stats.rs

// Copyright 2012 The Rust Project Developers. See the COPYRIGHT
// file at the top-level directory of this distribution and at
// http://rust-lang.org/COPYRIGHT.
//
// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
// <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your
// option. This file may not be copied, modified, or distributed
// except according to those terms.

#![allow(missing_docs)]
#![allow(deprecated)] // Float

use std::cmp::Ordering::{self, Equal, Greater, Less};
use std::mem;

fn local_cmp(x: f64, y: f64) -> Ordering {
    // arbitrarily decide that NaNs are larger than everything.
    if y.is_nan() {
        Less
    } else if x.is_nan() {
        Greater
    } else if x < y {
        Less
    } else if x == y {
        Equal
    } else {
        Greater
    }
}

fn local_sort(v: &mut [f64]) {
    v.sort_by(|x: &f64, y: &f64| local_cmp(*x, *y));
}

/// Trait that provides simple descriptive statistics on a univariate set of numeric samples.
pub trait Stats {
    /// Sum of the samples.
    ///
    /// Note: this method sacrifices performance at the altar of accuracy
    /// Depends on IEEE-754 arithmetic guarantees. See proof of correctness at:
    /// ["Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates"]
    /// (http://www.cs.cmu.edu/~quake-papers/robust-arithmetic.ps)
    fn sum(&self) -> f64;

    /// Minimum value of the samples.
    fn min(&self) -> f64;

    /// Maximum value of the samples.
    fn max(&self) -> f64;

    /// Arithmetic mean (average) of the samples: sum divided by sample-count.
    ///
    /// See: https://en.wikipedia.org/wiki/Arithmetic_mean
    fn mean(&self) -> f64;

    /// Median of the samples: value separating the lower half of the samples from the higher half.
    /// Equal to `self.percentile(50.0)`.
    ///
    /// See: https://en.wikipedia.org/wiki/Median
    fn median(&self) -> f64;

    /// Variance of the samples: bias-corrected mean of the squares of the differences of each
    /// sample from the sample mean. Note that this calculates the _sample variance_ rather than the
    /// population variance, which is assumed to be unknown. It therefore corrects the `(n-1)/n`
    /// bias that would appear if we calculated a population variance, by dividing by `(n-1)` rather
    /// than `n`.
    ///
    /// See: https://en.wikipedia.org/wiki/Variance
    fn var(&self) -> f64;

    /// Standard deviation: the square root of the sample variance.
    ///
    /// Note: this is not a robust statistic for non-normal distributions. Prefer the
    /// `median_abs_dev` for unknown distributions.
    ///
    /// See: https://en.wikipedia.org/wiki/Standard_deviation
    fn std_dev(&self) -> f64;

    /// Standard deviation as a percent of the mean value. See `std_dev` and `mean`.
    ///
    /// Note: this is not a robust statistic for non-normal distributions. Prefer the
    /// `median_abs_dev_pct` for unknown distributions.
    fn std_dev_pct(&self) -> f64;

    /// Scaled median of the absolute deviations of each sample from the sample median. This is a
    /// robust (distribution-agnostic) estimator of sample variability. Use this in preference to
    /// `std_dev` if you cannot assume your sample is normally distributed. Note that this is scaled
    /// by the constant `1.4826` to allow its use as a consistent estimator for the standard
    /// deviation.
    ///
    /// See: http://en.wikipedia.org/wiki/Median_absolute_deviation
    fn median_abs_dev(&self) -> f64;

    /// Median absolute deviation as a percent of the median. See `median_abs_dev` and `median`.
    fn median_abs_dev_pct(&self) -> f64;

    /// Percentile: the value below which `pct` percent of the values in `self` fall. For example,
    /// percentile(95.0) will return the value `v` such that 95% of the samples `s` in `self`
    /// satisfy `s <= v`.
    ///
    /// Calculated by linear interpolation between closest ranks.
    ///
    /// See: http://en.wikipedia.org/wiki/Percentile
    fn percentile(&self, pct: f64) -> f64;

