#### Introduction to the chi-square distribution

Basic Definitions:

Chi-square distribution: It is a distribution obtained from multiplying the ratio of sample variance to the population variance by the degree of freedom whenever random samples are chosen from normally distributed population.

Contingency Table: The data arranged in a tabular form for the chi-square independence test.

Expected Frequency: The frequency obtained by computation is termed as expected frequency.

Goodness-of-fit Test: It is a test to see when a sample comes from a population by given distribution.

Independence Test: It is a test to see when the column and row variables are independent.

Observed Frequency: It is the frequency obtained by the observation. These are sample frequencies.

Chi-Square Distribution:

The chi-square (χ2) distribution is received from the values of the ratio of sample variance and population variance multiplied by the degree of freedom. This takes place whenever the population is generally distributed with population variance sigma^2.

Properties of the Chi-Square:

a) The Chi-square is non-negative. It is the ratio of two non-negative values, so should be non-negative itself.
b) Chi-square is non-symmetric.
c) There are numerous distinct chi-square distributions, one for each and every degree of freedom.
d) The degree of freedom whenever working with the single population variance is (n-1).

Chi-Square Probabilities:

As the chi-square distribution is not symmetric, the technique for looking up left-tail values is distinct from the technique for looking up right tail values.

a) Area to right - just utilize the region given.

b) Area to left - the table needs the region to the right, therefore subtract the given region from one and look this region up in the table.

c) Area in both tails - divide the region by two. Look-up this region for right critical value and one minus this region for the left critical value.

DF which aren't in the table:

Whenever the degrees of freedom are not listed in table, there are a couple of selections that you have.

a) We can interpolate. This is most likely the more precise way. Interpolation includes estimating the critical value by figuring how far the provided degrees of freedom are between the two df in table and going that far among the critical values in the table. Most of the people born in 70's did not have to learn interpolation in high school as they had calculators that would do logarithms.

b) We can go with the critical value that is less likely to cause you to refuse in error (that is, type I error). For a right tail test, this is critical value moreover to the right (maximum). For left tail test, this is the value further to the left (least). For a two-tail test, it is the value further to left and the value further to right. Note that it is not the column with degree of freedom further to right; it is the critical value that is further to right.

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