Confidence intervals

A sample mean is a guess. A confidence interval adds an honest margin of error.

For a large enough sample, a 95% confidence interval for the mean is:

mean ± 1.96 × s ÷ √n

where s is the sample standard deviation and s ÷ √n the standard error. (1.96 comes from the normal curve: 95% of it lies within 1.96 SD of the centre.)

from math import sqrt
import statistics as st

heights = [162, 170, 168, 175, 159, 181, 166, 172, 169, 177,
           164, 171, 174, 160, 168, 173, 167, 179, 165, 170,
           176, 163, 169, 172, 171]
n = len(heights)
mean = st.mean(heights)
se = st.stdev(heights) / sqrt(n)
print("n =", n, " mean =", round(mean, 1), " standard error =", round(se, 2))
print("95% CI:", round(mean - 1.96 * se, 1), "to", round(mean + 1.96 * se, 1))

What "95%" means

If you repeated the whole study many times, about 95% of the intervals built this way would contain the true mean. It does not mean "the true mean is in this interval with probability 95%": the true mean is fixed; it is the interval that varies.

Narrower intervals

The margin shrinks with √n: four times the data halves the margin. Asking for 99% confidence instead widens it (2.58 instead of 1.96).

For small samples (n below about 30), statisticians replace 1.96 with a slightly larger number from the t distribution.

Mark lesson complete