    /// Quartiles of the sample: three values that divide the sample into four equal groups, each
    /// with 1/4 of the data. The middle value is the median. See `median` and `percentile`. This
    /// function may calculate the 3 quartiles more efficiently than 3 calls to `percentile`, but
    /// is otherwise equivalent.
    ///
    /// See also: https://en.wikipedia.org/wiki/Quartile
    fn quartiles(&self) -> (f64, f64, f64);

    /// Inter-quartile range: the difference between the 25th percentile (1st quartile) and the 75th
    /// percentile (3rd quartile). See `quartiles`.
    ///
    /// See also: https://en.wikipedia.org/wiki/Interquartile_range
    fn iqr(&self) -> f64;
}

/// Extracted collection of all the summary statistics of a sample set.
#[derive(Clone, PartialEq)]
#[allow(missing_docs)]
pub struct Summary {
    pub sum: f64,
    pub min: f64,
    pub max: f64,
    pub mean: f64,
    pub median: f64,
    pub var: f64,
    pub std_dev: f64,
    pub std_dev_pct: f64,
    pub median_abs_dev: f64,
    pub median_abs_dev_pct: f64,
    pub quartiles: (f64, f64, f64),
    pub iqr: f64,
}

impl Summary {
    /// Construct a new summary of a sample set.
    pub fn new(samples: &[f64]) -> Summary {
        Summary {
            sum: samples.sum(),
            min: samples.min(),
            max: samples.max(),
            mean: samples.mean(),
            median: samples.median(),
            var: samples.var(),
            std_dev: samples.std_dev(),
            std_dev_pct: samples.std_dev_pct(),
            median_abs_dev: samples.median_abs_dev(),
            median_abs_dev_pct: samples.median_abs_dev_pct(),
            quartiles: samples.quartiles(),
            iqr: samples.iqr(),
        }
    }
}

impl Stats for [f64] {
    // FIXME #11059 handle NaN, inf and overflow
    fn sum(&self) -> f64 {
        let mut partials = vec![];

        for &x in self {
            let mut x = x;
            let mut j = 0;
            // This inner loop applies `hi`/`lo` summation to each
            // partial so that the list of partial sums remains exact.
            for i in 0..partials.len() {
                let mut y: f64 = partials[i];
                if x.abs() < y.abs() {
                    mem::swap(&mut x, &mut y);
                }
                // Rounded `x+y` is stored in `hi` with round-off stored in
                // `lo`. Together `hi+lo` are exactly equal to `x+y`.
                let hi = x + y;
                let lo = y - (hi - x);
                if lo != 0.0 {
                    partials[j] = lo;
                    j += 1;
                }
                x = hi;
            }
            if j >= partials.len() {
                partials.push(x);
            } else {
                partials[j] = x;
                partials.truncate(j + 1);
            }
        }
        let zero: f64 = 0.0;
        partials.iter().fold(zero, |p, q| p + *q)
    }

    fn min(&self) -> f64 {
        assert!(!self.is_empty());
        self.iter().fold(self[0], |p, q| p.min(*q))
    }

    fn max(&self) -> f64 {
        assert!(!self.is_empty());
        self.iter().fold(self[0], |p, q| p.max(*q))
    }

    fn mean(&self) -> f64 {
        assert!(!self.is_empty());
        self.sum() / (self.len() as f64)
    }

    fn median(&self) -> f64 {
        self.percentile(50 as f64)
    }

    fn var(&self) -> f64 {
        if self.len() < 2 {
            0.0
        } else {
            let mean = self.mean();
            let mut v: f64 = 0.0;
            for s in self {
                let x = *s - mean;
                v += x * x;
            }
            // NB: this is _supposed to be_ len-1, not len. If you
            // change it back to len, you will be calculating a
            // population variance, not a sample variance.
            let denom = (self.len() - 1) as f64;
            v / denom
        }
    }

