Percentiles

Precise definition

The $p$th percentile is a value at or below which approximately $p\%$ of observations fall, with the exact finite-sample definition depending on convention. More formally, a population $p$-quantile can be defined as $q_p=\inf\{x:F(x)\ge p\}$.

Notation and mathematical language

A percentile is a value; percentile rank describes a value's relative position. The median is the 50th percentile, and quartiles correspond roughly to 25%, 50%, and 75%. Software methods differ in index formulas and interpolation, especially for small samples.

Conceptual picture

Percentiles convert position in an ordered distribution into a common 0–100 scale. They are robust to extreme magnitudes because they use order, but they do not show how far apart neighbouring scores are.

Conditions and key results

Ties can produce ranges of percentile ranks. Percentiles from different populations, years, or tests are comparable only if reference distributions and definitions align. Being at the 90th percentile does not mean scoring 90% or being 90% better.

A reliable strategy

1. Order the data and state the percentile convention or software method.
2. Convert $p$ to the prescribed rank and interpolate only as that convention directs.
3. Report the percentile value with the reference population and time period.
4. Check by counting observations below and at the value and note ties or small-sample ambiguity.

Fully worked example

Interpretation and application

Percentiles report growth charts, test norms, response times, and incomes. They describe rank within a reference distribution; causal or ability claims require much more evidence, and norm changes can shift ranks without individual change.

Common mistakes

Verification and reasonableness

• Count ordered positions on both sides of the reported value.
• Compare results from two documented algorithms and explain any discrepancy.
• Use a box plot or empirical cumulative distribution as a visual check.

Practice

1. What percentile is the median?
2. Does 90th percentile mean 90% correct?
3. Why may software disagree?

Further deduction

For a continuous strictly increasing distribution, $q_p=F^{-1}(p)$ and $P(X\le q_p)=p$. With atoms or flat portions, an inverse may not be unique and the infimum definition handles jumps. This distinction explains ties and why exact equality to $p$ can fail for discrete data.

Percentile changes can be unintuitive. Moving from the 50th to 60th percentile means rank changed relative to the reference group, not that the raw value rose 10% or by a fixed number of units. In a dense centre the raw change may be small; in a sparse tail the same percentile shift may require a much larger raw change.

Sample percentile conventions differ when the desired rank lies between observations. Nearest-rank, linear interpolation, and statistical software defaults can return different values for a small dataset while all preserve the ordering idea. A reproducible report names the convention or software and sample size rather than presenting the last decimal as universal. For large smooth samples the differences often shrink, but at decision thresholds a convention change can alter classification.

Related topics

• Box Plots
• Z-Scores
• Frequency Tables