Chi-Squared Distributions

CHI-SQUARED DISTRIBUTIONS

What is a chi-squared distribution?

We have previously discussed the Standard Normal curve and the associated distribution. Ð'ÐŽÐ'§ZÐ'ÐŽÐ'Ё was the term for the standard normal random variable. How is distributed? The answer is a chi-squared distribution. The importance of the Chi-square distribution stems from the fact that it describes the distribution of the Variance of a sample taken from a Normal distributed population.

The chi-squared distribution arises frequently in applications because of its close association with the normal distribution. The test is used for nominal data (two nominal variables Ð'ÐŽV see a separate post on more explanation of nominal data) that are independent. All events in the table should be independent. This means that no two frequencies can be based on the same individual.

Unlike the normal and t-distributions, the Ð'Јq2-distribution is not symmetric. Like the t-distribution, the Ð'Јq2-distribution consists of a whole family of distributions distinguished by a single whole number parameter, called the number of degrees of freedom

What do we mean by Degrees of Freedom?

Statisticians use the terms "degrees of freedom" to describe the number of values in the final calculation of a statistic that are free to vary. Degrees of freedom can be described as:

Ð'ÐŽÐ'§Degrees of freedom is a measure of how much precision an estimate of variation has. A general rule is that the degrees of freedom decrease when we have to estimate more parameters.Ð'ÐŽÐ'Ё

(Ref: http://www.cmh.edu/stats/ask/df.asp)

Consider this application of the degrees of freedom:

Sample standard deviation is calculated using the following formula:

Sample standard deviation, S = sqrt [Ð'Ñ"Ð"Ñ" (Ð'Ñ"nx Ð'ÐŽV x )2]

n-1

Population Standard Deviation,Ð'Ñ"n Ð'Ñ"Ð"Ј = sqrt [Ð'Ñ"Ð"Ñ" ( x Ð'ÐŽV Ð'Ñ"Ð"Ñœ)2]

N

Where:

X Ð'ÐŽV sample mean

Ð'Ñ"Ð"Ñœ Ð'ÐŽV population mean

Why do we divide by n-1 for the sample standard deviation?

We divide by n-1 because we use the mean to calculate the standard deviation. The final squared deviation from the mean must include the one value of X such that the sum of all the Xs divided by n will equal the obtained mean of the sample. All of the other (n - 1) squared deviations from the mean can, theoretically, have any values whatsoever.

In more complex situations, like Analysis of Variance and Multiple Linear Regression, we usually have to estimate more