A Components of Variance (COV) study partitions total process variation into the contribution of each layer in a sampling hierarchy — for example, lot-to-lot, hour-to-hour, and within-hour. This guide explains the Sanders R̄/d₂ method and shows a worked example.
Why decompose variation?
If 80% of your part-to-part variation comes from lot-to-lot drift, optimising within-lot sampling is wasted effort. COV tells you where to look first.
Nested vs crossed designs
A study is nested when each lower-level factor is unique to one higher-level factor (e.g., Hour 1 inside Lot A is a different hour from Hour 1 inside Lot B). It is crossed when the same lower-level factor appears under every higher-level factor (e.g., the same five Operators measure the same ten Parts). The Sanders R̄/d₂ decomposition described here applies to nested designs only; crossed designs should be analysed with control charts and main-effect plots, or with ANOVA-based variance components.
The Sanders R̄/d₂ method
For each layer from the bottom up:
- Compute the mean within each lower-level group (e.g., the mean of the replicates within each Hour).
- Compute the average range (R̄) of those means within each parent group.
- Estimate the standard deviation as σ = R̄ / d₂ for the corresponding subgroup size.
- For the apparent variance of the layer's means, subtract the contribution of the layers below: σ²true = σ²apparent − σ²below / nbelow.
- For the top layer, when there are few groups, use the Moving Range (MR̄) of the top-level means with d₂(n=2) = 1.128.
Worked example: 4 lots × 6 hours × 5 samples
| Layer | R̄ or MR̄ | n / d₂ | σ² | % of total |
|---|---|---|---|---|
| Within hour (replicates) | R̄ = 3.79 | n = 5, d₂ = 2.326 | 2.66 | 31.6% |
| Hour within lot | R̄ = 3.10 | n = 6, d₂ = 2.534 | 0.97 | 11.5% |
| Lot to lot (MR method) | MR̄ = 2.53 | n = 2, d₂ = 1.128 | 4.79 | 57.0% |
The conclusion: more than half the variation is lot-to-lot, so process knowledge work should focus there, not on within-hour repeatability.
Common mistakes
- Ignoring the layers-below correction. The variance of higher-level means already contains the lower-level noise divided by the subgroup size; failing to subtract it overstates the higher layer.
- Using the straight range with very few top-level groups. With only 3–5 lots, R̄/d₂ has poor degrees of freedom; switch to the Moving Range with d₂(n=2) = 1.128.
- Treating a crossed study as nested. The R̄/d₂ decomposition will produce nonsense if the lower-level factor is actually the same across parents.
How Ops Excellence handles this
The COV tool inside Ops Excellence implements the Sanders method end-to-end, automatically switches to the Moving Range estimator when the top-level group count is small, refuses to compute nested variance components for crossed designs, and renders X̄/R control charts at every rollup level so you can see stability layer by layer. Try it on your own data.