A sample of 51 units has a variance of 56.25. Find a 90% Standard Devi

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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/χ²_α/2) < σ < Square Root((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 = 51 - 1
Degrees of Freedom = 50

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 = 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/χ²_α/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:
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 = 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/χ²_{1 - 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/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

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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