Three tests, because each one fails differently
The IQR fences and the modified z-score are built on quartiles and the median, so they are resistant to the outliers they are looking for. The plain z-score uses the mean and sample standard deviation and is the one most readers know. Showing all three lets you see when they disagree, which usually means the distribution is skewed or the sample is small. The fence rule is Tukey's box plot rule: 1.5 × IQR for inner fences and 3 × IQR for outer fences, as in the NIST/SEMATECH e-Handbook of Statistical Methods. Quartiles here use linear interpolation between ranks, the same as Excel QUARTILE.INC.
When to use it
Run it on a population before choosing items to test: disbursements, expense reports, credit memos, inventory adjustments, revenue entries near period end. Do it by account or by vendor rather than on a mixed ledger, because a $40,000 invoice is normal for one vendor and extreme for another. Items flagged are candidates for vouching, in addition to whatever your sampling plan selects.
Limits
Amounts that span orders of magnitude flag the top of the range every time; stratify first, or work from a logarithmic view in your own tool. If more than half of the values are identical the IQR is zero and the fence tests stop. Small items can matter too: a $42 amount among payments that vary a little around $1,000 flags, and a fraud made of many small items will not.
Illustrative figures only. Check the result against your firm's methodology before relying on it.
Common questions
How does the IQR rule find outliers?
Tukey’s fences: compute the first and third quartiles (Q1, Q3) and the interquartile range IQR = Q3 − Q1. A value below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR is a mild outlier, and beyond 3 × IQR from the quartiles it is an extreme outlier. Quartiles are less affected by the outliers themselves than the mean and standard deviation are.
When is the z-score unreliable?
The mean and standard deviation are pulled by the very values you are hunting for. In a small sample with one huge item, the item inflates the standard deviation and may not reach z = 3. Transaction amounts are also usually right-skewed, so many legitimate large items will flag. That is why this tool also reports the median-based modified z-score.
What is the modified z-score?
M = 0.6745 × (x − median) ÷ MAD, where MAD is the median absolute deviation. Iglewicz and Hoaglin recommend labeling items with |M| above 3.5 as potential outliers; the NIST engineering statistics handbook describes the same rule. It uses medians, so a few extreme values do not hide each other.
Is an outlier a misstatement?
No. It is a statistically unusual amount. Large legitimate transactions are outliers by definition. Each flagged item is a lead to vouch, and the lack of flags does not show the population is free of error or fraud.
How does this fit into audit procedures?
It supports analytical procedures under AU-C 520 and the selection of specific items for testing under AU-C 500 and AU-C 530. Selecting outliers for testing is a targeted selection of high-risk items and does not by itself support a conclusion about the remaining population.
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