Quantile

What Is Quantile?

A quantile is a statistical term used in quantitative analysis that divides the range of a probability distribution into continuous intervals with equal probabilities. Essentially, quantiles are cut points that partition a data set into groups of roughly equal size. While often used interchangeably, common quantiles like quartiles, deciles, and percentiles offer more granular insights into data distribution than simple averages. For instance, the median is a specific type of quantile, representing the 0.5 (or 50%) quantile, dividing a data set into two equal halves. Understanding quantiles is crucial for comprehending the spread and characteristics of data, extending beyond just central tendency.

History and Origin

The concept of quantiles, particularly the percentile, has roots in the late 19th century through the work of Sir Francis Galton. A polymath with significant contributions to statistical inference, Galton introduced the term "percentile" in 1885 while conducting his extensive studies on heredity.²⁴,²³ His pioneering efforts laid the groundwork for modern statistical methods, including concepts like correlation and regression, that revolutionized how data in various scientific fields, including early social sciences, were analyzed.²²,²¹ Galton's work emphasized the importance of understanding the distribution of data points, not just their average, which directly led to the development and adoption of quantiles as fundamental financial metrics.

Key Takeaways

• Quantiles divide a data set or probability distribution into segments containing an equal proportion of observations.
• Common types of quantiles include the median (0.5 quantile), quartile (0.25, 0.50, 0.75 quantiles), and decile (0.10, 0.20, ..., 0.90 quantiles).
• They provide a more comprehensive view of data spread and distribution compared to single summary statistics like the mean.
• Quantiles are widely used in finance for risk management, performance evaluation, and understanding income distribution.
• The method of calculating quantiles can vary slightly, especially for smaller data sets, which can lead to minor differences in results across different statistical software.

Formula and Calculation

For a given data set sorted in ascending order, the k-th quantile ($Q_k$) can be calculated. While there are several methods, a common approach involves finding the position of the quantile within the ordered data.

Let $N$ be the total number of data points in the set.
To find the position ($P$) of the $p$-th quantile (where $p$ is the desired quantile as a decimal, e.g., 0.25 for the first quartile, 0.50 for the median, 0.90 for the ninth decile):

Once $P$ is calculated:

• If $P$ is an integer, the quantile is the value at that position in the sorted data.
• If $P$ is not an integer, the quantile is found by interpolating between the two surrounding data points. For example, if $P = X.Y$, where $X$ is the integer part and $Y$ is the fractional part, the quantile is given by:

Here, $\text{Data}[X]$ refers to the value at the $X$-th position in the sorted data set, and $\text{Data}[X+1]$ is the value at the $(X+1)$-th position. This formula helps determine a specific point within the range of a probability distribution.

Interpreting the Quantile

Interpreting a quantile provides insights into the relative standing of a data point within a distribution. For example, if an investment return falls at the 0.75 quartile, it means that 75% of the observed returns were at or below that value, and 25% were above it. This offers a clearer picture than just knowing the average return, especially when analyzing distributions that are not symmetric. In data analysis, quantiles help identify central tendencies, spread, and potential outliers without being heavily influenced by extreme values. They are particularly useful for understanding the dispersion of data and how different segments of a population or data set perform.

Hypothetical Example

Consider a hypothetical portfolio's annual returns over 10 years, sorted in ascending order:

To find the 0.75 quantile (or third quartile) for this data set ($N=10$):

1. Calculate the position ($P$):
   $P = 0.75 \times (10 + 1) = 0.75 \times 11 = 8.25$

2. Since $P$ is not an integer, we interpolate between the 8th and 9th values.

   • The 8th value (Data²⁰) is 9.0%.
   • The 9th value (Data¹⁹) is 10.2%.
   • The fractional part ($Y$) is 0.25.

   $Q_{0.75} = \text{Data}¹⁸ + 0.25 \times (\text{Data}¹⁷ - \text{Data}¹⁶)$
   $Q_{0.75} = 9.0 + 0.25 \times (10.2 - 9.0)$
   $Q_{0.75} = 9.0 + 0.25 \times (1.2)$
   $Q_{0.75} = 9.0 + 0.3$
   $Q_{0.75} = 9.3%$

This indicates that 75% of the portfolio's annual returns were 9.3% or less over this 10-year period. This specific quantile provides a more nuanced understanding of performance than a simple average return.

