3.B The Normal Curve and Percentile Rank

Measure almost any human trait across a large group — height, reaction time, test performance — and most people cluster near the average, with fewer and fewer people as you move toward the extremes in either direction. That produces a symmetric, bell-shaped curve: the normal curve. It shows up this often because most traits are the sum of many small, independent influences (genes, environment, chance), and adding up lots of independent factors tends to produce this exact shape.

The 68-95-99.7 rule carves that bell into bands based on standard deviations from the mean. About 68 percent of scores fall within one standard deviation of the mean, about 95 percent fall within two, and about 99.7 percent fall within three. Because the curve is symmetric, you can split each band in half to find percentile rank — the percentage of scores at or below a given point. A score exactly at the mean sits at the 50th percentile; a score one standard deviation above it sits near the 84th, since 50 percent lie below the mean and another 34 percent (half of the middle 68 percent) lie between the mean and that score.

Not every distribution is normal. Skewed distributions pile up on one side with a long tail trailing off the other — the tail names the skew. Bimodal distributions have two humps, hinting that two distinct groups got lumped into one dataset, which can distort or destroy the clean bell shape. This matters directly for intelligence testing: IQ scores are built to form a normal curve with a fixed mean and standard deviation, so a person's raw score only means something once it's placed on that curve and turned into a percentile.