A sample of 51 units has a variance σ2 of 56.25. Find a 90% confidence interval of the standard deviation σ

Confidence Interval Formula for σ is as follows:
Square Root((n - 1)s2/χ2α/2) < σ < Square Root((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 = 51 - 1
Degrees of Freedom = 50

Calculate α:
α = 1 - confidence%
α = 1 - 0.9
α = 0.1

Find low end confidence interval value:
αlow end = α/2
αlow end = 0.1/2
αlow end = 0.05

Find low end χ2 value for 0.05
χ20.05 = 67.5048 <--- Value can be found on Excel using =CHIINV(0.05,50)

Calculate low end confidence interval total:
Low End = Square Root((n - 1)s2/χ2α/2)
Low End = √(50)(56.25)/67.5048)
Low End = √2812.5/67.5048
Low End = √41.663703914388
Low End = 6.4547

Find high end confidence interval value:
αhigh end = 1 - α/2
αhigh end = 1 - 0.1/2
αhigh end = 0.95

Find high end χ2 value for 0.95
χ20.95 = 34.7643 <--- Value can be found on Excel using =CHIINV(0.95,50)

Calculate high end confidence interval total:
High End = Square Root((n - 1)s2/χ21 - α/2)
High End = √(50)(56.25)/34.7643)
High End = √2812.5/34.7643
High End = √80.901959769073
High End = 8.9946

Now we have everything, display our interval answer:

6.4547 < σ < 8.9946 <---- This is our 90% confidence interval

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What this means is if we repeated experiments, the proportion of such intervals that contain σ would be 90%

What is the Answer?

6.4547 < σ < 8.9946 <---- This is our 90% 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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