    fn std_dev(&self) -> f64 {
        self.var().sqrt()
    }

    fn std_dev_pct(&self) -> f64 {
        let hundred = 100 as f64;
        (self.std_dev() / self.mean()) * hundred
    }

    fn median_abs_dev(&self) -> f64 {
        let med = self.median();
        let abs_devs: Vec<f64> = self.iter().map(|&v| (med - v).abs()).collect();
        // This constant is derived by smarter statistics brains than me, but it is
        // consistent with how R and other packages treat the MAD.
        let number = 1.4826;
        abs_devs.median() * number
    }

    fn median_abs_dev_pct(&self) -> f64 {
        let hundred = 100 as f64;
        (self.median_abs_dev() / self.median()) * hundred
    }

    fn percentile(&self, pct: f64) -> f64 {
        let mut tmp = self.to_vec();
        local_sort(&mut tmp);
        percentile_of_sorted(&tmp, pct)
    }

    fn quartiles(&self) -> (f64, f64, f64) {
        let mut tmp = self.to_vec();
        local_sort(&mut tmp);
        let first = 25f64;
        let a = percentile_of_sorted(&tmp, first);
        let secound = 50f64;
        let b = percentile_of_sorted(&tmp, secound);
        let third = 75f64;
        let c = percentile_of_sorted(&tmp, third);
        (a, b, c)
    }

    fn iqr(&self) -> f64 {
        let (a, _, c) = self.quartiles();
        c - a
    }
}


// Helper function: extract a value representing the `pct` percentile of a sorted sample-set, using
// linear interpolation. If samples are not sorted, return nonsensical value.
fn percentile_of_sorted(sorted_samples: &[f64], pct: f64) -> f64 {
    assert!(!sorted_samples.is_empty());
    if sorted_samples.len() == 1 {
        return sorted_samples[0];
    }
    let zero: f64 = 0.0;
    assert!(zero <= pct);
    let hundred = 100f64;
    assert!(pct <= hundred);
    if pct == hundred {
        return sorted_samples[sorted_samples.len() - 1];
    }
    let length = (sorted_samples.len() - 1) as f64;
    let rank = (pct / hundred) * length;
    let lrank = rank.floor();
    let d = rank - lrank;
    let n = lrank as usize;
    let lo = sorted_samples[n];
    let hi = sorted_samples[n + 1];
    lo + (hi - lo) * d
}


/// Winsorize a set of samples, replacing values above the `100-pct` percentile
/// and below the `pct` percentile with those percentiles themselves. This is a
/// way of minimizing the effect of outliers, at the cost of biasing the sample.
/// It differs from trimming in that it does not change the number of samples,
/// just changes the values of those that are outliers.
///
/// See: http://en.wikipedia.org/wiki/Winsorising
pub fn winsorize(samples: &mut [f64], pct: f64) {
    let mut tmp = samples.to_vec();
    local_sort(&mut tmp);
    let lo = percentile_of_sorted(&tmp, pct);
    let hundred = 100 as f64;
    let hi = percentile_of_sorted(&tmp, hundred - pct);
    for samp in samples {
        if *samp > hi {
            *samp = hi
        } else if *samp < lo {
            *samp = lo
        }
    }
}

// Test vectors generated from R, using the script src/etc/stat-test-vectors.r.

#[cfg(test)]
mod tests {
    use stats::Stats;
    use stats::Summary;
    use std::f64;
    use std::io::prelude::*;
    use std::io;

    macro_rules! assert_approx_eq {
        ($a:expr, $b:expr) => ({
            let (a, b) = (&$a, &$b);
            assert!((*a - *b).abs() < 1.0e-6,
                    "{} is not approximately equal to {}", *a, *b);
        })
    }

    fn check(samples: &[f64], summ: &Summary) {

        let summ2 = Summary::new(samples);

        let mut w = io::sink();
        let w = &mut w;
        (write!(w, "\n")).unwrap();

        assert_eq!(summ.sum, summ2.sum);
        assert_eq!(summ.min, summ2.min);
        assert_eq!(summ.max, summ2.max);
        assert_eq!(summ.mean, summ2.mean);
        assert_eq!(summ.median, summ2.median);

