The journalist's article was illustrated with a photo of a businessman. Is that the best choice, given the study? Photo credit: Yuri Arcurs/Getty Images

Are you familiar with proper numbers? How about declarative fractions?

If you claimed that you knew these concepts, you're probably fooling yourself....because declarative fractions and proper numbers are not real math things.

A Washington Post journalist summarized a study on who is more likely to falsely claim expertise, or as some folks might put it: to "BS." The article's headline reads, "Rich guys are most likely to have no idea what they are talking about, study suggests."

What did the study do?

Study participants were asked to assess their knowledge of 16 math topics on a five-point scale ranging from “never heard of it” to “know it well, understand the concept.” Crucially, three of those topics were complete fabrications: “proper numbers,” “subjunctive scaling” and “declarative fractions.” Those who said they were knowledgeable about the fictitious topics were categorized as BSers.

The sample for the study was international:

The study drew from the Program for International Student Assessment, which is administered to tens of thousands of 15-year-olds worldwide. The test included a background questionnaire that captures demographic information, along with students’ attitudes toward the subjects they study in school. That section of the test included the questions about math knowledge.

Here is the pattern of results for gender (You can see tables of the data in the journalist's article).

The data revealed that boys across all nine countries were significantly more likely than girls to pretend expertise, with the difference between the two working out to nearly half a standard deviation in some countries — a big gap, statistically speaking.

And here's the description of the results for income:

There’s also a significant class-based difference, with respondents from the wealthiest households showing a greater proclivity toward overstatement than those from the poorest.

a) Here's a statistical validity question. When the authors say that the boys are about a half a standard deviation higher than girls in some countries and report it's "a big gap, statistically speaking," they are referring to effect size. In fact, if two groups are half a standard deviation apart, the effect size is probably about d = 0.5. Is that considered a small, medium, or large effect according to Cohen's conventions?

b) Here's an external validity question. What were the age and income levels of the students in the study? Reflect on the headline, which refers to "rich guys," and the photo accompanying the story, which shows an adult man. What do you think--what external validity issues are raised here?

c) Here's a construct validity question. The construct the journalist claims is being measured here is the "tendency to BS." How was that construct operationalized in this study? What do you think of that operationalization? Can you think of other ways to operationalize it?

d) You might think we're ready for an internal validity question. However, internal validity doesn't seem relevant here, because nobody seems to be making a causal claim....or are they? Could you make a causal claim here?

e) Advanced question: You might wish to investigate the table of data in the article that depicts the "BS gaps between boys and girls." Do you think that this table really needed separate columns for boys, girls, and "gap"? Why was it formatted this way? How else might you present these data?

f) At the end of the article, the scientists are quoted about the study's implications and limitations. One of the study's original authors says,

The authors also point out that their study was narrowly defined and restricted to the realm of mathematics. “Ideally,” they wrote, “future research should try to include a greater number of fake constructs in order to maximize precision of the bulls--- scale.”

Which of the four big validities seems to be addressed in this quote?