English
العربية
Deutsch
Español
Français
Italiano
Português
Русский
Türkçe
Tiếng Việt
Confidence Interval Calculator
Find the margin of error around a sample mean or proportion.
A survey says 62% of customers are satisfied. On its own that number is almost meaningless, because it came from a sample rather than everybody. The confidence interval is what tells you how much to trust it.
Calculating an interval
- Enter your sample size.
- Enter the sample mean or the observed proportion.
- Enter the standard deviation if you are working with a mean.
- Choose a confidence level — 95% is the near-universal default.
- Read the margin of error and the resulting interval.
What a 95% confidence interval actually means
It does not mean there is a 95% chance the true value sits inside your particular interval. The true value is fixed; it is your sample that is random.
What it means is this: if you repeated the whole exercise many times, drawing a fresh sample each time and computing an interval each time, about 95% of those intervals would contain the true value. It is a statement about the reliability of the method, not about the one number in front of you. This distinction gets flattened constantly in reporting, and it is worth holding onto.
Sample size and margin of error
| Sample size | Margin of error at 95% | Comment |
|---|---|---|
| 100 | ±9.8% | Indicative only |
| 400 | ±4.9% | Typical small survey |
| 1,000 | ±3.1% | Standard opinion poll |
| 2,000 | ±2.2% | Diminishing returns begin |
| 10,000 | ±1.0% | Rarely worth the cost |
Why bigger samples help less than you would hope
Margin of error shrinks with the square root of the sample size, not with the sample size itself. To halve your margin of error you need four times as many responses.
That is why opinion polls settle around a thousand people. Going from 1,000 to 2,000 buys you less than a percentage point of precision at double the cost, and long before that point the real threat to accuracy is sampling bias rather than sample size. A biased sample of 100,000 is worse than a representative sample of 500.
Confidence interval questions
What does a 95% confidence level mean?
That the method produces intervals containing the true value about 95% of the time across repeated sampling. It is a property of the procedure rather than a probability attached to your single result.
How do I choose a confidence level?
95% is the convention in most fields. Use 99% when being wrong is costly, such as in medical research, accepting a wider interval in exchange. Use 90% for exploratory work where a narrower interval is more useful than extra certainty.
What sample size do I need?
For a margin of error around 3% at 95% confidence you need roughly 1,000 responses. For 5%, about 400. Quadrupling the sample only halves the margin, so there is a practical ceiling.
What is the difference between margin of error and standard deviation?
Standard deviation describes how spread out the individual values are. Margin of error describes how uncertain your estimate of the average is. Larger samples shrink the margin of error but leave the standard deviation roughly where it was.
Does a confidence interval account for bad sampling?
No, and this is the most important caveat. It assumes your sample was drawn randomly and representatively. If the sampling method is biased, the interval is calculated correctly around the wrong number.