        // We needed a few more digits to get exact equality on these
        // but they're within float epsilon, which is 1.0e-6.
        assert_approx_eq!(summ.var, summ2.var);
        assert_approx_eq!(summ.std_dev, summ2.std_dev);
        assert_approx_eq!(summ.std_dev_pct, summ2.std_dev_pct);
        assert_approx_eq!(summ.median_abs_dev, summ2.median_abs_dev);
        assert_approx_eq!(summ.median_abs_dev_pct, summ2.median_abs_dev_pct);

        assert_eq!(summ.quartiles, summ2.quartiles);
        assert_eq!(summ.iqr, summ2.iqr);
    }

    #[test]
    fn test_min_max_nan() {
        let xs = &[1.0, 2.0, f64::NAN, 3.0, 4.0];
        let summary = Summary::new(xs);
        assert_eq!(summary.min, 1.0);
        assert_eq!(summary.max, 4.0);
    }

    #[test]
    fn test_norm2() {
        let val = &[958.0000000000, 924.0000000000];
        let summ = &Summary {
            sum: 1882.0000000000,
            min: 924.0000000000,
            max: 958.0000000000,
            mean: 941.0000000000,
            median: 941.0000000000,
            var: 578.0000000000,
            std_dev: 24.0416305603,
            std_dev_pct: 2.5549022912,
            median_abs_dev: 25.2042000000,
            median_abs_dev_pct: 2.6784484591,
            quartiles: (932.5000000000, 941.0000000000, 949.5000000000),
            iqr: 17.0000000000,
        };
        check(val, summ);
    }
    #[test]
    fn test_norm10narrow() {
        let val = &[966.0000000000,
                    985.0000000000,
                    1110.0000000000,
                    848.0000000000,
                    821.0000000000,
                    975.0000000000,
                    962.0000000000,
                    1157.0000000000,
                    1217.0000000000,
                    955.0000000000];
        let summ = &Summary {
            sum: 9996.0000000000,
            min: 821.0000000000,
            max: 1217.0000000000,
            mean: 999.6000000000,
            median: 970.5000000000,
            var: 16050.7111111111,
            std_dev: 126.6914010938,
            std_dev_pct: 12.6742097933,
            median_abs_dev: 102.2994000000,
            median_abs_dev_pct: 10.5408964451,
            quartiles: (956.7500000000, 970.5000000000, 1078.7500000000),
            iqr: 122.0000000000,
        };
        check(val, summ);
    }
    #[test]
    fn test_norm10medium() {
        let val = &[954.0000000000,
                    1064.0000000000,
                    855.0000000000,
                    1000.0000000000,
                    743.0000000000,
                    1084.0000000000,
                    704.0000000000,
                    1023.0000000000,
                    357.0000000000,
                    869.0000000000];
        let summ = &Summary {
            sum: 8653.0000000000,
            min: 357.0000000000,
            max: 1084.0000000000,
            mean: 865.3000000000,
            median: 911.5000000000,
            var: 48628.4555555556,
            std_dev: 220.5186059170,
            std_dev_pct: 25.4846418487,
            median_abs_dev: 195.7032000000,
            median_abs_dev_pct: 21.4704552935,
            quartiles: (771.0000000000, 911.5000000000, 1017.2500000000),
            iqr: 246.2500000000,
        };
        check(val, summ);
    }
    #[test]
    fn test_norm10wide() {
        let val = &[505.0000000000,
                    497.0000000000,
                    1591.0000000000,
                    887.0000000000,
                    1026.0000000000,
                    136.0000000000,
                    1580.0000000000,
                    940.0000000000,
                    754.0000000000,
                    1433.0000000000];
        let summ = &Summary {
            sum: 9349.0000000000,
            min: 136.0000000000,
            max: 1591.0000000000,
            mean: 934.9000000000,
            median: 913.5000000000,
            var: 239208.9888888889,
            std_dev: 489.0899599142,
            std_dev_pct: 52.3146817750,
            median_abs_dev: 611.5725000000,
            median_abs_dev_pct: 66.9482758621,
            quartiles: (567.2500000000, 913.5000000000, 1331.2500000000),
            iqr: 764.0000000000,
        };
        check(val, summ);
    }
    #[test]
    fn test_norm25verynarrow() {
        let val = &[991.0000000000,
