Confidence Intervals

A single sample mean is just an estimate — a confidence interval gives a range that's likely to contain the true population value, instead of one specific number pretending to be exact.

import math

def margin_of_error(std, n, z=1.96):
    return z * (std / math.sqrt(n))

z=1.96 corresponds to a 95% confidence level for a normal distribution — it's the number of standard errors on each side of the mean that captures the middle 95% of the distribution.

The correct interpretation of "95% confidence" is about the procedure, not this one specific interval: if you repeated the whole sampling process many times and built a new interval each time, about 95% of those intervals would contain the true population value. Once you have one actual interval in front of you, the true value either is or isn't in it — there's no probability left to talk about for that specific interval.

A wider interval (bigger z, like 2.58 for 99% confidence) trades precision for more confidence that you've captured the true value — you can always be "more confident" by being willing to say less precisely where the value is.