Confidence Interval to Standard Deviation Calculator
Convert a confidence interval into a standard deviation and standard error. Enter the interval bounds, sample size, and confidence level to get σ and SE with the steps shown.
Confidence Interval to Standard Deviation Calculator
Work out the standard error and standard deviation behind a reported confidence interval. Enter the interval bounds, the sample size, and the confidence level.
Step by step
- Interval width = -0.01 − -5.2 = 5.19
- Critical value for 95% confidence: Z = 1.96
- Standard error = width ÷ (2 × Z) = 5.19 ÷ (2 × 1.96) = 1.324
- Standard deviation = SE × √n = 1.324 × √78 = 11.6933
Formula
CI = estimate ± Z × (σ / √n), so σ = (upper − lower) ÷ (2 × Z) × √n
Documentation
This calculator converts a reported confidence interval back into a standard error and a standard deviation. It needs the two interval bounds, the sample size, and the confidence level. Papers often report a mean with its confidence interval but leave out the standard deviation, which meta-analysis and power calculations need.
What the calculator does
A confidence interval for a mean is normally built like this:
CI = estimate ± Z × (σ / √n)
The part after the ± sign is the margin of error. It equals the critical value Z times the standard error. Because the interval is symmetric, half its width is the margin of error. Reversing the formula recovers the standard error, and then the standard deviation.
Formula
width = upper bound − lower bound
SE = width ÷ (2 × Z)
σ = SE × √n
SE is the standard error of the mean. σ is the standard deviation. Z is the two-sided critical value of the standard normal distribution for the confidence level.
Critical values
| Confidence level | Z |
|---|---|
| 90% | 1.645 |
| 95% | 1.960 |
| 99% | 2.576 |
For any other level, Z is the value that leaves half the remaining probability in each tail. The calculator computes it for whatever level is entered.
Example
A study reports a mean difference of −2.6 with a 95% confidence interval of −5.2 to −0.01, based on 78 participants.
- Width = −0.01 − (−5.2) = 5.19
- Z for 95% confidence = 1.96
- SE = 5.19 ÷ (2 × 1.96) = 1.324
- σ = 1.324 × √78 = 1.324 × 8.832 = 11.69
The standard deviation is about 11.69.
When this method is approximate
The formula assumes the interval is symmetric around the estimate and was built from the normal distribution. Two situations break that assumption.
Small samples often use the t distribution instead of the normal one. The t critical value is larger, so using Z slightly underestimates the standard deviation. The gap shrinks as the sample grows and is small above roughly 30 observations.
Intervals for ratios, such as odds ratios or hazard ratios, are symmetric on the logarithmic scale, not the original scale. Those bounds must be converted with logarithms before this formula applies.
If the point estimate is not at the midpoint of the interval, the calculator flags the interval as asymmetric and the result should be treated as an estimate.
Frequently asked questions
What is the difference between standard error and standard deviation?
The standard deviation describes how spread out individual values are. The standard error describes how precisely the mean is known. They are linked by SE = σ / √n, so the standard error shrinks as the sample grows while the standard deviation does not.
Why divide by 2Z instead of Z?
The full width covers both sides of the estimate. One margin of error is half the width, so the half-width is divided by Z, which is the same as dividing the full width by 2Z.
Can this be used with a t-based interval?
Yes, but replace Z with the t critical value for n − 1 degrees of freedom. Using Z on a small sample gives a slightly low standard deviation.
What if only the margin of error is reported?
Double it to get the width, then follow the same steps.
Does the point estimate change the answer?
No. Only the width, the confidence level, and the sample size enter the formula. The estimate is used to check that the interval is symmetric.