Chi-square
Calculating the observed chi-square
Determining degrees of freedom
Interpreting the test statistic
Some limitations of chi-square
Steps for using chi-square for hypothesis testing
The chi-square tests the null hypothesis that the difference between the expected frequencies and observed frequencies in a table is significantly different from zero. Another formulation asks: is the difference between observed and expected so extreme that it is improbable that it was produced by chance?
How do we do we calculate and use chi-square?
In reduced form: we calculate the observed chi-square value for the table we have produced from our sample, and then we compare it to the expected chi-square value which is given by a theoretical chi-square distribution for tables of similar dimensions as ours – that is, tables with the same degrees of freedom.
The theoretical chi-square distribution is based on the idea of taking an infinite number of repeated samples from the same population and calculating an observed chi-square each time, under the assumption that the null hypothesis is true. Thus, if the null hypothesis were true, we would expect that the observed chi-square for our particular table would fall within the range of expected values for tables with the same degrees of freedom as ours. Conversely, if the observed chi-square value for our particular table is higher than the expected range of chi-square values given by the theoretical chi-square distribution for tables like our, then we can decide that our findings are sufficiently improbable that we can reject the null hypothesis. In this case, our research hypothesis stands.
Researchers usually define in advance what they mean by “sufficiently improbable” by specifying a cut-off point in the theoretical distribution below which the observed test statistic must fall to reject the null hypothesis. This cutoff point below which our observed chi-square statistic must fall is called alpha and denoted by the Greek letter a, is customarily set at .05, .01, or .001.
This overall process and each of these terms are elaborated below for an example from 2000 MA census data.
Calculating the observed chi-square
Determining degrees of freedom
Interpreting the test statistic
Some limitations of chi-square
Steps for using chi-square for hypothesis testing