How this calculation works
This tool uses a normal z interval, requiring a known population standard deviation and a suitable independent random sample. A 95% procedure covers the fixed population mean in 95% of repeated samples under its assumptions.
Interval = sample mean ± z × population standard deviation ÷ √nA worked example
Mean 100, known standard deviation 15 and n = 100 gives a 95% interval of approximately 97.060054 to 102.939946.
Good to know
Not a t interval when population standard deviation is unknown. Normality or a justified large-sample approximation is needed; dependence and biased sampling invalidate this simple interval.
The confidence level describes a repeated procedure
A 95% confidence procedure is designed to cover the fixed population mean in 95% of repeated samples when its assumptions hold. After one interval is calculated, it is not a statement that 95% of individual observations lie inside those endpoints.
Increasing the selected confidence level widens the interval because the critical value grows. Greater coverage comes with less numerical precision, even when the same data and sample size are used.
Known population spread is a strong requirement
This is a z interval with a supplied population standard deviation. When population spread is unknown and estimated from a small sample, a suitable t interval is usually a different calculation. Do not relabel a sample spread as known merely to fit the form.
Independence and a suitable sampling design matter. Normality or an appropriate large-sample justification is also needed. The endpoints cannot repair selection bias or missing portions of the target population.
A few questions, answered
Is this a prediction interval for a new observation?
No. It estimates a population mean, not the location of an individual future value.
Why does a larger sample narrow the interval?
With spread and confidence level fixed, the standard error decreases as 1/√n.