A chi-square goodness-of-fit test compares observed categorical counts with expected counts and evaluates whether their discrepancy is statistically significant. For Benford’s law, the null hypothesis states that leading-digit counts follow the Benford distribution. section 12.5 section 12.5.1
Apply the test by following these steps:
- Create expected counts by multiplying each expected proportion by the total observed count. section 12.5.1
- Compute the chi-square statistic by summing (observed count − expected count)² / expected count across categories. section 12.5.1
- Compare the p-value with the 5% threshold: reject conformity when p < 0.05; otherwise, fail to reject the null hypothesis. section 12.5.1
- Check that the record count is sufficient before treating the result as meaningful. section 12.5.1
>>> bf_populations = \
>>> bf_populations.assign(Expected_Counts = \
>>> bf_populations.Expected * \
>>> sum(bf_populations.Counts))
>>> print(bf_populations)
Counts Found Expected Expected_Counts
First_1_Dig
1 70 0.297872 0.301030 70.742049
2 37 0.157447 0.176091 41.381446
3 30 0.127660 0.124939 29.360603
4 21 0.089362 0.096910 22.773853
5 25 0.106383 0.079181 18.607593
6 19 0.080851 0.066947 15.732496
7 7 0.029787 0.057992 13.628108
8 14 0.059574 0.051153 12.020843
9 12 0.051064 0.045757 10.753010
Builds expected leading-digit counts from proportions and the observed total before running the test. section 12.5.1
World population leading digits
For 235 population counts, the test produces a statistic of 7.193 and a p-value of 0.516. Because the p-value exceeds 5%, the result is not statistically significant, so the null hypothesis is not rejected. section 12.5.1
