# How to analyze contingency table results? Help with wording to describe observations

I'm learning about contingency tables and proportions. Need help interpreting this table of students interested in one of three subjects by gender.

Male Female
Science 20 (30.8%) 20 (30.8%)
Math 30 (46.2%) 15 (23.1%)
Art 15 (23.1%) 30 (46.2%)
Total 65 (100.0%) 65 (100.0%)

chi2 statistic = 10, p-value= 0.007

Calling out differences between genders, are any of these correct?

• Males are twice as likely to prefer math than females.
• Most males (46%) prefer math, compared to only 23% of females.
• Students are equally likely to prefer science, regardless of gender.
• Males showed a higher rate of interest in math than females by 23.1 percentage-points.
• Males prefer science half as much as females prefer science.
• Males prefer science 50% less than females do.
• Males prefer math 2x as much as females do.

Any other examples/ideas on wording appreciated.

You have counts, and a contingency table TAB as follows:

m = c(20, 30, 15);  f = c(20, 15, 30)
TAB = rbind(m, f);  TAB
[,1] [,2] [,3]
m   20   30   15
f   20   15   30


A chi-squared test shows significantly different interests for makes and females.

chisq.test(TAB)

Pearson's Chi-squared test

data:  TAB
X-squared = 10, df = 2, p-value = 0.006738


If you are going to make specific comments about those differences, it is best to focus on differences that can be confirmed by ad hoc tests. For example, looking just at Math and Science, we have the following 2-by-2 table, which shows no significant difference between genders in preferences between math and science.

SM = TAB[ ,1:2];  MS
[,1] [,2]
m   20   30
f   20   15
chisq.test(SM)

Pearson's Chi-squared test with
Yates' continuity correction

data:  SM
X-squared = 1.7892, df = 1, p-value = 0.181


In the population, there may be a real difference between males and females as to math and science, but you have no evidence of that in you your data.

So before making summary comments, it would be best to see if there are significant differences revealed in other 2-by-2 sub-tables. Perhaps look at math and art.

MA = TAB[ ,2:3]
MA
[,1] [,2]
m   30   15
f   15   30
chisq.test(MA)

Pearson's Chi-squared test
with Yates' continuity correction

data:  MA
X-squared = 8.7111, df = 1, p-value = 0.003163


Note: With all 'counts' ending in 0 or 5, these are pretty clearly fake data contrived for an exercise. It would be more interesting to discuss differences revealed by real data--maybe even 'data' unencumbered by classical prejudices against mathematical interest among females (or disinterest in art by men).