On a bounded scale — a 1-to-5 Likert item, a percentage, a 0-to-60 test — the mean and the standard deviation are not free of each other: one constrains the other. Some pairs no sample can produce.
The easy bound. With values between a and b, the standard deviation can never exceed (b − a) / 2, and only a sample split exactly between the two endpoints gets there. On a 1-to-5 scale, that is 2 points.
The bound that actually bites. Bhatia-Davis: SD ≤ √((b − M)(M − a)). It depends on the mean and narrows sharply as the mean approaches an endpoint. With M = 4.5 on a 1-to-5 scale, the ceiling drops from 2 to 1.32.
That is the one that finds real cases, because ceiling-effect items — satisfaction, adherence, agreement — are exactly where means cluster near an endpoint and where an SD of 1.8 slips into a table unexamined.
It is the standard error, not the deviation. The number one cause. The standard error is the SD divided by the square root of N, so it is much smaller; swapping them the other way produces impossible deviations. And in print, SE and SD differ by one letter.
The stated scale is not the one used. The text says "a 1-to-5 scale" and reports the sum of twenty items, which runs from 20 to 100. The mean and SD are fine; the range printed next to them is not.
Reverse items left unrecoded. They produce odd means and inflated deviations. They show up earlier as a negative item-total correlation, covered in reverse items and scale scores.
A possible mean and SD are not thereby correct: only that no arithmetic contradicts them. This is a test of impossibility, not of truth.
And it does not work on unbounded scales. A reaction time, a count or an income has no maximum, so there is no bound to apply.
Where it works very well is reviewing someone else’s Table 1 in a journal club: five minutes, and no data request needed. How to build that table properly is in Table 1 in APA 7.
A result bounding the variance of any variable whose values lie in an interval: the variance cannot exceed (b − M)(M − a). It is exact and assumes no particular distribution, so it applies equally to Likert data, percentages and test scores.
Yes, and that is where it bites hardest, because a 1-to-5 item has a very short range. With means above 4 or below 2, the maximum possible SD drops sharply and impossible tables become obvious.
Then the range is the sum’s, not the item’s: twenty items scored 1 to 5 run from 20 to 100. Using the item range with the sum’s mean is exactly what produces false positives here.
Write to the authors flagging the inconsistency without attributing intent. The most frequent explanation is that a standard error was published as a standard deviation, and that is fixed with a corrigendum.