A percentile is a statistical measure indicating the value below which a given percentage of observations in a data set falls. For a data set sorted in ascending order, the p‑th percentile corresponds to the point at which p % of the data are less than or equal to that value, where p is a number between 0 and 100.
Definition and Calculation
- Ordered Data: Arrange the observations $x_1, x_2, \dots, x_n$ in non‑decreasing order.
- Rank Position: Compute the rank $R = \frac{p}{100},(n+1)$.
- Interpolation:
- If $R$ is an integer, the percentile is the value at position $R$.
- If $R$ is not an integer, interpolate linearly between the surrounding observations:
$$ \text{Percentile}(p) = x_{\lfloor R \rfloor} + (R - \lfloor R \rfloor),(x_{\lceil R \rceil} - x_{\lfloor R \rfloor}) $$
Various conventions exist for handling endpoints, ties, and small sample sizes; the method above is commonly used in software packages such as R (type 7) and Excel (PERCENTILE.INC).
Common Percentiles
- 25th percentile (first quartile, Q1) – the value below which 25 % of observations fall.
- 50th percentile (median, Q2) – the middle value; 50 % of observations are below it.
- 75th percentile (third quartile, Q3) – the value below which 75 % of observations fall.
- 90th, 95th, 99th percentiles – often used in performance testing, risk assessment, and health metrics.
Applications
- Education: Test scores are frequently reported as percentiles to compare a student’s performance with a reference population.
- Medicine: Growth charts for children use percentiles (e.g., the 5th and 95th percentiles) to assess development.
- Finance: Value‑at‑Risk (VaR) is expressed as a percentile of a loss distribution (e.g., the 95th percentile).
- Quality Control: Percentile ranks help identify outliers and set specification limits.
Related Concepts
- Quantile: A more general term for dividing a distribution into equally sized intervals; percentiles are the 100‑quantiles.
- Cumulative Distribution Function (CDF): The percentile corresponds to the inverse of the CDF for a given probability p/100.
- Z‑score: The standard normal equivalent of a percentile can be obtained via the inverse standard normal CDF.
Historical Note
The concept of dividing a data set into equal parts dates back to early descriptive statistics in the 19th century. The specific term “percentile” entered statistical literature in the early 20th century, parallel to the development of standardized testing and epidemiological growth charts.
Limitations
- Sample Size Sensitivity: In small samples, percentile estimates can be unstable and heavily dependent on the interpolation method used.
- Assumption of Order: Percentiles convey relative position but provide no information about the spacing between data points; distributions with identical percentiles can have markedly different shapes.
References
- Tukey, J. W. (1977). Exploratory Data Analysis. Addison‑Wesley.
- Hyndman, R. J., & Fan, Y. (1996). "Sample Quantiles in Statistical Packages". The American Statistician, 50(4), 361‑365.
- National Center for Health Statistics. (2000). Growth Charts. CDC.
