Chi-square Distribution

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Distribution - Chi-square

References: Dr. Mikel J. Harry, Juran's Quality Control Handbook


The Chi-square distribution generates curves representing the function of the chi-square statistic This statistic is used in the chi-square test (see Chi-square homogeneity test and Chi-square goodness of fit test).

Chi-square Distribution Tests

Use the Chi-square test to see whether there is a significant relationship between two categorical variables.

Chi-square tests are used for testing the "goodness of fit" and to decide whether there is any difference between the observed experimental/sample values and the expected theoretical values.

For example given a sample, we may like to test if it has been drawn from a normal population.

It tells us how likely it is that an observed distribution is due to chance. It's called a "goodness of fit" statistic, because it measures how well the observed distribution of data fits with the distribution that is expected if the variables are independent.


the chi square distributionChi-square Distribution

Free Chi-square Distribution Training


A. Vertical axis - Scale to measure the probability at different values of chi-square p(X-sq).

B. Horizontal axis - Scale of measure of the chi-square statistic X-sq.

C. Curve of the Chi-square distribution for various degrees of freedom. The number of degrees of freedom is represented by the Greek letter v(nu). The probability that a value X is less than a specified value X-sq, is the area under this curve up to the point X-sq.

When the number of degrees of freedom (v) is small the density function is severely asymmetric. As the number of degrees of freedom increases (v), the line becomes more symmetric. As the number of observations (n) becomes very large, the curve resembles a normal distribution. The mean and variance of the X-sq distribution are v and 2v respectively
The highest value of p ( X-sq ) in the curve occurs at X-sq = v - 2 for v > = 2.

Major Considerations

Applicable to continuous and discrete variables. The number of degrees of freedom is given by the formula v = n – 1 where n = number of observations in the sample

Application Cookbook

1. Use Excel functions to calculate the probability and the inverse of the chi-square distribution.

2. In Minitab use the following menu to generate a chi square distribution. CALC>PROBABILITY DISTRIBUTIONS>CHI-SQUARE

3. Alternatively, the tables printed at the end of most statistics books can be used.

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