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Traffic Stops in Ferguson, MO

Explain it to me as if I liked math

On its face, racial profiling seems like an efficient way to catch traffic violators. This is real data about traffic stops from 2013.

Among the drivers that got tickets, 93% of them were black

% of the drivers on the road are black. While there are more black drivers, they are overrepresented among those that got tickets.

There are several reasons why they could be overrepresented. One is that more of them are caught because of racial profiling. Black drivers are indeed more likely to be stopped.

% black drivers stopped by police

% white drivers stopped by police

However, black drivers who are stopped are still more likely to be ticketed, which implies that black drivers really violate traffic laws at a greater rate the white drivers. On the other hand, it could instead mean that determining whether there was a traffic violation is not objective either, and that black drivers are just more likely to get tickets.

The proportion of black drivers stopped who got ticketed
The proportion of white drivers stopped who got ticketed

True rates of traffic violations.

Unfortunately, it's not possible from this information to determine the true rates of traffic violations. Instead, we can backsolve the true rates of traffic violations in the case that the outcome is fair.

In this case, the demographics of true traffic violators should be the same as the demographics of the traffic violators that are caught and ticketed.

It's reasonable to assume that there are some white drivers who violate traffic who avoid being stopped and ticketed. Then the true rate of offenders must be somewhat more than 4%. Let's say it's 5%. This implies that 35% of black drivers violate traffic.

The assumptions of the true rates of traffic violations among the drivers are displayed in the chart below.

These assumptions ensure that the true demographics of traffic violators including those that are not stopped is the same as the demographics of the ones that are ticketed.

These assumptions are sometimes referred to as the base rates because they are the true rates which may not be directly observable. The difference between black and white drivers seems too large to be realistic. If these rates are true then we can also calculate how many traffic offenders of each race are not stopped at all. The below chart shows the size of these groups.

is the probability that a black driver who is not stopped by police violated traffic.

is the probability that a white driver who is not stopped by police violated traffic.

One could interpret the differences in the rates to mean that black drivers should be stopped more.


These are a few important metrics around traffic policing practices. Traffic police have limited resources and so the amount of effort involved in policing as well as efficiency of the policing effort are certainly important metrics. At the same time, the fairness of the policing is also important for the community as well as the individuals.

Effort: % drivers stopped. This represents the level of surveillance on drivers.

Efficiency: drivers who got a citation per driver stopped

Unfairness: % overrepresentation of black violators among drivers stopped

Alternative Base Rates

If the base rate of violating traffic for white drivers is a slightly higher rate, 7%, then 50% of black drivers violate traffic which is surely not possible. If white drivers violate traffic at 15%, then 100% of black drivers are violating traffic. Play with the sliders of the base rates of offending traffic. What do you think are reasonable rates?

% of white drivers violate traffic

% of black drivers violate traffic

Almost certainly you find that it is more realistic for the rates of violating traffic to be more similar across demographics. For example, lowering the rate of traffic violating for black drivers to 13% the result is that all the black drivers that violated traffic have already been caught. If the rate for white drivers is kept at 5%, then by what percent are black drivers overrepresented among those ticketed compared to the true rates of traffic offenders? This is the Unfairness metric. Did the amount of Effort change? How about Efficiency?

Interaction With Confirmation Bias

This is a separate effect from confirmation bias, which is where you are more likely to notice "facts" that are consistent with your world view. This result shows that even without confirmation bias, which we all have, we are likely to believe things that aren't true simply because we are also bad at reasoning about probability.