Field note · March 21, 2026
The interval is the result
A point estimate of 9.7%, the interval around it, and what a tighter one would cost.
A worked interval, on returned in retail-orders: 677 of 7,000 rows, so 9.7%.
The standard error on a proportion is sqrt(p(1-p)/n). Here that is 0.35 percentage points, so the 95% interval runs 9.0% to 10.4% — a width of 1.4 points.
import math
n, k = 7000, 677
p = k / n
se = math.sqrt(p * (1 - p) / n)
print(f"{p:.3%} [{p - 1.96*se:.3%}, {p + 1.96*se:.3%}]")That interval is tight, which is what 7,000 rows buys you. It is worth knowing why it is tight, so you recognise the cases where it is not.
Getting the interval down to ±0.5 points would need about 13,425 rows. Precision costs sample size quadratically — halving the width costs four times the data — which is the single most useful fact for anyone about to promise a more precise answer next week.
Normal approximation, and it starts lying at small counts or proportions near the boundaries; the Wilson interval behaves there. The pattern has both.