PRACTICAL STATISTICS
RMS, RMSE, SD and RSS: four different questions
Similar-looking square roots do not make these quantities interchangeable.
Revised 30 September 2026. The original publication date and address are retained.
Start with the quantity you need
| Calculation | Question |
|---|---|
| RMS of raw values | What is the quadratic mean of the values? |
| RMSE about a stated reference or prediction | How large are the discrepancies from that reference? |
| Sample SD | How dispersed are the observations around their sample mean? |
| RSS of uncertainty contributions | How do specified contributions combine under an appropriate model? |
A worked example, with the arithmetic visible
Use the five synthetic values 99, 101, 100, 98 and 102 units. Their arithmetic mean is 100. The squared raw values sum to 50,010. The residuals about 100 are −1, +1, 0, −2 and +2; their squared values sum to 10.
Raw RMS
√(50,010 / 5) = 100.010 units
RMSE about 100
√(10 / 5) = 1.414 units
Sample SD
√(10 / 4) = 1.581 units
SE of the mean
1.581 / √5 = 0.707 units, assuming independent observations
Raw RMS stays close to 100 because the location dominates the squared raw values. It does not isolate their spread. RMSE depends on the reference or prediction being assessed; it is not generally the same as the SD of residuals. A residual standard deviation from a fitted model also needs the appropriate residual degrees of freedom.
The step that must not be skipped
Square the residuals before averaging them for RMSE. Averaging signed residuals allows positive and negative discrepancies to cancel. Taking the square root of that signed average is not an RMSE calculation.
RSS belongs to a measurement model
Suppose three independent standard uncertainty contributions, already expressed in output units and including their sensitivity factors, are 1, 2 and 1.5 units. Their combined standard uncertainty is √(1² + 2² + 1.5²) = 2.69 units. These must be standard uncertainties—not unidentified ± ranges. Correlation, overlap between contributions or non-unit sensitivity changes the calculation.
RSS being smaller than the arithmetic sum does not prove it is the right answer. The model and assumptions decide that. Likewise, none of the four quantities is a universal replacement for a clearly defined reported result.
Sources & further reading
- NIST/SEMATECH — sample variance and standard deviation
- JCGM 100 — combining standard uncertainty contributions
Educational material. Examples are synthetic. Consult applicable full guidance and your locally approved procedure for operational decisions.
Take the next step
Choose a resource for the question you want to understand.
- SD and standard error guide — Separate the spread of results from the precision of a mean.
- Distribution viewer — Explore how the shape and spread of a distribution change its summary measures.
- Measurement model guide — Read the assumptions behind combining uncertainty contributions.