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Sample quantiles

Sample Quantiles: Definition, Formula, Example, and FAQs

What Is Sample Quantiles?

Sample quantiles are values that divide a data set into equal-sized, ordered subgroups, providing a snapshot of the probability distribution of the observed data. They are a fundamental concept within statistical analysis, a branch of quantitative finance. Unlike the theoretical quantiles of a known population distribution, sample quantiles are derived directly from a specific collection of empirical observations. They help to understand the spread and central tendency of data without assuming a particular underlying distribution, making them a crucial tool in non-parametric statistics.

History and Origin

The concepts underlying quantiles can be traced back to the late 19th and early 20th centuries, deeply intertwined with the pioneering work of Sir Francis Galton. Galton, a prominent Victorian polymath, made significant contributions to the field of statistics, including coining terms like "quartile," "decile," and "percentile" around 1885. His work on heredity and anthropometrics necessitated methods to divide and analyze large datasets, leading to the formalization of these distributional measures. While not explicitly using the term "sample quantile" in its modern form, Galton's developments laid the groundwork for the empirical estimation of these statistical divisions4.

Key Takeaways

  • Sample quantiles divide an ordered data set into equal proportions, offering insights into its distribution.
  • They are estimated from observed data, providing a practical tool for data analysis in various fields, including finance.
  • Common examples include the median (0.5 quantile), quartiles (0.25, 0.5, 0.75 quantiles), deciles (0.1, 0.2, ..., 0.9 quantiles), and percentiles (0.01, 0.02, ..., 0.99 quantiles).
  • Their calculation involves ordering the data and identifying values at specific positions, often requiring interpolation for non-integer ranks.
  • Sample quantiles are widely used in risk management, economic analysis, and performance evaluation.

Formula and Calculation

Calculating a sample quantile involves two primary steps: ordering the data and then determining the position of the quantile.
Given a sorted data set (X_1, X_2, \dots, X_n), where (X_1 \le X_2 \le \dots \le X_n), the (p^{th}) sample quantile (where (0 < p < 1)) is typically estimated as follows:

  1. Calculate the index: Compute (k = p \times (n - 1) + 1).

    • (p) = the desired quantile (e.g., 0.25 for the first quartile, 0.50 for the median).
    • (n) = the total number of observations in the sample.
    • (k) = the calculated rank or position.

  2. Determine the quantile value:

    • If (k) is an integer, the sample quantile is simply the value (X_k).
    • If (k) is not an integer, it means the quantile falls between two observations. In this common scenario, interpolation is used. A widely accepted method is linear interpolation:
      Let (i = \lfloor k \rfloor) (the integer part of (k)) and (f = k - i) (the fractional part of (k)).
      The (p^{th}) sample quantile, denoted as (Q_p), is calculated as:
      Qp=Xi+f×(Xi+1−Xi)Q_p = X_i + f \times (X_{i+1} - X_i)
      This method is one of several standard approaches for estimating sample quantiles, ensuring that a quantile can be found even when its position does not align perfectly with an existing data point. This calculation relies on the concept of order statistics.

Interpreting the Sample Quantiles

Interpreting sample quantiles provides crucial insights into the distribution and characteristics of a data set. For instance, if the 0.25 sample quantile (first quartile) for a stock's daily returns is -0.5%, it means 25% of the observed daily returns were -0.5% or lower. Similarly, the 0.75 sample quantile (third quartile) at +1.0% indicates that 75% of returns were +1.0% or lower, meaning 25% were higher.

The most common sample quantile is the median, which is the 0.5 sample quantile. It represents the middle value of a data set, with half the observations falling below it and half above, making it robust to outliers. The difference between the 0.75 and 0.25 sample quantiles is known as the interquartile range, which is a measure of statistical dispersion, describing the spread of the central 50% of the data. By examining various sample quantiles, one can reconstruct the approximate shape of the empirical distribution of the data, highlighting skewness or the presence of heavy tails.

Hypothetical Example

Consider a small portfolio of 10 different stock returns over a specific quarter, sorted in ascending order:
Returns (in %): ([-3.2, -2.5, -1.8, -0.7, 0.1, 0.5, 1.2, 1.9, 2.8, 3.5])

We want to find the 0.90 sample quantile (the 90th percentile) to understand the upper end of the return distribution.

  1. Calculate the index (k):
    (n = 10) (number of observations)
    (p = 0.90) (desired quantile)
    (k = 0.90 \times (10 - 1) + 1 = 0.90 \times 9 + 1 = 8.1 + 1 = 9.1)

  2. Determine the quantile value using interpolation:
    Since (k = 9.1) is not an integer, we use linear interpolation.
    (i = \lfloor 9.1 \rfloor = 9)
    (f = 9.1 - 9 = 0.1)

    The 9th observation ((X_9)) is 2.8%.
    The 10th observation ((X_{10})) is 3.5%.

