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Univariate Statistics

Source: R/univariate_statistics.R
univariate_statistics.Rd

Functions to compute robust univariate statistics from a dataset.

Central Tendency:

  • mean_boost: Computes the arithmetic mean using Higham's numerically stable algorithm.

  • median_boost: Computes the median (robust to outliers).

  • mode: Computes the mode(s) of the dataset.

Dispersion (Spread):

  • variance: Computes the population variance using Higham's algorithm.

  • sample_variance: Computes the sample variance (unbiased estimator).

  • mean_and_sample_variance: Efficiently computes both mean and sample variance in one pass.

  • median_absolute_deviation: Computes the Median Absolute Deviation (MAD), a robust measure of variability.

  • interquartile_range: Computes the Interquartile Range (IQR = Q3 - Q1), robust to outliers.

Shape:

  • skewness: Measures the asymmetry of the distribution (Pebay's algorithm).

  • kurtosis: Measures the "tailedness" of the distribution (Pebay's algorithm).

  • excess_kurtosis: Kurtosis minus 3 (Normal distribution has 0 excess kurtosis).

  • first_four_moments: Computes Mean, Variance, Skewness, and Kurtosis in a single pass.

Inequality:

  • gini_coefficient: Computes the Gini coefficient (population). range \([0, 1 - 1/n]\).

  • sample_gini_coefficient: Computes the sample Gini coefficient. range \([0, 1]\).

Usage

mean_boost(x)

# Default S3 method
variance(x, ...)

sample_variance(x)

mean_and_sample_variance(x)

# Default S3 method
skewness(x, ...)

# Default S3 method
kurtosis(x, ...)

excess_kurtosis(x)

first_four_moments(x)

median_boost(x)

median_absolute_deviation(x)

interquartile_range(x)

gini_coefficient(x)

sample_gini_coefficient(x)

# Default S3 method
mode(x, ...)

Arguments

x

A numeric vector containing the dataset.

...

Additional arguments (for S3 compatibility, e.g., with defaults).

Value

A numeric value (or vector for moments/mode) with the computed statistic.

Details

These functions are designed to be numerically stable and efficient. Most implementations follow algorithms described by Higham (Accuracy and Stability of Numerical Algorithms) or Pebay (Sandia Labs) for one-pass parallel computation.

References

Higham, N. J. (2002). Accuracy and stability of numerical algorithms. SIAM. Pebay, P. P. (2008). Formulas for Robust, One-Pass Parallel Computation of Covariances and Arbitrary-Order Statistical Moments. Sandia Report.

See also

Boost Documentation

Examples

data <- c(1, 2, 3, 4, 100) # Dataset with an outlier

# --- Central Tendency ---
mean_boost(data)
#> [1] 22
median_boost(data) # Less affected by 100
#> [1] 3
mode(c(1, 2, 2, 3))
#> [1] 2

# --- Dispersion ---
variance(data)
#> [1] 1522
sample_variance(data)
#> [1] 1902.5
median_absolute_deviation(data) # Robust
#> [1] 1
interquartile_range(data)       # Robust
#> [1] 50.5

# --- Shape ---
skewness(data)
#> [1] 1.497537
excess_kurtosis(data)
#> [1] 0.2467165
first_four_moments(data)
#> [1]      22    1522   88920 7520967

# --- Inequality ---
gini_coefficient(c(1, 0, 0, 0)) # High inequality
#> [1] 0.75
# Gini Coefficient
gini_coefficient(c(1, 2, 3, 4, 5))
#> [1] 0.2666667
# Sample Gini Coefficient
sample_gini_coefficient(c(1, 2, 3, 4, 5))
#> [1] 0.3333333
# Mode
mode(c(1, 2, 2, 3, 4))
#> [1] 2