Often times we think that data favours our argument. It is common for a speaker to spin information to suit their purpose. However, doing an analysis on the data provided gives a clear picture of the information.

In the case of discrimination against women in an organization, at first glance, it did not look like there was a case for discrimination. In a total of 3240 staff, 2120 were men and 1120 were women. Out of which, 1,450 men were promoted and 510 women were promoted. It is not odd to expect more men to be promoted given that there are more men in the office than women. Interestingly, with further analysis of these numbers, a case for discrimination showed.

The analysis is simple. First you determine the probability of being a man and that of being a woman in the organization. 65% for men and 35% for women.

Let the event of being a man be M

Let the event of being woman be W

Let the event of being promoted B

Let the event of not being promoted be B’

M

W

Marginal probability

B

1450

510

0.654321

B’

670

610

0.345679

Total

2120

1120

3240

1

Marginal probability table

Next, you identify the joint probability of being promoted and being man as well as the joint probability of being promoted and being a woman. To do this, divide the total number of promoted women by the total number of the staff. Do the same for men.

M

W

B

0.447531

0.157407

0.604938

B’

0.20679

0.188272

0.395062

1

Contingency or Joint probability table

Now to the matter at hand; to find the probability of being promoted given that you are a man and the probability of being promoted given that you are a woman. This is called a joint probability.

The probability of being promoted given that it is a man is:

P(B|M) =P(BnM)/P(M)

= 0.45/0.65

= 0.68

The probability of being promoted given that it is a woman is:

P(B|W) = P(BnW)/P(W)

= 0.16/0.34

= 0.455

This is where the bias becomes obvious. So long as the probability of being promoted for a man is not the same as the probability of being promoted for a woman, the numbers will suggest that there is a bias against the side with the lower likelihood of promotion.

So we see that using probabilities to analyse numbers sometimes brings out the obscured information and a new line of thought is given credence.