Confidence Interval Calculator: Reading the Margin of Error

A single percentage from a sample is a guess with a known amount of slack in it. The confidence interval is how you report the slack, and the size of it usually matters more than the headline figure.

What the interval is telling you

Take that poll of 400 people where 52% said yes. The standard error is the square root of (0.52 x 0.48 / 400), which is 0.0250. Multiply by 1.96 for 95% confidence and the margin of error is 0.049, or 4.9 percentage points. The interval runs 47.1% to 56.9%. Since 50% sits inside it, anyone reporting that result as a lead is reading more into it than the data supports.

The 95% describes the method, not this particular interval. If you repeated the whole survey many times and built an interval each time, about 95 in every 100 of those intervals would contain the true population value. This one either does or does not.

Proportions and means use different arithmetic

For an average rather than a percentage, the calculator needs the sample mean and the standard deviation. Time 30 deliveries, get a mean of 42 minutes with a standard deviation of 9 minutes, and the standard error is 9 divided by the square root of 30, which is 1.64. With 29 degrees of freedom the t value at 95% is 2.045, giving a margin of 3.36 minutes and an interval from 38.6 to 45.4 minutes. If pulling a standard deviation out of raw data is the awkward part, our standard deviation walkthrough covers it.

Using the confidence interval calculator

First choose proportion or mean, because that decides which formula runs. For a proportion you need the sample size and the proportion itself, entered as either 52 or 0.52 depending on the field label. For a mean you need sample size, sample mean and standard deviation. Confidence level is a dropdown, normally 90, 95 or 99.

Sample size means the number of usable responses you actually received, not the number of people you contacted. Mixing those up is the most common input error, and it always makes your interval look better than it is. Descriptive statistics from your dataset, including the mean, are covered in our mean, median and mode guide.

Choosing 90, 95 or 99

Higher confidence means a wider interval. The same 400-person poll at 99% uses 2.576 instead of 1.96, giving a margin of 6.4 points and a range of 45.6% to 58.4%. You bought more certainty and paid for it with a vaguer answer. Use 95% unless you have a reason not to, and say which you used.

Where intervals get misread

Doubling the sample does not halve the margin. You have to quadruple it. Going from 400 to 1,600 respondents takes that 4.9 point margin down to 2.4 points, which is why national polls cluster around 1,000 to 1,200 people and stop there. The next improvement costs four times as much for half the gain.

The interval also only covers sampling error. It says nothing about a leading question, people who refused to answer, or a sample drawn from the wrong group. A biased survey of 10,000 has a tiny margin of error and a large wrong answer. Finally, two intervals that overlap slightly do not prove there is no difference between the groups. That question needs a test of its own.

Common questions

What sample size do I need for a 3 point margin? About 1,067 at 95% confidence for a proportion near 50%, which is the worst case. Results nearer 10% or 90% need fewer people for the same margin.

Is the margin of error the same as the confidence interval? The margin is half the width. The interval is the estimate plus and minus that margin, so 52% with a 4.9 point margin gives an interval of 47.1% to 56.9%.

Can I use this on a sample of 12? You can, using the t distribution, but the interval will be wide and the result leans on the underlying data being roughly normal. Treat anything under about 30 as indicative.

Does a 95% interval mean a 95% chance the true value is inside it? Not under the standard definition. The true value is fixed, and the interval is the thing that varies from sample to sample. The 95% belongs to the procedure.