Practical Applications

Quantiles are indispensable tools across various financial disciplines. In risk management, they are fundamental to calculating measures like Value at Risk (VaR), which estimates the maximum potential loss of a portfolio over a specific timeframe at a given confidence level, typically expressed as a high quantile (e.g., 95th or 99th percentile).¹⁵,¹⁴ This allows financial institutions to set capital requirements and assess their exposure to market risk. The Basel Committee on Banking Supervision, for instance, uses quantile-based measures like stressed VaR in its regulatory frameworks for banks to ensure adequate capital buffers.¹³

Beyond risk, quantiles are used in financial modeling to understand the distribution of asset returns, predict financial crises, and optimize portfolios.¹²,¹¹ They are also vital in analyzing income inequality and wealth distribution, with organizations like the Federal Reserve using income percentiles to track economic disparities.¹⁰,⁹ This helps policymakers and economists understand how income and wealth are distributed across different segments of the population.

Limitations and Criticisms

While highly versatile, quantiles, especially when applied in measures like Value at Risk, have certain limitations. One significant criticism of VaR is that it does not provide information about the potential magnitude of losses that exceed the specified confidence level, meaning it doesn't quantify "tail risk" or what happens in extreme market events.⁸,⁷,⁶ For example, a 99% VaR tells you the loss not expected to be exceeded 99% of the time, but it doesn't tell you if the 1% of times it is exceeded, the loss could be catastrophic.⁵

Furthermore, VaR may not always encourage proper diversification as it is not "subadditive," meaning the VaR of a combined portfolio could theoretically be greater than the sum of the Va VaRs of its individual components, which contradicts the principle that diversification reduces overall risk.⁴ In regression analysis using quantiles (quantile regression), interpretability can be challenging when comparing coefficients across different quantiles, and the computational complexity can be higher than traditional regression techniques.³,² Also, calculating quantiles for small data sets can sometimes lead to varying results depending on the specific method used.¹

Quantile vs. Percentile

The terms "quantile" and "percentile" are intimately related and often used interchangeably, but it's helpful to understand their specific relationship. A quantile is a general term for any point that divides a distribution into equal parts. Percentiles are a specific type of quantile, where a data set is divided into 100 equal parts. For example, the 25th percentile is the 0.25 quantile, the 50th percentile is the 0.50 quantile (also known as the median), and the 90th percentile is the 0.90 quantile.,

Therefore, all percentiles are quantiles, but not all quantiles are necessarily percentiles. Other common quantiles include quartiles (dividing data into four parts), deciles (dividing into ten parts), and quintiles (dividing into five parts). The confusion often arises because the underlying mathematical concept is the same: identifying points in a distribution below which a certain proportion of data falls.

FAQs

What is the primary purpose of using quantiles?

The primary purpose of using quantiles is to understand the spread and distribution of a data set more thoroughly than traditional averages. They help identify points below which a certain percentage of data falls, providing insights into variations and extremes.

How do quantiles differ from the mean?

The mean (average) represents the central tendency of a data set. A quantile, conversely, identifies a specific point in the data's ordered distribution, dividing it into segments of equal probability. For example, the median is a quantile that shows the exact middle point, which can be more representative than the mean in skewed distributions.

Can quantiles be used with any type of data?

Quantiles are most effectively used with numerical data that can be ordered, such as financial returns, income figures, or asset prices. They provide meaningful insights into the distribution of continuous or ordinal data. For data analysis of categorical data, other statistical measures are generally more appropriate.

Why is the 99th percentile relevant in finance?

In finance, the 99th percentile is often used in risk management to calculate Value at Risk (VaR). This represents a point where, with 99% confidence, losses are not expected to exceed a certain value. It helps institutions quantify potential downsides and adhere to regulatory capital requirements.