z) critical values when the population standard deviation is unknown and the sample is small.
Correction: When σ\sigma is unknown and you use ss in its place, the test statistic follows a t-distribution, not a z-distribution. The t critical values are larger, reflecting the added uncertainty from estimating σ\sigma.
Mistake: Setting degrees of freedom equal to nn instead of n−1n - 1 in a one-sample t-test.
Correction: For a one-sample t-test, degrees of freedom are ν=n−1\nu = n - 1 because one degree of freedom is lost when estimating the mean from the sample.

Related Terms