                    1018.0000000000,
                    998.0000000000,
                    1013.0000000000,
                    974.0000000000,
                    1007.0000000000,
                    1014.0000000000,
                    999.0000000000,
                    1011.0000000000,
                    978.0000000000,
                    985.0000000000,
                    999.0000000000,
                    983.0000000000,
                    982.0000000000,
                    1015.0000000000,
                    1002.0000000000,
                    977.0000000000,
                    948.0000000000,
                    1040.0000000000,
                    974.0000000000,
                    996.0000000000,
                    989.0000000000,
                    1015.0000000000,
                    994.0000000000,
                    1024.0000000000];
        let summ = &Summary {
            sum: 24926.0000000000,
            min: 948.0000000000,
            max: 1040.0000000000,
            mean: 997.0400000000,
            median: 998.0000000000,
            var: 393.2066666667,
            std_dev: 19.8294393937,
            std_dev_pct: 1.9888308788,
            median_abs_dev: 22.2390000000,
            median_abs_dev_pct: 2.2283567134,
            quartiles: (983.0000000000, 998.0000000000, 1013.0000000000),
            iqr: 30.0000000000,
        };
        check(val, summ);
    }
    #[test]
    fn test_exp10a() {
        let val = &[23.0000000000,
                    11.0000000000,
                    2.0000000000,
                    57.0000000000,
                    4.0000000000,
                    12.0000000000,
                    5.0000000000,
                    29.0000000000,
                    3.0000000000,
                    21.0000000000];
        let summ = &Summary {
            sum: 167.0000000000,
            min: 2.0000000000,
            max: 57.0000000000,
            mean: 16.7000000000,
            median: 11.5000000000,
            var: 287.7888888889,
            std_dev: 16.9643416875,
            std_dev_pct: 101.5828843560,
            median_abs_dev: 13.3434000000,
            median_abs_dev_pct: 116.0295652174,
            quartiles: (4.2500000000, 11.5000000000, 22.5000000000),
            iqr: 18.2500000000,
        };
        check(val, summ);
    }
    #[test]
    fn test_exp10b() {
        let val = &[24.0000000000,
                    17.0000000000,
                    6.0000000000,
                    38.0000000000,
                    25.0000000000,
                    7.0000000000,
                    51.0000000000,
                    2.0000000000,
                    61.0000000000,
                    32.0000000000];
        let summ = &Summary {
            sum: 263.0000000000,
            min: 2.0000000000,
            max: 61.0000000000,
            mean: 26.3000000000,
            median: 24.5000000000,
            var: 383.5666666667,
            std_dev: 19.5848580967,
            std_dev_pct: 74.4671410520,
            median_abs_dev: 22.9803000000,
            median_abs_dev_pct: 93.7971428571,
            quartiles: (9.5000000000, 24.5000000000, 36.5000000000),
            iqr: 27.0000000000,
        };
        check(val, summ);
    }
    #[test]
    fn test_exp10c() {
        let val = &[71.0000000000,
                    2.0000000000,
                    32.0000000000,
                    1.0000000000,
                    6.0000000000,
                    28.0000000000,
                    13.0000000000,
                    37.0000000000,
                    16.0000000000,
                    36.0000000000];
        let summ = &Summary {
            sum: 242.0000000000,
            min: 1.0000000000,
            max: 71.0000000000,
            mean: 24.2000000000,
            median: 22.0000000000,
            var: 458.1777777778,
            std_dev: 21.4050876611,
            std_dev_pct: 88.4507754589,
            median_abs_dev: 21.4977000000,
            median_abs_dev_pct: 97.7168181818,
            quartiles: (7.7500000000, 22.0000000000, 35.0000000000),
            iqr: 27.2500000000,
        };
        check(val, summ);
    }
    #[test]
    fn test_exp25() {
        let val = &[3.0000000000,
                    24.0000000000,
                    1.0000000000,
                    19.0000000000,
                    7.0000000000,
