Normal Distribution and Percentiles

GRE Quantitative · Axiom Academy

LESSON Normal Distribution and Percentiles The bell curve, the 68-95-99.7 rule, and how to rank one value against a whole data set. A normal distribution is a symmetric, bell-shaped curve centered on the mean (μ). Data doesn't scatter evenly — it piles up near the mean and thins out symmetrically as you move away from it in either direction, measured in units of the standard deviation (σ). For any normal distribution, a fixed share of the data always falls within 1, 2, and 3 standard deviations of the mean — this is called the empirical rule . Watch each band shade in around the mean: A set of exam scores is normally distributed with mean and standard deviation . So about 68% of scores fall between 85 and 115, about 95% fall between 70 and 130, and about 99.7% fall between 55 and 145. A score of 130 sits right at the edge of the 95% band — only about 2.5% of scores are higher. A percentile describes a value's position relative to the rest of a data set — it does NOT require a normal distribution at all. The k th percentile is the value below which of the data falls. Ten students' scores, sorted: 50, 55, 60, 65, 70, 75, 80 , 85, 90, 95. Exactly 6 of the 10 scores are below 80, so: A score of 80 is at the 60th percentile — 60% of the class scored lower. 25th = first quartile (Q1) · 50th = median (Q2) · 75th = third quartile (Q3) A student at the 90th percentile scored better than 90% of the class — NOT the top 90%.

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