Chi square

The chi-square distribution has one parameter k, the degrees of freedom.  The chi squared distribution becomes more symmetric as k increases.  The mean and variance of the chi square distribution also increase as k increases and the mean = k and variance = 2k.

Chi-square test explained

Chi-square test is based on the chi square distribution.  It is used for the goodness-of-fit test because it asks whether there is a good fit between the data (O = Observed frequency) and the theory (E = Expected frequency).  Chi-square determines whether the differences between the observed and expected scores can be attributed to some actual difference in behavior or if this difference between the scores is caused by chance using the formula chi square = sum{ (O-E)^2) / E}.   

Determining significance of chi square

If we wish to reject H_o at the .05 level, we will determine if our value of chi square is greater than the critical value of chi square that cuts off the upper 5% of the distribution at our particular degrees of freedom value.  If our value of chi square from the formula is greater  than the critical value of chi square, we reject H_o and conclude that the obtained frequencies differ from the expected frequencies more than would be predicted by chance.  

Two variable chi square

The two variable chi-square is used to determine whether one variable is contingent on a second variable.  So, we construct a contingency table that shows the distribution of one variable at each level of the other variable. For a contingency table, the expected frequency for a given cell is obtained by multiplying the totals for the row and column in which the cell is located and dividing by total sample size.