Known Sigma Mean Difference Interval Calculator
Builds a two-sided interval for the difference between independent means when both population standard deviations are treated as known. The form displays (x̄1 − x̄2) ± z*√(σ1²/n1 + σ2²/n2) beside mean difference interval, using a worked condition that can be recalculated with the labeled inputs.
Enter the statistical summaries
Mean difference interval
Scope of the known sigma mean difference interval method
The known sigma mean difference interval page builds a two-sided interval for the difference between independent means when both population standard deviations are treated as known.
Mean difference interval is limited to the statistical quantity named by the result panel. The mean difference interval calculation does not silently add a population, time horizon, causal direction, or decision threshold that is absent from the fields.
Before entering the known sigma mean difference interval data
- Group 1 mean: The displayed group 1 mean is 52; retain its unit, group, and statistical role when reproducing mean difference interval.
- Known group 1 sigma: The displayed known group 1 sigma is 10 units; retain its unit, group, and statistical role when reproducing mean difference interval.
- Group 1 size: The displayed group 1 size is 100 observations; retain its unit, group, and statistical role when reproducing mean difference interval.
- Group 2 mean: The displayed group 2 mean is 48; retain its unit, group, and statistical role when reproducing mean difference interval.
- Known group 2 sigma: The displayed known group 2 sigma is 12 units; retain its unit, group, and statistical role when reproducing mean difference interval.
- Group 2 size: The displayed group 2 size is 120 observations; retain its unit, group, and statistical role when reproducing mean difference interval.
- Critical z value: The displayed critical z value is 1.96; retain its unit, group, and statistical role when reproducing mean difference interval.
The entries used for mean difference interval must refer to one coherent analysis condition. Combining incompatible populations, periods, or measurement definitions can produce valid mean difference interval arithmetic for a nonexistent study.
The arithmetic used for mean difference interval
For mean difference interval, match every symbol in the relationship to a labeled field before substituting numbers. Mean difference interval is reported in the scale implied by the inputs and formula.
While checking mean difference interval, inspect every denominator in (x̄1 − x̄2) ± z*√(σ1²/n1 + σ2²/n2). For mean difference interval, a zero or near-zero denominator can make mean difference interval undefined or unstable.
Checking the displayed example
The default mean difference interval condition is Group 1 mean = 52, Known group 1 sigma = 10 units, Group 1 size = 100 observations, Group 2 mean = 48, Known group 2 sigma = 12 units, Group 2 size = 120 observations, Critical z value = 1.96.
The example difference is 4.00 and the 95% interval is approximately 1.09 to 6.91.
The live calculator reports Estimate 4 · Lower bound 1.0928502 · Upper bound 6.9071498 · Margin 2.9071498 · Standard error 1.4832397. Repeating one intermediate step from (x̄1 − x̄2) ± z*√(σ1²/n1 + σ2²/n2) provides a fixed mean difference interval reference check for later code changes.
Statistical context for known sigma mean difference interval
Independence and credible known sigmas are part of the model; estimated standard deviations call for a t method.
For known sigma mean difference interval, the interval is produced by a repeated-sampling procedure; it is not the probability that a fixed parameter lies inside these particular endpoints.
A second check on mean difference interval
When interpreting known sigma mean difference interval, coverage depends on the stated standard-error model, critical value, independence conditions, and any approximation used by the method.
As a second check for mean difference interval, reversing the numerator and denominator answers a different question, so retain the direction printed in (x̄1 − x̄2) ± z*√(σ1²/n1 + σ2²/n2).
A nearby method may answer the next question: welch mean difference interval.
What to record with mean difference interval
Report mean difference interval using (x̄1 − x̄2) ± z*√(σ1²/n1 + σ2²/n2), followed by the entered values, units, exclusions, and analysis date. Name the mean difference interval population or dataset boundary instead of leaving it implicit.
Keep the full calculator output with the record, including Estimate 4 · Lower bound 1.0928502 · Upper bound 6.9071498 · Margin 2.9071498 · Standard error 1.4832397. A later mean difference interval review can then distinguish a changed input from a different convention or software implementation.
Questions about mean difference interval
How should mean difference interval be rounded?
Keep the unrounded mean difference interval for subsequent arithmetic, then report only the precision supported by the source measurements and the decision context. Extra digits in mean difference interval do not correct sampling or model error.
Which input deserves the closest boundary check?
For mean difference interval, start with critical z value and then group 1 mean. Confirm the mean difference interval units and allowed domain because a valid-looking entry can still describe the wrong statistical setup.
Why could another program report a different mean difference interval?
A different convention for rounding, tails, ties, interpolation, parameterization, or missing values can change mean difference interval. Compare the printed mean difference interval formula and its input definitions before treating either output as wrong.
What does mean difference interval represent on this page?
It is the quantity produced by (x̄1 − x̄2) ± z*√(σ1²/n1 + σ2²/n2) from the displayed group 1 mean, known group 1 sigma, group 1 size, group 2 mean, known group 2 sigma, group 2 size, critical z value. This page builds a two-sided interval for the difference between independent means when both population standard deviations are treated as known.