Pie chart

A pie chart encodes each category's relative frequency as a share of one circle, so a category's angle is its relative frequency times 360 degrees. The whole circle stands for every observational unit exactly once, which forces the categories to be mutually exclusive and exhaustive with shares summing to 1.

If 40 of 200 students choose band, that slice is $40 / 200 = 0.20$ of the circle, or $0.20 \times 360 = 72$ degrees. Each percentage point is worth $360 / 100 = 3.6$ degrees, so two categories three points apart are separated by only 10.8 degrees out of 360. Drawn as bars, those two categories stand three units apart against a labeled scale and the gap is read straight off, which is the case against the pie chart in one line.

The misreading is about counts. "School A's band slice is bigger than school B's, so school A has more band students." The geometry of a pie chart encodes shares, never counts. Thirty percent of school A's 100 students is 30 students in a 108-degree slice, while 25 percent of school B's 400 students is 100 students in a 90-degree slice. A's slice is visibly larger and A has 70 fewer band students. Two pies are comparable only when both totals are stated, and even then you are comparing proportions.

The circle discards $n$ entirely: 40 of 200 and 4 of 20 draw the identical 72-degree slice. The exhaustive requirement bites as well. Ask students to check every sport they play and the shares sum past 1, so those responses have no valid pie chart at all, only a bar graph of the separate percentages.

A pie chart of a quantitative variable is a category error, because that is a histogram. Bar charts and pie charts are topic 1.4, Graphical Representations for One Categorical Variable.