05quantile

The 0.05 quantile, also known as the 5th percentile, is a statistical threshold representing the value below which 5% of observations in a given distribution fall. It is the p-th quantile with p equal to 0.05, and it can be defined as the inverse of the cumulative distribution function: q0.05 = F^(-1)(0.05). In some contexts, the term 05quantile is used informally to denote this same value.

In a continuous distribution, the 0.05 quantile is unique. In discrete or heavily tied data, several nearby values may satisfy the percentile criterion, and interpolation between data points is often used to obtain a precise estimate. The practical estimate of the 0.05 quantile from a sample relies on order statistics and chosen interpolation rules, which can differ across software packages.

How the 0.05 quantile is estimated varies by method. Common approaches include empirical quantiles computed directly from sorted data and interpolated quantiles that blend adjacent order statistics. Software implementations come with different default formulas, commonly referred to as quantile types or methods in statistical languages such as R, Python, and Julia.

Interpretation and applications: the 0.05 quantile marks the lower tail of the distribution and is frequently used in risk assessment, quality control, and thresholding decisions. For a normal distribution with mean μ and standard deviation σ, the 0.05 quantile is μ + z0.05·σ, where z0.05 ≈ -1.64485. This value helps describe how far an observation is from the center in the lower tail.

Limitations: quantiles are robust to outliers and summarize tail behavior, but their estimation can be imprecise in small samples. Discrete data and ties can complicate interpretation, and different interpolation methods yield slight variations in q0.05.