Transparent method
The formula
Worked example
Example inputs
How to interpret the result
A 95% confidence interval does not mean there's a 95% chance the true population mean falls inside this particular interval: the true mean either is or isn't in there, with no probability left to assign once the sample is drawn. What 95% actually describes is the method: run the same sampling process many times, and about 95% of the intervals it produces would contain the true mean.
Before you rely on it
What to check
State the interpretation carefully as being about the long-run reliability of the method, not the probability attached to this one specific interval.
The common error
Where people go wrong with confidence interval calculator
Reading a 95% interval as 95% likely to contain the true value. That statement describes a different, more casual idea than what the confidence level is actually a guarantee about.
Sensitivity evidence
How sample mean changes the lower bound
| Sample mean | Lower bound | Upper bound |
|---|---|---|
| 80 | 74.6 | 85.4 |
| 90 | 84.6 | 95.4 |
| 100worked example | 94.6 | 105.4 |
| 110 | 104.6 | 115.4 |
| 120 | 114.6 | 125.4 |
Every input, tested
Which input moves the lower bound most
| Input | Tested from | To | Lower bound at each end | Swing |
|---|---|---|---|---|
| Sample mean | 90 | 110 | 84.6 to 104.6 | 20 (21%) |
| Confidence level | 90% | 99% | 95.5 to 92.9 | 2.5 (2.7%) |
| Standard deviation | 14 | 17 | 95 to 93.9 | 1.1 (1.1%) |
| Sample size | 27 | 33 | 94.3 to 94.9 | 0.54 (0.6%) |
Two variables at once
Lower bound by sample mean and standard deviation
| Sample mean \ Standard deviation | 12 | 15 | 18 |
|---|---|---|---|
| 80 | 75.7 | 74.6 | 73.6 |
| 90 | 85.7 | 84.6 | 83.6 |
| 100 | 95.7 | 94.6 | 93.6 |
| 110 | 105.7 | 104.6 | 103.6 |
| 120 | 115.7 | 114.6 | 113.6 |
Step by step
The worked example, input by input
| Input | Value used | What it means |
|---|---|---|
| Sample mean | 100 | |
| Standard deviation | 15 | |
| Sample size | 30 | |
| Confidence level | 95% | How confident the interval should be. |
| Lower bound | 94.6 | |
| Upper bound | 105.4 | |
Inputs, definitions and assumptions
Sample mean
Standard deviation
Sample size
Confidence level
How confident the interval should be.
Where the inputs come from
Enter the sample mean, the standard deviation, the sample size and the z value for your confidence level: 1.645 for 90%, 1.96 for 95% and 2.576 for 99%. The margin of error is z times the standard deviation divided by the square root of the sample size. Use the population standard deviation if known, or a large enough sample to use the sample deviation. Samples should be random and representative.
When the result stops applying
For small samples with an unknown standard deviation, the t distribution is more appropriate than z, and the interval would be wider. The method assumes random sampling and roughly normal means. Bias in the sample is not fixed by a larger sample. Confidence is about the procedure, not the probability that this particular interval is right.
What to compare it with
Plan how many observations you need on the sample size calculator, find the standard deviation on its page, and test claims on the z-score and p-value calculators. The mean and median page gives the sample mean, and the normal probability calculator handles areas under the curve.
People also ask
Frequently asked questions
What does a 95% confidence interval mean?
If you repeated the sampling many times, about 95% of the intervals built this way would contain the true population value.
How can I make an interval narrower?
Increase the sample size, reduce variability in the measurements or accept a lower confidence level, which trades certainty for precision.
When should I use the t distribution?
For small samples when the population standard deviation is unknown, since the t distribution accounts for the extra uncertainty.