Confidence Interval for Variance and Standard Deviation Calculator

Publish date: 2024-07-16

A sample of 16 units has a variance σ2 of 4.84. Find a 99% confidence interval of the variance σ2

Confidence Interval Formula for σ2 is as follows:
(n - 1)s2/χ2α/2 < σ2 < (n - 1)s2/χ21 - α/2 where:
(n - 1) = Degrees of Freedom, s2 = sample variance and α = 1 - Confidence Percentage

First find degrees of freedom:
Degrees of Freedom = n - 1
Degrees of Freedom = 16 - 1
Degrees of Freedom = 15

Calculate α:
α = 1 - confidence%
α = 1 - 0.99
α = 0.01

Find low end confidence interval value:
αlow end = α/2
αlow end = 0.01/2
αlow end = 0.005

Find low end χ2 value for 0.005
χ20.005 = 32.8015 <--- Value can be found on Excel using =CHIINV(0.005,15)

Calculate low end confidence interval total:
Low End = (n - 1)s2/χ2α/2
Low End = (15)(4.84)/32.8015
Low End = 72.6/32.8015
Low End = 2.2133

Find high end confidence interval value:
αhigh end = 1 - α/2
αhigh end = 1 - 0.01/2
αhigh end = 0.995

Find high end χ2 value for 0.995
χ20.995 = 4.6009 <--- Value can be found on Excel using =CHIINV(0.995,15)

Calculate high end confidence interval total:
High End = (n - 1)s2/χ21 - α/2
High End = (15)(4.84)/4.6009
High End = 72.6/4.6009
High End = 15.7795

Now we have everything, display our interval answer:

2.2133 < σ2 < 15.7795 <---- This is our 99% confidence interval

What this means is if we repeated experiments, the proportion of such intervals that contain σ2 would be 99%

What is the Answer?

2.2133 < σ2 < 15.7795 <---- This is our 99% confidence interval

How does the Confidence Interval for Variance and Standard Deviation Calculator work?

Free Confidence Interval for Variance and Standard Deviation Calculator - Calculates a (95% - 99%) estimation of confidence interval for the standard deviation or variance using the χ2 method with (n - 1) degrees of freedom.
This calculator has 3 inputs.

What 4 formulas are used for the Confidence Interval for Variance and Standard Deviation Calculator?

Degrees of Freedom = n - 1
Square Root((n - 1)s2/χ2α/2) < σ < Square Root((n - 1)s2 / χ21 - α/2)

Square Root((n - 1)s2/χ2α/2) < σ2 < Square Root((n - 1)s2 / χ21 - α/2)

For more math formulas, check out our Formula Dossier

What 5 concepts are covered in the Confidence Interval for Variance and Standard Deviation Calculator?

confidence intervala range of values so defined that there is a specified probability that the value of a parameter lies within it.confidence interval for variance and standard deviationa range of values that is likely to contain a population standard deviation or variance with a certain level of confidencedegrees of freedomnumber of values in the final calculation of a statistic that are free to varystandard deviationa measure of the amount of variation or dispersion of a set of values. The square root of variancevarianceHow far a set of random numbers are spead out from the mean

Example calculations for the Confidence Interval for Variance and Standard Deviation Calculator