Variance Confidence Interval Calculator
Applies chi-square quantiles to estimate a normal population variance. The form displays ((n−1)s²/χ²upper, (n−1)s²/χ²lower) beside variance confidence interval, using a worked condition that can be recalculated with the labeled inputs.
Enter the statistical summaries during an independent check
Variance confidence interval
The question behind variance confidence interval
The variance confidence interval page applies chi-square quantiles to estimate a normal population variance.
Variance confidence interval is limited to the statistical quantity named by the result panel. The variance confidence interval calculation does not silently add a population, time horizon, causal direction, or decision threshold that is absent from the fields.
Reading the variance confidence interval fields
- Sample variance: For variance confidence interval, the worked value for sample variance is 25 squared units. Treat the sample variance entry (25 squared units) explicitly as a count, proportion, rate, estimate, or model parameter before comparing variance confidence interval conditions. The form enforces minimum 0.
- Sample size: For variance confidence interval, the worked value for sample size is 20 observations. Treat the sample size entry (20 observations) explicitly as a count, proportion, rate, estimate, or model parameter before comparing variance confidence interval conditions. The form enforces minimum 2.
- Lower-tail chi-square quantile: For variance confidence interval, the worked value for lower-tail chi-square quantile is 8.907. Treat the lower-tail chi-square quantile entry (8.907) explicitly as a count, proportion, rate, estimate, or model parameter before comparing variance confidence interval conditions. The form enforces minimum 1e-06.
- Upper-tail chi-square quantile: For variance confidence interval, the worked value for upper-tail chi-square quantile is 32.852. Treat the upper-tail chi-square quantile entry (32.852) explicitly as a count, proportion, rate, estimate, or model parameter before comparing variance confidence interval conditions. The form enforces minimum 1e-06.
The entries used for variance confidence interval must refer to one coherent analysis condition. Combining incompatible populations, periods, or measurement definitions can produce valid variance confidence interval arithmetic for a nonexistent study.
The arithmetic used for variance confidence interval
For variance confidence interval, match every symbol in the relationship to a labeled field before substituting numbers. Variance confidence interval is reported in squared units.
While checking variance confidence interval, inspect every denominator in ((n−1)s²/χ²upper, (n−1)s²/χ²lower). For variance confidence interval, a zero or near-zero denominator can make variance confidence interval undefined or unstable.
A fixed case for comparison
The default variance confidence interval condition is Sample variance = 25 squared units, Sample size = 20 observations, Lower-tail chi-square quantile = 8.907, Upper-tail chi-square quantile = 32.852.
For s²=25 and n=20, the entered 95% quantiles give limits near 14.46 and 53.33 squared units.
The live calculator reports Sample variance 25 squared units · Lower bound 14.458785 squared units · Upper bound 53.328842 squared units. Repeating one intermediate step from ((n−1)s²/χ²upper, (n−1)s²/χ²lower) provides a fixed variance confidence interval reference check for later code changes.
What the variance confidence interval arithmetic assumes
Normality matters strongly for this interval, and the lower and upper chi-square quantiles appear in reversed denominators.
For variance confidence 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 variance confidence interval
When interpreting variance confidence 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 variance confidence interval, reversing the numerator and denominator answers a different question, so retain the direction printed in ((n−1)s²/χ²upper, (n−1)s²/χ²lower).
Testing how stable variance confidence interval is
Change sample variance while holding the remaining entries fixed, then state why the direction and size of the variance confidence interval change are plausible from ((n−1)s²/χ²upper, (n−1)s²/χ²lower).
Repeat the variance confidence interval exercise with upper-tail chi-square quantile. If a modest defensible change materially alters the interpretation, report both conditions rather than presenting that variance confidence interval scenario as exact.
Where a plausible variance confidence interval can go wrong
Before accepting variance confidence interval, compare every entered value with its label, unit, and allowed domain after reading the printed relationship from left to right.
For variance confidence interval, do not move a number between fields merely because the units look compatible; each label gives the number a different statistical role.
Another variance confidence interval failure occurs when a rounded output is reused as though it were the original measurement. Carry guard digits through calculations that depend on variance confidence interval, then round only the reported value.
For a related comparison, continue with poisson rate confidence interval.
A reproducible record of variance confidence interval
Report variance confidence interval using ((n−1)s²/χ²upper, (n−1)s²/χ²lower), followed by the entered values, units, exclusions, and analysis date. Name the variance confidence interval population or dataset boundary instead of leaving it implicit.
Keep the full calculator output with the record, including Sample variance 25 squared units · Lower bound 14.458785 squared units · Upper bound 53.328842 squared units. A later variance confidence interval review can then distinguish a changed input from a different convention or software implementation.
Questions about variance confidence interval
What does variance confidence interval represent on this page?
It is the quantity produced by ((n−1)s²/χ²upper, (n−1)s²/χ²lower) from the displayed sample variance, sample size, lower-tail chi-square quantile, upper-tail chi-square quantile. This page applies chi-square quantiles to estimate a normal population variance.
What should be saved with variance confidence interval?
Save the entered values and units for sample variance, sample size, lower-tail chi-square quantile, upper-tail chi-square quantile, along with the analysis date, exclusions, software or formula version, and the relationship ((n−1)s²/χ²upper, (n−1)s²/χ²lower). That record is sufficient to rebuild this specific variance confidence interval calculation.
Does variance confidence interval establish a causal or population conclusion?
No. The displayed variance confidence interval value is conditional on the entered data and named method. The variance confidence interval design, measurement process, and assumptions determine what can be concluded beyond those values.