https://towardsdatascience.com/simpsons-paradox-d2f4d8f08d42

Simpson's Paradox diagram reference

In this example illustrating Simpson's paradox, height is seen as a a beneficial factor (positive slope) in each age group, but harmful (downward slope) overall.

University of Michigan - Ann Arbor

When groups of data show one particular trend, but this trend is reversed when the groups are combined together. (A/B) > (a/b) & (C/D) > (c/d) but (A+C)/(B+D) < (a+c)/(b+d)

Simpson's Paradox

Taking drug D lowers the risk of heart attack in people, however, it raises the risk of heart attack in both men and women. When accounting for gender it will raise the risk of heart attack, but when gender is not separated from the general it actually lowered the risk of heart attack. 
We found this to be a numerical calculation that are biases simply do not account for. We found that:
3/40 > 1/20 or (A/B > a/b),
8/20 > 12/40 or (C/D > c/d),
but that does not mean (3 + 8)/(40 + 20) > (1 + 12)/(20 + 40) or (A + C)/(B + D) > (a + c)/(b + d).
We found that (3 + 8)/(40 + 20) < (1 + 12)/(20 + 40)


Simpson's Paradox Example

Simpson's Paradox Diagram

In 1973, Eugene Hammel, an associate dean at the University of California, noticed a worrisome trend in the university’s admission rates for men and women. According to his data, it is shown that 44 percent of the men who applied to graduate school at Berkeley had been accepted, compared to only 35 percent of the women.
After a careful investigation, Eugene found out that even though for each apartment, the admissions decisions were consistently more favorable to women than to men, the admission rate for women is lower than that of men.
The causal story Eugene discovered is that women were rejected in greater numbers because they applied to harder departments to get into, like history and literature, therefore having a lower acceptance rate.

Berkeley Admissions Paradox

If you were to graph a scatterplot of cholesterol levels and amount exercise for a number of people it would appear that as exercise increases, so does cholesterol (graph on the right). When you account for age in labeling the data points however, it becomes obvious that exercise helps to instead reduce cholesterol (graph on the left).

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