    (Q_{0.90} = X_9 + f \times (X_{10} - X_9))
    (Q_{0.90} = 2.8 + 0.1 \times (3.5 - 2.8))
    (Q_{0.90} = 2.8 + 0.1 \times 0.7)
    (Q_{0.90} = 2.8 + 0.07)
    (Q_{0.90} = 2.87%)

This means that 90% of the observed quarterly returns were 2.87% or lower, based on this sample. This helps in understanding the distribution of returns and potential upward performance.

Practical Applications

Sample quantiles are extensively applied across finance, economics, and various data-driven fields.

  • Risk Management: In risk management, sample quantiles are essential for calculating risk measures such as Value at Risk (VaR). VaR, often specified as a high quantile (e.g., the 99th percentile), estimates the maximum potential loss over a specific period at a given confidence level. For example, a 99% VaR of $1 million implies that, under normal market conditions, there is a 1% chance of losing more than $1 million. Financial regulations, such as those set by the Basel Committee on Banking Supervision, incorporate VaR for determining regulatory capital requirements for banks to absorb unexpected losses3.
  • Economic Analysis: Economists and policymakers use quantiles to analyze and report on the distribution of income, wealth, and consumption within a population. For instance, the Federal Reserve provides data on the U.S. household wealth distribution by different percentile groups, revealing disparities in asset holdings across various segments of society2.
  • Performance Evaluation: In investment, sample quantiles can be used to compare the performance of funds or portfolios against peer groups. A fund manager might be interested in knowing if their fund's returns fall into the top quartiles relative to similar funds, providing a benchmark for performance.
  • Financial Modeling: Quantiles are crucial in stress testing and scenario analysis, helping modelers understand the potential impact of extreme market movements.

Limitations and Criticisms

While invaluable, sample quantiles come with certain limitations and criticisms:

  • Sensitivity to Sample Size: The accuracy of sample quantiles heavily depends on the size and representativeness of the data set. Small samples can lead to highly variable and imprecise quantile estimates, particularly for extreme quantiles (those very close to 0 or 1), where data points are sparse.
  • Estimation Methods: There are multiple methods for calculating sample quantiles, especially when the desired quantile falls between two data points (requiring interpolation). Different interpolation methods can yield slightly different results, which can be problematic in applications requiring high precision.
  • Lack of Smoothness: Empirical quantile functions are step functions, making them non-smooth. This can be a challenge for certain statistical techniques that assume continuity or differentiability.
  • Model Uncertainty: When sample quantiles are used to estimate parameters of a theoretical distribution or to inform financial modeling (e.g., in Value at Risk calculations), there is inherent model uncertainty. Incorrect modeling assumptions about the underlying data-generating process can lead to wrong quantile estimates, especially for extreme tail events1. This underscores the importance of rigorous model validation and understanding the data's true characteristics.

Sample Quantiles vs. Percentile

The terms "sample quantile" and "percentile" are often used interchangeably, but it is important to clarify their relationship. A percentile is a specific type of quantile, expressed as a percentage. For example, the 90th percentile is the 0.90 sample quantile. Therefore, all percentiles are quantiles, but not all quantiles are typically referred to as percentiles (e.g., the median is the 0.5 quantile, or 50th percentile).

The main point of confusion arises because "quantile" is a more general term that includes all ways to divide a data set into equal proportions, such as quartiles (divisions into four parts), deciles (divisions into ten parts), and percentiles (divisions into 100 parts). Percentiles explicitly state the proportion as a percentage (e.g., 25%, 75%). When discussing general divisions of data, "sample quantiles" is the more encompassing and precise statistical term.

FAQs

Q: What is the main purpose of calculating sample quantiles?
A: The main purpose of calculating sample quantiles is to understand the distribution of observed data without making assumptions about its underlying probability distribution. They provide specific values that divide the data set into equal proportions, revealing insights into central tendency, spread, and the presence of extreme values.

Q: How do sample quantiles differ from theoretical quantiles?
A: Sample quantiles are derived directly from actual observed data and are therefore estimates. Theoretical quantiles, on the other hand, are derived from a known or assumed theoretical probability distribution (e.g., the normal distribution) for an entire population. Sample quantiles approximate the theoretical quantiles based on the available data analysis.

Q: Can sample quantiles be used for forecasting?
A: While sample quantiles themselves are descriptive statistics of past data, they are widely used in models that inform forecasting, especially in risk management. For example, historical VaR, which relies on past sample quantiles of losses, can be used to project potential future losses under similar conditions. However, they do not inherently predict future values but rather characterize the distribution of historical outcomes.

Q: Why is interpolation sometimes needed when calculating sample quantiles?
A: Interpolation is needed when the calculated position for a specific sample quantile (like the 0.25 [quartiles] or 0.90 [percentile]) falls between two actual data points in the sorted list. Since the data is discrete, interpolation provides a statistically consistent way to estimate the value that would logically represent that quantile's position if the data were continuous.