Confidence Intervals For Variance And Standard Deviation - Chi Square Distribution

In this video we discuss how to construct confidence intervals for variance and standard deviation. We also cover what the chi-square distribution is, and how it is used to get critical values used for the intervals.

Transcript/notes (partial)
To construct confidence intervals for variance and standard deviation, a different statistical distribution is needed, and it is called the chi-square distribution. The symbol for chi-square is this x looking character raised to the 2, and it is the Greek letter chi.

The chi-square distribution is similar to the t distribution in that it is a family of curves based on the degrees of freedom.

Here is an example of a chi-square distribution with 4 degrees of freedom, and here is one with 10 degrees of freedom, and here is one with 15 degrees of freedom.

A couple of characteristics about the chi-square distribution. The area under each distribution curve is equal to 1 or 100%. The chi-square distributions are positively skewed so, the distribution is not symmetric. The degrees of freedom are equal to n minus 1, or the sample size minus 1, and as the degrees of freedom increase, the chi square distribution approaches a normal distribution.

As far as the 90%, 95% or 99% confidence intervals, they have a similar concept for the chi square distribution, however, since the chi square distribution is not symmetric, the critical values will not be the same.

So, lets say that this is a chi square distribution with 20 degrees of freedom, and we want to construct a 90% confidence interval with a sample size of 22. So, 90% of the area under the curve lies between these 2 values, which I have marked with red lines. And these lines, which represent critical values, are noted as chi square left and chi square right. And since 90% of the area lies between the lines, 10% lies outside of the lines, so, we have 5% under the curve to the left of chi square left, and 5% under the curve to the right of chi square right.

Now, we need to use a chi square distribution table. This table is based on the degrees of freedom and an area or percentage to the right of a chi square value. So to find the value of chi square left, we need to know the total area to the right of that value.

And in our graph, we have 5% to the left, so, 100% minus 5% gives us a total area of 95% to the right of the value. Since our sample size is 22, n minus 1, 22 – 1 equals 21 for the degrees of freedom. So, in the table, we want to find the 95% or 0.95 column here near the top, and under the degrees of freedom column we go down to the 21 row. And the value listed where these intersect is 11.591, and that is our value for chi square left.

Next, for chi square right, again we need to know the area or percentage to the right of chi square right, and we already know that from earlier, it is 5% or 0.05, and again the degrees of freedom is 21. In the table we find the 0.05 column near the top, and under the degrees of freedom column we go down to the 21 row. And the value listed where these intersect is 32.671, and that is our value for chi square right.

And the formula for the confidence interval for a standard deviation is square root of the quantity n minus 1 times s squared, divided by chi square right, less than sigma, less than the square root of the quantity n minus 1 times s squared, divided by chi square left.

In both of these formulas, n is the sample size, s squared is the sample variance, chi square right and left are critical values determined by the confidence level and degrees of freedom, and sigma squared is the population variance, and sigma is the population standard deviation.

Ok, example time. Lets say that a protein powder manufacture randomly selects and weighs 25 samples of its top selling product. The sample standard deviation is 4.8 grams. Assuming the weights are normally distributed, construct 95% confidence intervals for the population variance and standard deviation.

So, we need to find chi square left and right. We are using a 95% confidence interval and the degrees of freedom are n minus 1, 25 minus 1, so, 24 degrees of freedom. Here is an example of a chi square curve with 24 degrees of freedom. In this middle section we have 95%, our confidence interval, these 2 red lines are our critical values, chi square left and right. And we have 2.5% outside each of these lines.

Now we can plug and calculate into the variance formula. So, we have 25 minus 1 times 4.8 squared divided by 39.364, less than sigma squared, less than 25 -1 times 4.8 squared divided by 12.401. And our final result is 14.05 less than sigma squared, less than 44.59.

Timestamps
0:00 What Is The Chi Square Distribution?
1:12 Example Of 90% Confidence Interval For A Chi Square Distribution
1:46 How To Use A Chi Square Distribution Table To Find Critical Values
3:02 Formula For The Confidence Interval For Variance
4:12 Example Problem For Finding The Confidence Interval For Variance

[EmekaOgugua]

I don't get why a video of this level of value is getting not so much likes

[neosharkey]

I don't know how professional college lecturers can make even the most simple thing sound confusing. I took stat 2 18 months after stat 1 and I remember nothing. Professor makes it seem like rocket science, Youtuber makes it seem like child's play. Thank you.

[carlosrussobaez]

Incredible! Makes so much sense. Thank you so much

[lakshikatharejaa]

Very well explained in a short video. Keep uploading more topics!

[danebryant]

Awesome Video, spent hours looking for how to work with all the variables. Thank you!

[我从通化来]

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[Bubumeow]

wow the best video for beginner like me .... perfect sir..thankssss

[neelabhgaur]

amazing video, thank you so much sir for making me understand this concept in such a little time

[dr.robinsonpaul]

Excellent | Short | Perfect | Easy = ESPE