Benfords

Benfords provides a series of functions intended to make Benford's Law research and usage more convenient.

Usage

With Benfords you can conduct quick comparisons to Benfords law, create outputs like charts and .csv files, generate random Benford-variates, as well as calculate the expected probabilities or extract your own empirical digit frequencies.

Comparisons to Benford's Law

The benfords() function allows you to quickly compare your data's digits to Benford's Law:

import benfords

#Generate some random data
test_data = variate(1000)

#Compare to Benford's Law
benfords(test_data)
#  Digit  Expected Value  Actual Value  Difference
#0     1        0.301030         0.316    0.014970
#1     2        0.176091         0.178    0.001909
#2     3        0.124939         0.113   -0.011939
#3     4        0.096910         0.097    0.000090
#4     5        0.079181         0.075   -0.004181
#5     6        0.066947         0.072    0.005053
#6     7        0.057992         0.059    0.001008
#7     8        0.051153         0.041   -0.010153
#8     9        0.045757         0.049    0.003243

Use test() to calculate test statistics:

test(benfords(test_data), test_statistic='d')
# 0.01914657325117365

Currently, test() supports only the d statistic described in Cho and Gaines, 2012. Future releases will include additional test statistics.

benfords() can also output charts and csv:

benfords(test_data, output_csv=True, output_plot=True, filename='2021-01-31 Analysis')

There are also parmeters for examining the digits beyond the first, as well as multiple digits at a time. This example uses the second and third digits:

benfords(test_data, start_position=2, length=2)
#   Digit  Expected Value  Actual Value  Difference
#0      0        0.119679         0.000   -0.119679
#1      1        0.113890         0.000   -0.113890
#2      2        0.108821         0.000   -0.108821
#3      3        0.104330         0.000   -0.104330
#4      4        0.100308         0.000   -0.100308
#..   ...             ...           ...         ...
#95    95        0.027760         0.002   -0.025760
#96    96        0.027558         0.009   -0.018558
#97    97        0.027358         0.005   -0.022358
#98    98        0.027162         0.004   -0.023162
#99    99        0.026969         0.007   -0.019969
#
#[100 rows x 4 columns]

Generate random Benford-distributed digits.

You can generate random Benford-distributed digits with the variate() function. Just specify how many you want:

variate(5)

# [6.494198781949683, 5.511615661880242, 7.311726835973362, 1.6809486480388234, 8.877345103827716]

Variates are generated according to the method in Jamain, 2001.

Expected Probabilities

Calculate the probabilities expected under Benford's Law with expectation(). What's the expected probability that the second digit will be 5?

expectation(5, 2)

# 0.09667723580232242

Extract the significant digits from your data

fsd() and nsd() return the significant digits of your input data. They currently enjoy scalars (integers and floats), lists, 1d numpy arrays, and pandas dataframes.

fsd() returns the first significant digit according to the significand formula provided by Berger and Hill, 2015.

test = [5, 0.321, -2989.2, -0.00001]

fsd(test) 
#array([5., 3., 2., 1.])

#NSD() returns the nth digit of your data. It always returns a numpy array containing strings. It includes parameters to select the second, third, and higher digits, as well as control the number of digits. This example shows the 4th significant digits.

nsd(test, 4) 
#array(['0', '1', '8', '0']

Citation

If you use this work in your own research, please cite it in your publications:

McCarville, Daniel. Benford's Law, (2021). https://github.com/danielmccarville/Benfords

License

MIT