Key takeaway: The t-distribution is a symmetric, bell-shaped family of curves with heavier tails than the normal curve, used for sample means when the population standard deviation must be estimated from the data.
Grade 12 – AP StatisticsAP Statistics Unit 4Statistics and Data
Definition of t-Distribution
The t-distribution corrects for an extra source of uncertainty. When the population spread is unknown, the sample standard deviation replaces it, and that estimate itself varies, especially in small samples. The result is a curve that is still centered at 0 but has more area in the tails.
Each curve is labeled by its degrees of freedom. With few degrees of freedom the tails are fat; as the count grows, the curve approaches the standard normal curve. Critical values \(t^{*}\) are therefore larger than z-values for the same confidence level.
Formula and notation
The standardized sample mean follows a t curve with \(n-1\) degrees of freedom:
\[ t=\frac{\,\overline{x}-\mu}{s/\sqrt{n}},\qquad \,\overline{x}\pm t^{*}\frac{s}{\sqrt{n}}. \]
The second expression is the confidence interval for a mean.
Key facts
- Its center is 0 and its shape is symmetric, like the normal curve.
- Heavier tails mean larger critical values than z for small samples.
- At df = 11 and 95% confidence, t* is 2.201 compared with z* of 1.96.
- Beyond about 30 degrees of freedom the curves are nearly indistinguishable.
t-Distribution example
A sample of 12 students has a mean quiz time of 74.5 seconds with standard deviation 8.4 seconds. Build a 95% interval for the true mean time.
-
Degrees of freedom: \(12-1=11\), so the table gives \(t^{*}=2.201\).
-
Standard error: \(\frac{8.4}{\sqrt{12}}\approx 2.425\).
-
Margin: \(2.201\times 2.425\approx 5.34\).
-
Interval: \(74.5\pm 5.34\), or \((69.16,\ 79.84)\).
We are 95% confident that the true mean is between 69.16 and 79.84 seconds.
Common mistake
Watch out
A student uses \(z^{*}=1.96\) for the same sample and gets a margin of \(1.96\times 2.425\approx 4.75\).
Why it is wrong
With only 12 observations and an estimated standard deviation, the normal curve understates the uncertainty. The t-distribution with 11 degrees of freedom needs 2.201, which widens the margin to 5.34. The narrower interval would claim more precision than the data support.
Related terms and lessons
Browse the glossary: A–Z list · AP Statistics vocabulary · Statistics and Data