R, Rbar and Rmax: how the range shows the spread

Same average, more scatter

A grinding operation has produced shafts with a steady average for weeks. One afternoon, the hourly subgroup still averages close to 48.00 mm, but the five values are spread from 47.94 to 48.07. The mean gives no warning. Something has happened to the spread, perhaps a loose fixture or a wheel that needs dressing, and it shows up only if you look at the variation inside the subgroup.

The simplest measure of that variation is the range.

R: the range of a subgroup

The range of a subgroup is

R = largest value - smallest value

Worked example (illustrative values). A subgroup of five shafts measures 48.02, 47.98, 48.01, 48.04 and 47.99 mm. The largest value is 48.04 and the smallest 47.98, so R = 0.06 mm. The afternoon subgroup above has R = 48.07 - 47.94 = 0.13 mm, more than twice as much.

The range is quick to calculate by hand, which is why it became the standard measure of spread on shop-floor control charts. For small subgroups, typically two to nine values, it carries almost as much information as the standard deviation.

R̄: the normal spread

R̄, read “R-bar”, is the average of the ranges over many subgroups:

R̄ = (R1 + R2 + … + Rk) / k

R̄ is the centre line of the R chart. It describes the short-term spread the process normally has, when only the everyday sources of variation are at work.

Rmax: the upper control limit for R

The R chart has its own control limits, calculated from R̄ with constants that depend on the subgroup size n:

• Upper control limit: UCL(R) = D4 × R̄
• Lower control limit: LCL(R) = D3 × R̄

The upper limit is often called Rmax. For subgroups of five, D4 = 2.114 and D3 = 0, so the lower limit is zero. The constants come from published tables in, for example, the AIAG SPC manual.

Continuing the example: if R̄ = 0.050 mm over 25 subgroups, then Rmax = 2.114 × 0.050 = 0.106 mm. The afternoon subgroup with R = 0.13 is above Rmax, which is a signal that the spread has changed.

In the figure each bar is the range of one subgroup. Subgroup 5 rises above Rmax: the spread in that hour was larger than the process normally has, and the cause should be investigated even though the mean looked fine.

Why R̄ matters beyond the R chart

R̄ is used in two other calculations, so its quality affects them both.

• Estimating the standard deviation. The short-term standard deviation is estimated as σ̂ = R̄ / d2. For n = 5, d2 = 2.326. With R̄ = 0.050, σ̂ = 0.050 / 2.326 ≈ 0.0215 mm. This is the σ used in Cp and Cpk, see process capability.
• Limits for the X̄ chart. The limits for the subgroup means are X̿ ± A2 × R̄, with A2 = 0.577 for n = 5.

For this reason, the R chart is read first. If the spread is not stable, the limits on the X̄ chart and the capability figures are built on a moving foundation.

Limitations and misunderstandings

• Rmax is not a tolerance. It is calculated from process data and says nothing about whether the parts meet the requirement.
• The range uses only two values. For larger subgroups, roughly ten or more, the standard deviation of the subgroup (the X̄-s chart) is more efficient.
• A falling R can also be a signal. Values below the centre line for a long run may mean a real improvement, or a gauge with poor resolution that rounds everything to the same value.
• Mixed subgroups inflate R. If a subgroup contains parts from two cavities, the range includes the difference between the cavities.

In short

R shows the spread in one subgroup, R̄ the normal spread, and Rmax the point where the spread is larger than the process can explain. Watch the R chart first: a stable spread is the foundation for everything else on the chart.