I have a few measures based on probabilities. I am looking for a way to compute their central tendencies / summaries / averages.
Let's say that, I have 2 people: A and B. I have probabilities that they were late to work month-over-month, calculated from some actual or sample data. Let each row in the table below represent each month. Now, I can calculate conditional probabilities (P(B/A) and P(A/B)) and degree of co-occurrence (P(B/A) / P(B) or P(A/B) / P(A) or P(A n B) / (P(A)P(B))).
P(A) P(B) P(A n B) P(B/A) P(A/B) P(A n B)/(P(A)P(B))
month1 0.3 0.2 0.1 0.333 0.5 1.667
month2 0.2 0.25 0.13 0.65 0.52 2.6
month3 0.18 0.22 0.16 0.889 0.727 4.04
Since we used sample data (or assumed that there are exactly the same number of working days each month), we can use arithmetic mean for P(A), P(B) and P(A n B). Beyond that, it's not clear what is the right 'average'. The following table represents arithmetic mean of each of the columns from the above table:
P(A) P(B) P(A n B) P(B/A) P(A/B) P(A n B)/(P(A)P(B))
Average 0.227 0.223 0.13 0.624 0.582 2.769
These numbers are inconsistent. If I performed conditional probability and co-occurrence calculations from the probabilities, I would get P(B/A) = 0.573, P(A/B) = 0.583, and P(A n B)/(P(A)P(B)) = 2.572. The results look vastly different. Geometric mean leads to similar problems. Is there a principled way to calculate 'average' of conditional probabilities and degree of co-occurrences without having to convert them back to original data?
Also note, if this is easy to calculate, I am looking to also find central tendency of (P(A n B) - P(A)P(B)) / P(A), which is defined as degree of implication and is a ratio metric that ranges between [-1, 1].