Z-Statistic
Definition
A per-digit test that calculates whether the deviation between an observed frequency and the Benford expected frequency for that single digit is statistically significant, allowing the analyst to pinpoint which digits are anomalous.
- Tests
- One digit position at a time
- Compares
- Observed frequency vs Benford expected frequency
- Output
- Whether a single digit's deviation is significant
- Use
- Pinpoints which digits are anomalous
Common questions
How does the digit-level z-statistic complement a chi-square test on the full digit distribution?+
A chi-square test tells the analyst whether the overall digit distribution departs from Benford's law but not which digit is responsible, while the per-digit z-statistic isolates each digit, so the two are used together, chi-square to flag a dataset and the z-statistic to identify which digits drove the result.
Does a significant z-statistic on one digit prove fraud?+
No, a significant deviation flags a digit worth investigating further, since legitimate datasets can depart from Benford's law for structural reasons such as price rounding or a non-naturally-occurring number range, so the statistic is a screening tool rather than direct proof of manipulation.
Related terms
- Benford's Law
- An empirical regularity in naturally occurring numerical datasets: the leading digit follows a logarithmic distribution, with 1 appearing about 30% of the...
- Chi-Square Test
- A goodness-of-fit test comparing the observed digit frequencies in a dataset to the expected Benford frequencies, producing a test statistic that is...
- Mean Absolute Deviation (MAD)
- A practical Benford conformity measure that averages the absolute differences between observed and expected proportions across all leading digits, with Nigrini benchmarks...
- Second-Digit Test
- Analysis of the second significant digit in isolation, where 0 is expected to appear about 11.97% of the time. A spike in...
- Two-Digit Test
- An extension of Benford's Law to the first two significant digits, producing 90 possible combinations each with a predicted frequency, which is...