                    5.0000000000,
                    30.0000000000,
                    39.0000000000,
                    31.0000000000,
                    13.0000000000,
                    25.0000000000,
                    48.0000000000,
                    1.0000000000,
                    6.0000000000,
                    42.0000000000,
                    63.0000000000,
                    2.0000000000,
                    12.0000000000,
                    108.0000000000,
                    26.0000000000,
                    1.0000000000,
                    7.0000000000,
                    44.0000000000,
                    25.0000000000,
                    11.0000000000];
        let summ = &Summary {
            sum: 593.0000000000,
            min: 1.0000000000,
            max: 108.0000000000,
            mean: 23.7200000000,
            median: 19.0000000000,
            var: 601.0433333333,
            std_dev: 24.5161851301,
            std_dev_pct: 103.3565983562,
            median_abs_dev: 19.2738000000,
            median_abs_dev_pct: 101.4410526316,
            quartiles: (6.0000000000, 19.0000000000, 31.0000000000),
            iqr: 25.0000000000,
        };
        check(val, summ);
    }
    #[test]
    fn test_binom25() {
        let val = &[18.0000000000,
                    17.0000000000,
                    27.0000000000,
                    15.0000000000,
                    21.0000000000,
                    25.0000000000,
                    17.0000000000,
                    24.0000000000,
                    25.0000000000,
                    24.0000000000,
                    26.0000000000,
                    26.0000000000,
                    23.0000000000,
                    15.0000000000,
                    23.0000000000,
                    17.0000000000,
                    18.0000000000,
                    18.0000000000,
                    21.0000000000,
                    16.0000000000,
                    15.0000000000,
                    31.0000000000,
                    20.0000000000,
                    17.0000000000,
                    15.0000000000];
        let summ = &Summary {
            sum: 514.0000000000,
            min: 15.0000000000,
            max: 31.0000000000,
            mean: 20.5600000000,
            median: 20.0000000000,
            var: 20.8400000000,
            std_dev: 4.5650848842,
            std_dev_pct: 22.2037202539,
            median_abs_dev: 5.9304000000,
            median_abs_dev_pct: 29.6520000000,
            quartiles: (17.0000000000, 20.0000000000, 24.0000000000),
            iqr: 7.0000000000,
        };
        check(val, summ);
    }
    #[test]
    fn test_pois25lambda30() {
        let val = &[27.0000000000,
                    33.0000000000,
                    34.0000000000,
                    34.0000000000,
                    24.0000000000,
                    39.0000000000,
                    28.0000000000,
                    27.0000000000,
                    31.0000000000,
                    28.0000000000,
                    38.0000000000,
                    21.0000000000,
                    33.0000000000,
                    36.0000000000,
                    29.0000000000,
                    37.0000000000,
                    32.0000000000,
                    34.0000000000,
                    31.0000000000,
                    39.0000000000,
                    25.0000000000,
                    31.0000000000,
                    32.0000000000,
                    40.0000000000,
                    24.0000000000];
        let summ = &Summary {
            sum: 787.0000000000,
            min: 21.0000000000,
            max: 40.0000000000,
            mean: 31.4800000000,
            median: 32.0000000000,
            var: 26.5933333333,
            std_dev: 5.1568724372,
            std_dev_pct: 16.3814245145,
            median_abs_dev: 5.9304000000,
            median_abs_dev_pct: 18.5325000000,
            quartiles: (28.0000000000, 32.0000000000, 34.0000000000),
            iqr: 6.0000000000,
        };
        check(val, summ);
    }
    #[test]
    fn test_pois25lambda40() {
        let val = &[42.0000000000,
                    50.0000000000,
                    42.0000000000,
                    46.0000000000,
                    34.0000000000,
