These are actual reported costs (60) for a series of projects for two years, say USD (though actually not) totalling $57,975,403.24:
(274,680.90) 3,900.13 103,999.61 512,918.71 1,079,734.06 (97,570.45) 4,218.45 110,499.74 516,501.94 1,249,999.48 (43,261.81) 9,951.36 116,287.74 531,650.58 1,398,872.64 (41,314.45) 14,191.48 124,999.48 551,420.65 1,999,996.52 (34,843.48) 34,999.09 145,808.39 574,996.39 2,249,270.19 (1,305.16) 35,218.84 215,999.87 605,869.55 3,255,268.13 (634.19) 42,007.48 227,045.03 715,369.04 3,758,841.42 0.00 47,045.03 262,369.29 749,881.94 3,897,417.42 60,000.00 54,999.09 347,790.19 794,996.65 3,925,001.81 960,000.00 74,175.23 359,948.78 800,321.94 4,266,898.32 39,999.74 * 77,998.97 359,998.06 824,731.87 7,494,344.91 59,999.48 * 404,998.07 849,997.80 10,646,324.13 919,938.97
Some negatives, the occasional zero and a high incidence of round thousands is to be expected. The two values asterisked are probably a single project. The values are all plausibly accurate but I would like to know whether I have grounds for suspecting the values have been manipulated as they seem to end just short of a round thousand far more often than I would expect:
In particular, 15 of the 60 are less than USD 4 short of the next round thousand. Excluding 7 negatives, 1 zero and 2 round thousands, 50 values spread across ten centuries averages out at about 5 per band. A very slight propensity to fall just short of a round number is to be expected – either as a result of something like a currency conversion approximation or the rare purchase of a low value consumer item (such as a book or shoes that mostly seem to be priced $x.99) but there may also be a slight propensity to just exceed a band.
- In this context, is the fact that the top band contains 20 examples at all significant?
- And if not is there a sample size at which such an anomaly would be statistically significant (say 95%)?
- Is there something like Benford’s Law that applies to trailing rather than leading digits?
I now have most of the values that aggregated as above. Of the 20 values that seemed odd to me, I do not have details for 4 but the sums of the components for 9 of the 20 are round thousands. In effect, based on actual underlying values, 9 of the 20 in the range
900>999.99 should be moved to the range
Roundk and hence change the chart appearance to something I would deem plausible.
I have also investigated the distribution of the last integer in the details that were represented by the remaining 7 in the range 900>999.99 which excluding zero (as likely to be favoured), is:
There were 973 data points of which 157 were ‘0’ and, though not quite horizontal, a linear trend line shows only relatively minor departure from a simple average of just under 91 instances per final integer.
So, my conclusion is that the reporting of aggregate values (which takes place at intervals) may for one or more reporting periods have been rounded*. This I think confirms that the results were not as should have been expected but, in this case, provides a wholly innocuous explanation.
Therefore I’d suggest A1: Yes, A2: n/a, A3: No – other than random might be expected (uniform distribution).
*Current Total reported as Previous total + Current Period change, rather than as Current Cumulative Total.