Statistical
Confidence interval for a mean with known population standard deviation
Calculate a normal-theory confidence interval with known population SD: standard error sigma/SQRT(n), margin Z*SE, and lower/upper bounds.
From sample spread to 95% confidence bounds
For independent observations from a normal population with known population standard deviation sigma, the standard error of the sample mean is sigma/SQRT(n).
Multiply by the two-sided normal critical value, such as 1.96 for 95%, to obtain the margin. If SD is estimated from the sample, use an appropriate Student t critical value with n-1 degrees of freedom instead; substituting sample SD with Z is only an approximation.
| Confidence Level | Critical Z | SE (SD = 12, n = 36) | Margin of Error |
|---|---|---|---|
| 90% | 1.645 | 12 / 6 = 2.00 | ± 3.29 |
| 95% | 1.960 | 12 / 6 = 2.00 | ± 3.92 |
| 99% | 2.576 | 12 / 6 = 2.00 | ± 5.15 |
Margin of error and confidence interval bound formulas
The worked example uses a known population SD of 12, sample mean 80 and 36 independent observations.
=ROUND(D2 * (B2 / SQRT(C2)), 2)95% Lower and Upper Confidence Bounds
With Mean = 80, SD = 12, n = 36, and Z = 1.96: SE = 2, MoE = 3.92, spanning [76.08, 83.92].
=ROUND(A2 - D2 * (B2 / SQRT(C2)), 2)Upper confidence bound
Add the margin to the sample mean.
=ROUND(A2 + D2 * (B2 / SQRT(C2)), 2)Inputs and interpretation
A 95% procedure covers the fixed population mean in 95% of repeated samples under its assumptions; it does not assign a 95% probability to that fixed parameter after observing the interval. This is not a prediction interval for an individual observation or a proportion interval. Sampling bias and dependence are not corrected by increasing n. Require positive integer n and nonnegative known SD.
Try the example
Use the original values or change them to test your understanding.