How it works

A useful distinction

More confidence produces wider intervals

Using the same sample, raising confidence from 95% to 99% increases the critical value and widens the interval. A narrower interval is not automatically better if it was obtained simply by accepting lower coverage.

Interpreting a mean confidence interval

With a sample mean of 100, known population standard deviation of 15, and sample size of 100, the standard error is 15 ÷ √100 = 1.5. A 95% normal-z interval is approximately 100 ± 1.96 × 1.5, or 97.06 to 102.94. If the standard deviation is estimated from the sample, select Student t instead. The interval estimates the population mean; it is not a range containing 95% of individual observations. Its coverage relies on the sampling and distribution assumptions.

Mean confidence interval: formula and assumptions

Interval = mean ± critical value × standard deviation / √n.

Use Student t when the population standard deviation is estimated from a sample. Normality or sufficient sample size is required.

Worked example

Using the Mean confidence interval example:

Inputs

Sample mean: 50
Standard deviation: 10
Sample size: 25
Confidence level (%): 95
Method: Student t (sample standard deviation)

Method

Interval = mean ± critical value × standard deviation / √n.

Result

Example result

Lower bound: 45.87

Upper bound: 54.13

Margin of error: 4.13

Critical value: 2.06

These are illustrative inputs. Change the values to explore a different scenario.

Formula references

References explain the underlying methods; they do not imply endorsement or individual professional review of Finflo.

Common questions

When should I use Student t instead of normal z?

Use Student t when the standard deviation is estimated from the sample; the z method assumes a known population standard deviation.

Is the interval a range containing most individual observations?

No. It describes uncertainty in the estimated mean under the model, rather than a prediction range for individual values.

Prepared by Finflo. See our methodology and editorial policy.

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