Ineffectiveness vs. effectiveness: the math behind the difference!

Let’s say you are shopping for a new home air filter. Comparing two models, you note that model A boasts that it blocks 99.9% of particulates; model B claims to block 99.8%.

That seems like a pretty minor difference. Would it surprise you to learn that model B is actually admitting twice as many particles as model A?

The reason lies in the terminology. When these machines claim to block N percent of particulates they are boasting about effectiveness. However, you don’t care about how effective the devices are — the true metric is how ineffective they are, because that is the part that impacts you. In the air-filter paradigm, the number of particulates that make it into your atmosphere is what matters.

So, you need to turn the number on its ear. An air filter that is 99.9% effective is 0.1% ineffective. One that is 99.8% effective is 0.2% ineffective. Let’s look at the numbers:

99.9% Effectiveness 99.8% Effectiveness
Total Particles Particles Blocked Particles Admitted Particles Blocked Particles Admitted
1,000 999 1 998 2
10,000 9,990 10 9,980 20
100,000 99,900 100 99,800 200


That’s a substantial — and probably unanticipated — difference! Our instinct is to think that model B would have needed to fall to 50% effectiveness to admit twice as many particles. (In fact, a model that was 50% effective would admit 500 times the particulates!)

Keep this in mind for anything that claims to be N percent effective at some task. Whether it’s birth control, vaccines, pain treatments, anti-bacterial soaps…it’s not the hits that count, it’s the misses.


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