To determine whether this value of x^{2} is statistically significant, first calculate the degrees of freedom (abbreviated df) for the problem. For a univariate table:

Next, go to appendix C, which is the distribution for x^{2}, and read down the left-hand margin to 13 df and across to find the critical value of x^{2} for any given level of significance. The levels of significance are listed across the top of the table. The greater the significance of a x^{2} value, the less likely it is that the distribution you are testing is the result of chance.

Considering the x^{2} value for the problem in table 20.12, the results look pretty significant. With 13 df, a x^{2} value of 22.362 is significant at the .05 level; a x^{2} value of 27.688 is significant at the .01 level; and a x^{2} value of 34.528 is significant at the .001 level.

With a x^{2} of 30.65, we can say that the distribution of the number of children across the 14 families is statistically significant at better than the .01 level, but not at the .001 level.

Statistical significance here means only that the distribution of number of children for these 14 families is not likely to be a chance event. Perhaps half the families happen to be at the end of their fertility careers and half are just starting. Perhaps half the families are members of a high-fertility ethnic group and half are not. The substantive significance of these data requires interpretation, based on your knowledge of what’s going on, on the ground.

Univariate numerical analysis—frequencies, means, distributions, and so on—and univariate graphical analysis—histograms, box plots, frequency polygons, and so on—tell us a lot. Begin all analysis this way and let all your data and your experience guide you in their interpretation. It is not always possible, however, to simply scan your data and use univariate, descriptive statistics to understand the subtle relations that they harbor. That will require more complex techniques, which we’ll take up in the next two chapters.

FURTHER READING

Cleaning data: Dijkers and Creighton (1994); Karmaker and Kwek (2007); Van den Broeck et al. (2005).

Exploring data: Hartwig and Dearing (1979); Pryjmachuk and Richards (2007); Tukey (1977).