Quantiles are cut points that divide a distribution by probability rather than by equal distances on the number line. They help describe where a value sits relative to the rest of the data, such as the median, quartiles, and percentiles. This matters because many real data sets are skewed or uneven, so typical positions are often more informative than raw distances.
Quantiles are used in test scores, income studies, weather records, medical measurements, and risk analysis.
The cumulative distribution function, or CDF, gives the probability that a random variable is less than or equal to a chosen value. The quantile function reverses this idea by taking a probability p and returning the data value x that has that much area to its left. On a graph, a quantile can be read by moving from a probability on the CDF up to the curve, then across to the x value.
For continuous distributions this is often written Q(p) = F^-1(p), where F is the CDF.
Key Facts
- A p-quantile is a value x such that P(X ≤ x) = p for a continuous distribution.
- The cumulative distribution function is F(x) = P(X ≤ x).
- The quantile function is Q(p) = F^-1(p), where 0 ≤ p ≤ 1.
- The median is the 0.50 quantile, so Q(0.50) is the middle value by probability.
- Quartiles are Q1 = Q(0.25), Q2 = Q(0.50), and Q3 = Q(0.75).
- A percentile converts probability to a percent: the 90th percentile is Q(0.90).
Vocabulary
- Quantile
- A quantile is a value that marks a specified cumulative probability in a distribution.
- Percentile
- A percentile is a quantile expressed as a percent, such as the 80th percentile for p = 0.80.
- Quartile
- A quartile is one of the three cut points that divide ordered data into four groups with about equal probability.
- Cumulative distribution function
- The cumulative distribution function F(x) gives the probability that a random variable is less than or equal to x.
- Quantile function
- The quantile function Q(p) gives the value x whose cumulative probability is p.
Common Mistakes to Avoid
- Confusing a percentile with a percent score: a percentile describes relative position in a distribution, not the fraction of questions answered correctly.
- Reading area to the right instead of area to the left: most CDFs and quantile functions use P(X ≤ x), so p is cumulative probability from the left.
- Assuming quartiles are equally spaced on the x-axis: quartiles split probability into equal parts, but the numerical distances between Q1, Q2, and Q3 can be unequal.
- Using Q(p) with p written as a whole percent like 75 instead of 0.75: the quantile function usually expects a probability between 0 and 1.
Practice Questions
- 1 For a distribution with CDF F(x) = x/20 on 0 ≤ x ≤ 20, find Q(0.25), Q(0.50), and Q(0.90).
- 2 A data set in sorted order is 4, 6, 7, 10, 12, 15, 18, 21, 24. Using the median of the full ordered list, find the 0.50 quantile.
- 3 A distribution is strongly right-skewed. Explain why the distance from Q(0.50) to Q(0.75) might be smaller than the distance from Q(0.75) to Q(0.95), even though both intervals cover probability ranges.