Which statement correctly defines the p-th percentile?

Which statement correctly defines the p-th percentile?

• A. The p-th percentile is the value x_p where the probability density function equals p/100.
• B. The p-th percentile is the value x_p where F(x_p) = p/100.
• C. The p-th percentile is the mean divided by p.
• D. The p-th percentile is the maximum value in the data set.

Answer: (B)

The p-th percentile is the value below which p percent of the data fall. In a distribution, this is defined using the cumulative distribution function F, where F(x) = P(X ≤ x). The p-th percentile is the value x_p that satisfies F(x_p) = p/100. This captures the idea of “splitting” the distribution at p percent. The intuition: if p = 50, you get the median; if p = 25, you get the value below which 25% of observations lie, and so on. The probability density function, by contrast, describes density, not cumulative probability, so equating it to p/100 does not define a percentile. The mean divided by p or the maximum value do not consistently identify a percentile across distributions.

The p-th percentile is the value below which p percent of the data fall. In a distribution, this is defined using the cumulative distribution function F, where F(x) = P(X ≤ x). The p-th percentile is the value x_p that satisfies F(x_p) = p/100. This captures the idea of “splitting” the distribution at p percent.

The intuition: if p = 50, you get the median; if p = 25, you get the value below which 25% of observations lie, and so on. The probability density function, by contrast, describes density, not cumulative probability, so equating it to p/100 does not define a percentile. The mean divided by p or the maximum value do not consistently identify a percentile across distributions.