In the previous exercises, we looked at an association between the influence and leader questions using a contingency table. We saw some evidence of an association between these questions.

Now, let’s take a moment to think about what the tables would look like if there were no association between the variables. Our first instinct may be that there would be .25 (25%) of the data in each of the four cells of the table, but that is not the case. Let’s take another look at our contingency table.

leader           no       yes
influence                    
no         0.271695  0.116518
yes        0.212670  0.399117

We might notice that the bottom row, which corresponds to people who think they have a talent for influencing people, accounts for 0.213 + 0.399 = 0.612 (or 61.2%) of surveyed people — more than half! This means that we can expect higher proportions in the bottom row, regardless of whether the questions are associated.

The proportion of respondents in each category of a single question is called a marginal proportion. For example, the marginal proportion of the population that has a talent for influencing people is 0.612. We can calculate all the marginal proportions from the contingency table of proportions (saved as influence_leader_prop) using row and column sums as follows:

leader_marginals = influence_leader_prop.sum(axis=0)
print(leader_marginals)
influence_marginals =  influence_leader_prop.sum(axis=1)
print(influence_marginals)

Output:

leader
no     0.484365
yes    0.515635
dtype: float64

influence
no     0.388213
yes    0.611787
dtype: float64

While respondents are approximately split on whether they see themselves as a leader, more people think they have a talent for influencing people than not.