A fictional neighborhood café, which we will call the Corner Table, is thinking of opening on Sundays. The owner wants to ask regular customers whether they would come in. Her question is simple: how many people do I need to ask? The answer depends on three choices, and none of them is "as many as possible".
Put those together and the standard formula for a large population is:
n = z² x p x (1 - p) / e²
With z = 1.96, p = 0.5 and e = 0.05: n = 3.8416 x 0.25 / 0.0025 = 384.16, which rounds up to 385. That is where the famous number 385 comes from. It is the sample you need for plus or minus 5 points at 95% confidence when the population is large and you know nothing about the likely answer.
The relationship between sample size and margin of error is not linear, and that is the single most useful thing to know. Halving the margin requires four times the sample, because error shrinks with the square root of n.
All at 95% confidence and p = 50%. Going from plus or minus 5 to plus or minus 3 costs you almost three times as many responses. Going to plus or minus 1 costs twenty-five times as many as plus or minus 5. This is why most national polls stop around 1,000 respondents: a margin near 3 points is the sweet spot between cost and usefulness. At 99% confidence the same targets need more: 664 for 5 points and 1,844 for 3 points.
The Corner Table has maybe 2,000 regular customers, not millions. For a finite population, you can apply a correction: n adjusted = n0 / (1 + (n0 - 1) / N). Starting from 384.16, a population of 2,000 gives about 323 responses. A population of 500 needs only 218. A population of 10,000 needs 370, barely below the large-population figure.
That shortcut matters for small groups such as a classroom, a club, or a single team inside a company. If you can reach most of a population of 500, you can get a good answer from fewer than half the people. The calculator on this site, the sample size calculator, includes the population field for exactly this reason.
Say the owner collects 323 responses and 58% say they would visit on Sundays. With a margin of about 5 points, the plausible range is roughly 53% to 63%. The confidence interval calculator has a proportion tab that gives the range for your own result and sample. Notice that the whole range is above half. If the result had been 52%, the range would run from 47% to 57%, and the honest summary would be "we cannot tell if it is more or less than half".
Subgroups are weaker than the whole. If only 120 of 1,000 respondents are students, the margin for the student answer is about plus or minus 9 points, not 3. Cutting the data into small slices is how people find "interesting differences" that are just noise.
Everything above assumes the people you ask are a fair random sample of the group you care about. If the owner hands out surveys only at the Saturday counter, she hears from Saturday regulars, who may be the keenest on weekend hours. If 20% of those asked reply, the 80% who stay silent may think differently. This is selection and non-response bias, and it does not shrink as the sample grows. A poll of 10,000 self-selected volunteers can be much worse than a careful 400.
Some practical ways to reduce bias without a big budget:
If your survey measures a number instead of a yes/no answer, such as spend per visit, the sample size depends on the standard deviation of that number, which you can estimate from a small pilot with the standard deviation calculator. For most everyday surveys, though, the three dials and the 385 rule will take you most of the way.
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