# Numerical blindness?

## I wouldn’t see a doctor, if I were you.

**When diagnosed with AIDS or breast cancer, most people feel as if they have been sentenced to death. Is this justified? Provided that he or she doesn’t belong to a high-risk group, a person only really carries the HIV virus 50 % of the time and actually has breast cancer in 9 % of the cases. The problem, however, is that most doctors tell a different story.**

Numerical blindness, better known as innumeracy, is a lack of understanding for numbers. Professor Gerd Gigerenzer, director of the Max-Planck-Institute for Human Development in Berlin, is an activist fighting this form of mathematical illiteracy.

In multiple studies, Gigerenzer has proved that risks in general are often communicated incorrectly, resulting in tragic results for those affected. The problem is as widespread as it is human. The most common cause is that we tend to argue based on probability, which is often misleading, instead of frequency, which is easier to understand. In a best case scenario, doctors (and, in turn, their patients) are informed of relationships as follows:

The probability that a woman in her early forties has breast cancer lies at 0.8 percent [1]. If she has breast cancer, than she has a 90 percent chance that her mammogram tests positive. If she doesn’t have breast cancer, there is a 7 percent likelihood that the test will still read positive. So what is the probability that the woman with a positive mammogram actually has breast cancer?

In short, it is also common to assume a basic share of population of 0.01, a sensitivity of 0.9 and a false positive rate of .09.

The same relationship can also be explained using actual incidents: Out of 1000 women, 8 will have breast cancer but only 7 will have a positive mammogram. Out of the 992 women who don’t have breast cancer, 70 will still have a positive mammogram. In other words, there are 77 positive mammograms. So, how many of the 77 women with a positive mammogram actually have breast cancer? [Solution]

Gigerenzer asked 48 doctors the same question. Half of them received the values as probability statistics while the other half received them as incident statistics. The results are shocking.

When the doctors were given probability statistics, their estimates that a woman with a positive mammogram actually has breast cancer ranged between 1 and 90 percent. In fact, a third of the doctors said 90 percent! Another third gave answers between 50 – 80 percent. The rest gave a probability of 10 percent or lower, whereby half of them gave an answer of 1 percent, which equals the basic share of population for breast cancer. The average was 70 percent [2].

When the doctors were given incident statistics, their results varied far less. Most doctors gave the correct answer or were very close. Only 5 gave answers over 50 percent.

The objective here is not to criticize doctors – or even lawyers, appraisers, or other people who incorrectly evaluate statistics on a regular basis. Gigerenzer and his colleagues have proven that innumeracy is a very widespread and common problem. In fact, we are subject to self-delusions, ranging from an AIDS test or DNA analysis to finger prints or the dangers of air travel, almost everywhere we look.

But these simple misunderstandings have real and major consequences for both us and our loved ones. The cure to fight this “social disease”, however, is simple as Gigerenzer describes in “Calculated Risks: How to Know When Numbers Deceive You“. This is an excellent read for anyone who has a reason to question probability statistics of any kind.

[1] These realistic but fictitious numbers are based on typical early detection examinations for woman between 40 and 50 with no visible symptoms.

[2] Gigerenzer 2004, page 68 f.