Confidence Interval for Variance and Standard Deviation Calculator

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Image to Crop 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

Confidence Interval for Variance and Standard Deviation Calculator Video


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