                    45.0000000000,
                    34.0000000000,
                    49.0000000000,
                    39.0000000000,
                    28.0000000000,
                    40.0000000000,
                    35.0000000000,
                    37.0000000000,
                    39.0000000000,
                    46.0000000000,
                    44.0000000000,
                    32.0000000000,
                    45.0000000000,
                    42.0000000000,
                    37.0000000000,
                    48.0000000000,
                    42.0000000000,
                    33.0000000000,
                    42.0000000000,
                    48.0000000000];
        let summ = &Summary {
            sum: 1019.0000000000,
            min: 28.0000000000,
            max: 50.0000000000,
            mean: 40.7600000000,
            median: 42.0000000000,
            var: 34.4400000000,
            std_dev: 5.8685603004,
            std_dev_pct: 14.3978417577,
            median_abs_dev: 5.9304000000,
            median_abs_dev_pct: 14.1200000000,
            quartiles: (37.0000000000, 42.0000000000, 45.0000000000),
            iqr: 8.0000000000,
        };
        check(val, summ);
    }
    #[test]
    fn test_pois25lambda50() {
        let val = &[45.0000000000,
                    43.0000000000,
                    44.0000000000,
                    61.0000000000,
                    51.0000000000,
                    53.0000000000,
                    59.0000000000,
                    52.0000000000,
                    49.0000000000,
                    51.0000000000,
                    51.0000000000,
                    50.0000000000,
                    49.0000000000,
                    56.0000000000,
                    42.0000000000,
                    52.0000000000,
                    51.0000000000,
                    43.0000000000,
                    48.0000000000,
                    48.0000000000,
                    50.0000000000,
                    42.0000000000,
                    43.0000000000,
                    42.0000000000,
                    60.0000000000];
        let summ = &Summary {
            sum: 1235.0000000000,
            min: 42.0000000000,
            max: 61.0000000000,
            mean: 49.4000000000,
            median: 50.0000000000,
            var: 31.6666666667,
            std_dev: 5.6273143387,
            std_dev_pct: 11.3913245723,
            median_abs_dev: 4.4478000000,
            median_abs_dev_pct: 8.8956000000,
            quartiles: (44.0000000000, 50.0000000000, 52.0000000000),
            iqr: 8.0000000000,
        };
        check(val, summ);
    }
    #[test]
    fn test_unif25() {
        let val = &[99.0000000000,
                    55.0000000000,
                    92.0000000000,
                    79.0000000000,
                    14.0000000000,
                    2.0000000000,
                    33.0000000000,
                    49.0000000000,
                    3.0000000000,
                    32.0000000000,
                    84.0000000000,
                    59.0000000000,
                    22.0000000000,
                    86.0000000000,
                    76.0000000000,
                    31.0000000000,
                    29.0000000000,
                    11.0000000000,
                    41.0000000000,
                    53.0000000000,
                    45.0000000000,
                    44.0000000000,
                    98.0000000000,
                    98.0000000000,
                    7.0000000000];
        let summ = &Summary {
            sum: 1242.0000000000,
            min: 2.0000000000,
            max: 99.0000000000,
            mean: 49.6800000000,
            median: 45.0000000000,
            var: 1015.6433333333,
            std_dev: 31.8691595957,
            std_dev_pct: 64.1488719719,
            median_abs_dev: 45.9606000000,
            median_abs_dev_pct: 102.1346666667,
            quartiles: (29.0000000000, 45.0000000000, 79.0000000000),
            iqr: 50.0000000000,
        };
        check(val, summ);
    }

    #[test]
    fn test_sum_f64s() {
        assert_eq!([0.5f64, 3.2321f64, 1.5678f64].sum(), 5.2999);
    }
    #[test]
    fn test_sum_f64_between_ints_that_sum_to_0() {
        assert_eq!([1e30f64, 1.2f64, -1e30f64].sum(), 1.2);
    }
}