Calculating a quantile seems straightforward. Sort your data, find the position corresponding to the desired percentile, and pick the value. But what happens when that position falls between two data points? The answer isn't a single operation, but a choice among several interpolation methods. The method you choose can subtly, and sometimes significantly, alter your results, especially with smaller datasets.

Most statistical software doesn't just pick one method at random. The various approaches were systematically classified by Rob J. Hyndman and Yanan Fan in 1996. They identified nine distinct formulas for estimating quantiles, many of which are now implemented in packages like R, NumPy, and Pandas. Understanding these types is crucial for any data scientist who needs to ensure their analysis is precise and reproducible.

The core of the problem lies in estimating the value for a quantile $p$ (where $0 \le p \le 1$) from a sample of size $N$. Let's denote the sorted sample as $x_1, x_2, ..., x_N$. The different methods first calculate an index, often a fractional one, based on $p$ and $N$. This index points to a location within the sorted data. If the index is an integer, the corresponding data point is used. If it's fractional, interpolation is required.

A common form for many of these estimators is a linear interpolation between two adjacent data points, $x_i$ and $x_{i+1}$:

The nine methods differ primarily in how they define the index and calculate $\gamma$. Some use a simple step-function approach (picking the 'lower' or 'higher' value), while others use linear interpolation. Let's visualize this with a simple example. Suppose we have the data {10, 20, 30, 40} and want to find the 40th percentile ($p=0.4$). Different methods will yield different results.

The differences arise from how each method defines the relationship between the quantile $p$ and the index in the sample. Below is a summary of all nine types.

With nine options, which one should you use? For most applications, the default Type 7 is a reasonable and widely understood choice. Its popularity in major software packages makes it a de facto standard, which helps with reproducibility. The formula for the index is $(N-1)p$, placing the 0th percentile at the first data point ($x_1$) and the 100th percentile at the last ($x_N$).

However, Hyndman and Fan's research showed that Type 8 provides an approximately unbiased estimate of the population quantile, regardless of the underlying distribution. This makes it a statistically robust choice, especially in formal research or when quantile accuracy is paramount. For normally distributed samples, Type 9 can also be a strong contender.

The distinction between methods also relates to the nature of your data. Step-function interpolations (like Types 1 and 2) are based on the empirical cumulative distribution function (ECDF), which treats the data discretely. Linear interpolation methods (like Types 4-9) implicitly assume the data comes from an underlying continuous distribution.

So, while you can often rely on the software default, a deeper understanding of quantile estimation allows you to make a more informed choice. When reporting results, especially with smaller datasets, it can be valuable to specify which interpolation method was used. This adds a layer of precision and transparency to your analysis.

Ultimately, quantiles are powerful tools for understanding distributions. Knowing how they are calculated under the hood empowers you to use them more effectively.