Definition: The t-distribution is a bell-shaped model with heavier tails than the normal, used for inference about a mean when the population standard deviation is unknown.
What is the t-distribution?
When you estimate the standard deviation with s instead of knowing sigma, extra uncertainty enters. The t-distribution accounts for it. It is centered at 0, symmetric, but with heavier tails than the normal curve. Its shape depends on degrees of freedom, df = n – 1 for a one-sample mean. As df grows, it approaches the normal curve.
How do you use it?
The interval for a mean is x-bar ± t* times s/square root of n.
Worked example: n = 16, x-bar = 52.3, s = 8. Then df = 15 and t* = 2.131 for 95% confidence. The standard error is 8/4 = 2, so the margin is 2.131 times 2 = 4.262. The interval is (48.04, 56.56).
A z-based interval would use 1.96 and come out too narrow.
What mistake is common?
Using the wrong degrees of freedom or using z* with a small sample. Mathbot reminds you to subtract 1 from n and to read t* from the correct row.