The GRIMMER test

GRIMMER (granularity-related inconsistency of means mapped to error repeats) takes the GRIM idea one step further, from the mean to the standard deviation.

The idea

An SD is computed from two whole numbers: the total of the answers and the total of their squares. Once GRIM has pinned down which totals fit the mean, only certain sums of squares remain, so only certain SDs can appear. Two more facts narrow it further. A square has the same parity as the number it came from (odd squared is odd, even squared is even), so the sum of squares has the same parity as the total. And when the mean is not a whole number, the answers cannot all be equal, which puts a floor under the SD.

How the checker does it

For every total that GRIM allows, it works out which whole-number sums of squares would give an SD that rounds to the reported one, keeps those with the right parity and at least the smallest possible spread, and reports consistent if any remain. All of it is integer arithmetic, so a tie on the rounding boundary is decided by the convention you choose, never by floating-point noise. This is the analytic form of the test described by Aurélien Allard and used in the rsprite2 and scrutiny R packages.

Worked example

A documented case from rsprite2: 18 people, mean 3.44, SD 2.47.

3.44 × 18 = 61.92 total must be 62 (62/18 = 3.444) SD 2.47 needs a sum of squares of 317 (2.4668) 317 is odd but the total 62 is even → not allowed 316 gives SD 2.4548 → 2.45; 318 gives 2.4787 → 2.48 → inconsistent

By contrast, 28 people with mean 5.21 and SD 1.60 pass: a total of 146 with a sum of squares of 830 gives 1.5953, which rounds to 1.60.

What a pass does not prove

GRIMMER checks conditions that every dataset must meet, but meeting them does not guarantee a dataset exists on the actual answer scale. Anaya's own example: five people, mean 2.0 and a variance of 100.8 pass GRIMMER, because nothing in the test knows the answers were, say, between 1 and 7. On a 1–7 scale that variance is far out of reach. Use SPRITE with the scale range when that matters.

Large SDs are also harder to test, because the square root squeezes neighbouring variances together: with five people the variances 100.56 and 100.64 both become an SD of 10.03.

Source

Anaya, J. (2016). The GRIMMER test: A method for testing the validity of reported measures of variability. PeerJ Preprints, 4:e2400v1. doi:10.7287/peerj.preprints.2400v1. Analytic version: Allard, A. (2018), Analytic-GRIMMER.

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