A sample of 70 units has a variance of 4.84. Find a 90% Variance Con

A sample of 70 units has a variance σ² of 4.84. Find a 90% confidence interval of the variance σ²

Confidence Interval Formula for σ² is as follows:
(n - 1)s²/χ²_α/2 < σ² < (n - 1)s²/χ²_{1 - 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 = 70 - 1
Degrees of Freedom = 69

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

Find low end confidence interval value:
α_{low end} = α/2
α_{low end} = 0.1/2
α_{low end} = 0.05

Find low end χ² value for 0.05
χ²_0.05 = 89.3912 <--- Value can be found on Excel using =CHIINV(0.05,69)

Calculate low end confidence interval total:
Low End = (n - 1)s²/χ²_α/2
Low End = (69)(4.84)/89.3912
Low End = 333.96/89.3912
Low End = 3.7359

Find high end confidence interval value:
α_{high end} = 1 - α/2
α_{high end} = 1 - 0.1/2
α_{high end} = 0.95

Find high end χ² value for 0.95
χ²_0.95 = 50.8792 <--- Value can be found on Excel using =CHIINV(0.95,69)

Calculate high end confidence interval total:
High End = (n - 1)s²/χ²_{1 - α/2}
High End = (69)(4.84)/50.8792
High End = 333.96/50.8792
High End = 6.5638

Now we have everything, display our interval answer:

3.7359 < σ² < 6.5638 <---- 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 σ² would be 90%

What is the Answer?

3.7359 < σ² < 6.5638 <---- 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/χ²_α/2) < σ < Square Root((n - 1)s² / χ²_{1 - α/2})

Square Root((n - 1)s²/χ²_α/2) < σ² < Square Root((n - 1)s² / χ²_{1 - 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 interval: a 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 deviation: a range of values that is likely to contain a population standard deviation or variance with a certain level of confidence
degrees of freedom: number of values in the final calculation of a statistic that are free to vary
standard deviation: a measure of the amount of variation or dispersion of a set of values. The square root of variance
variance: How far a set of random numbers are spead out from the mean