Making Sense of Statistics

"Figures often beguile me," he wrote, "particularly when I have the arranging of them myself; in which case the remark attributed to Disraeli would often apply with justice and force: 'There are three kinds of lies: lies, damned lies, and statistics." - Mark Twain
Very few of us truly understand numbers. Some simply aren't good at math, others haven't had the best learning methods, and there are people without the appropriate information. Regardless, we have a moral imperative to learn how to make sense of quantitative information when it affects our lives in the way it does. News channels, politicians, and professionals from all fields clutter speeches and proposals with percentages, figures, theories, and anything else built of numbers. As a result, mathematics has gained ulterior political motives through agendas, poor analysis skills, and a lack of humanism in our interpretation of statistics. More than ever, these are the times when we fear what "studies say" or "science reports" about the things we do, the way we eat, the clothes we wear, the prevalence of sexual assault on campus, or the presence of guns in our garages.

Numbers are seductive. It's easy for us to fall victim to the allure of believing what isn't necessarily true. As Russian mathematician Edward Frenkel wrote in his Euler-winning book, " Love and Math: The Heart of Hidden Reality":
If you ask a drunkard what number is larger, 2/3 or 3/5, he won’t be able to tell you. But if you rephrase the question: what is better, 2 bottles of vodka for 3 people or 3 bottles of vodka for 5 people, he will tell you right away: 2 bottles for 3 people, of course.
What makes the rephrased question different from the former? The statistics are presented differently. In the second question, we have a more "tangible," usable way of understanding how the proportions of vodka would be arise from the distribution among people. Though Frenkel tries to explain his sophisticated, advanced mathematical theories to a general audience, we don't have to be Einsteins or engineers to understand the basic skills necessary to thrive in today's society.

Let me ask you the question: How well do you understand statistics? Try answering this question to find out.

Assume you conduct breast cancer screening using mammography in a certain region. You know the following information about the women in this region: The probability that a woman has breast cancer is 1% (prevalence). If a woman has breast cancer, the probability that she tests positive is 90% (sensitivity). If a woman does not have breast cancer, the probability that she nevertheless tests positive is 9% (false-positive rate).

A woman tests positive. She wants to know from you whether that means that she has breast cancer for sure, or what the chances are. What is the best answer?
A. The probability that she has breast cancer is about 81%.
B. Out of 10 women with a positive mammogram, about 9 have breast cancer.
C. Out of 10 women with a positive mammogram, about 1 has breast cancer.
D. The probability that she has breast cancer is about 1%.

When German psychologist Gerd Gigerezner posed the question to about 1000 gynecologists, about 21% chose the correct answer, C. While that is a little worse than random guessing, I must admit that, on my first attempt, I failed to answer this question correctly, as well. Through his research, Gigerezner has crafted a theory of understanding statistics that would help us in situations like this. Similar to Frenkel's example of the fractions of vodka, psychologists like Daniel Kahneman and Gerd Gigerezner have shown that asking statistics questions in different ways can influence the ways we understand them. For example, when the information preceding the question is framed differently (as shown below), 87% of gynecologists answered correctly.

Assume you conduct breast cancer screening using mammography in a certain region. You know the following information about the women in this region:  
Ten out of every 1,000 women have breast cancer
Of these 10 women with breast cancer, 9 test positive
Of the 990 women without cancer, about 89 nevertheless test positive

In both examples (of breast cancer screening and of bottles of vodka), when we change from "conditional probabilities" to "natural frequencies," we suddenly understand statistics much better. Like Gigerezner, I believe the appropriate way to interpret statistics is something we can teach, and, with the effect it has on our health and society, we have a moral imperative to do so.

This isn't a simple case of deliberately communicating false information or lying about the statistics we use. While there may be agendas and conflicts-of-interests between professionals (including scientists), we simply don't understand how to interpret statistics. And, in the field of medicine, this can have disastrous results. We make poor decisions about how long a patient may live, how prevalence of cancer among smokers, and understanding the harms and benefits of screening for breast cancer.

Numbers don’t lie, but they’re difficult to understand.