BWS_TEST

Overview

The BWS_TEST function performs the Baumgartner-Weiss-Schindler (BWS) test to compare two independent samples. The BWS test is a nonparametric statistical test that evaluates whether two samples come from the same distribution, with particular sensitivity to differences in the tails of the distributions. Unlike parametric tests such as the t-test, the BWS test does not assume normality or equal variances, making it suitable for a wider range of data types. The test statistic is calculated as:

BWS = \frac{1}{n_x + n_y} \sum_{i=1}^{n_x + n_y} \left[ F_x(R_i) - F_y(R_i) \right]^2

where $F_x$ and $F_y$ are the empirical distribution functions of samples $x$ and $y$ , and $R_i$ are the ranks of the combined samples. For more details, see the scipy.stats.bws_test documentation .

This example function is provided as-is without any representation of accuracy.

Usage

To use the function in Excel:


=BWS_TEST(x, y, [alternative])

x (2D list, required): First sample. Must be a 2D array (column vector) with at least two rows, containing numeric values.
y (2D list, required): Second sample. Must be a 2D array (column vector) with at least two rows, containing numeric values.
alternative (string, optional, default=‘two-sided’): Defines the alternative hypothesis. Must be one of two-sided, less, or greater.

The function returns a single-row 2D array: [statistic, pvalue], where statistic is the BWS test statistic and pvalue is the p-value for the test. If the input is invalid, an error message (string) is returned instead.

Examples

Example 1: Basic Case

Inputs:

x	y
1.2	2.1
2.3	2.9
3.1	3.7
4.0	4.5

Excel formula:


=BWS_TEST({1.2;2.3;3.1;4.0}, {2.1;2.9;3.7;4.5})

Expected output:

statistic	pvalue
0.326	1.000

Example 2: With alternative = “less”

Inputs:

x	y	alternative
1.2	2.1	less
2.3	2.9
3.1	3.7
4.0	4.5

Excel formula:


=BWS_TEST({1.2;2.3;3.1;4.0}, {2.1;2.9;3.7;4.5}, "less")

Expected output:

statistic	pvalue
-0.326	0.357

Example 3: With alternative = “greater”

Inputs:

x	y	alternative
1.2	2.1	greater
2.3	2.9
3.1	3.7
4.0	4.5

Excel formula:


=BWS_TEST({1.2;2.3;3.1;4.0}, {2.1;2.9;3.7;4.5}, "greater")

Expected output:

statistic	pvalue
-0.326	0.657

Example 4: Different Samples

Inputs:

x	y
10.0	2.0
12.0	3.0
13.0	4.0
15.0	5.0

Excel formula:


=BWS_TEST({10.0;12.0;13.0;15.0}, {2.0;3.0;4.0;5.0})

Expected output:

statistic	pvalue
3.711	0.029

Python Code


from scipy.stats import bws_test as scipy_bws_test
from typing import List, Union
 
def bws_test(x: List[List[float]], y: List[List[float]], alternative: str = 'two-sided') -> Union[List[List[float]], str]:
    """
    Performs the Baumgartner-Weiss-Schindler test on two independent samples.
 
    Args:
        x: 2D list of float values. First sample.
        y: 2D list of float values. Second sample.
        alternative: Defines the alternative hypothesis ('two-sided', 'less', 'greater'). Default is 'two-sided'.
 
    Returns:
        2D list with a single row: [statistic, pvalue], or an error message (str) if input is invalid.
 
    This example function is provided as-is without any representation of accuracy.
    """
    # Validate x and y
    if not (isinstance(x, list) and all(isinstance(row, list) for row in x)) or len(x) < 2:
        return "Invalid input: x must be a 2D list with at least two rows."
    if not (isinstance(y, list) and all(isinstance(row, list) for row in y)) or len(y) < 2:
        return "Invalid input: y must be a 2D list with at least two rows."
    try:
        x_flat = [float(item) for row in x for item in row]
        y_flat = [float(item) for row in y for item in row]
    except Exception:
        return "Invalid input: x and y must contain only numeric values."
    if alternative not in ['two-sided', 'less', 'greater']:
        return "Invalid input: alternative must be 'two-sided', 'less', or 'greater'."
    try:
        result = scipy_bws_test(x_flat, y_flat, alternative=alternative)
    except Exception as e:
        return f"scipy.stats.bws_test error: {e}"
    stat, pvalue = result.statistic, result.pvalue
    # Convert to native float if needed
    try:
        stat = float(stat)
        pvalue = float(pvalue)
    except Exception:
        return "Invalid output: statistic or pvalue could not be converted to float."
    # Disallow nan/inf
    if any([isinstance(val, float) and (val != val or val in [float('inf'), float('-inf')]) for val in [stat, pvalue]]):
        return "Invalid output: statistic or pvalue is nan or inf."
    return [[stat, pvalue]]

Example Workbook

Link to Workbook