Biostatistics

688 CHAPTER 13 NONPARAMETRIC AND DISTRIBUTION-FREE STATISTICS
TABLE 13.5.2 Level of Mental Health Scores of
Junior High School Boys
Urban Rural Total
Number of scores above median 6 8 14
Number of scores below median 10 4 14
Total 16 12 28
and, because the total number of observations is even, obtaining the
mean of the two middle numbers. For our example the median is
ð33 þ 34Þ=2 ¼ 33:5.
We now determine for each group the number of observations
falling above and below the common median. The resulting frequencies
are arranged in a 2 2 table. For the present example we construct
Table 13.5.2.
If the two samples are, in fact, from populations with the same
median, we would expect about one-half the scores in each sample to be
above the combined median and about one-half to be below. If the
conditions relative to sample size and expected frequencies for a 2
2 contingency table as discussed in Chapter 12 are met, the chi-square test
with 1 degree of freedom may be used to test the null hypothesis of equal
population medians. For our examples we have, by Formula 12.4.1,
X 2 ¼ 28½ð6Þð4Þ
ð8Þð10Þ
ð16Þð12Þð14Þð14Þ
Š2
¼ 2:33
8. Statistical decision. Since 2:33 < 3:841, the critical value of x 2 with
a ¼ :05 and 1 degree of freedom, we are unable to reject the null
hypothesis on the basis of these data.
9. Conclusion. We conclude that the two samples may have been drawn
from populations with equal medians.
10. p value. Since 2:33 < 2:706, we have p >:10. &
Handling Values Equal to the Median Sometimes one or more observed
values will be exactly equal to the common median and, hence, will fall neither above nor
below it. We note that if n 1 þ n 2 is odd, at least one value will always be exactly equal to the
median. This raises the question of what to do with observations of this kind. One solution
is to drop them from the analysis if n 1 þ n 2 is large and there are only a few values that fall
at the combined median. Or we may dichotomize the scores into those that exceed the
median and those that do not, in which case the observations that equal the median will be
counted in the second category.
Median Test Extension The median test extends logically to the case where it is
desired to test the null hypothesis that k 3 samples are from populations with equal
medians. For this test a 2 k contingency table may be constructed by using the