Guide
How do you build and read an X-bar and R chart for a turned diameter?
An X-bar and R chart is a pair of Shewhart control charts that plot the average and the range of small subgroups of consecutive parts, so you can see whether a process is stable or drifting before parts go out of tolerance. For a turned diameter you measure about 5 consecutive parts at a fixed interval, set limits from the grand average and the average range, and react to the signals, not to individual readings.
By Saola Manufacturing engineering team · Published · Updated
This page walks through one example from data to decision. When lots are only a few hundred pieces, the textbook chart struggles; the SPC for low-volume machined parts guide covers the alternatives, and the Cpk vs Ppk guide covers capability numbers in depth.
On this page
When is an X-bar and R chart the right chart?
Use it for a measured characteristic, such as a diameter, made in a steady stream where you can take small subgroups of consecutive parts. It is the standard choice when subgroups are small, roughly 2 to 10 parts.
The NIST/SEMATECH e-Handbook notes that the range approach is quite satisfactory for subgroup sizes up to around 10, that larger subgroups are better served by a standard deviation chart, and that a typical subgroup size is 4 or 5 (NIST/SEMATECH e-Handbook, section 6.3.2.1). ASTM E2587, the ASTM practice for control charts in statistical process control, covers the same Shewhart charts for variables and attributes data (ASTM E2587-25 record).
A turned outside diameter on a CNC lathe is a good fit. Parts come off one at a time from a fixed setup, the main sources of variation (insert wear, thermal growth, bar variation, offset changes) act over time, and a micrometer or snap gauge gives a reading in seconds. If you only ever get one reading per interval, use an individuals and moving range chart instead.
How do you collect the subgroups?
Take a rational subgroup: several consecutive parts measured together, so the range inside the subgroup shows only short-term variation. Space the subgroups in time, for example every hour or every set number of parts, so the chart can see drift between them.
The point of the subgroup is that its parts were made under nearly identical conditions. Five parts in a row off the same spindle, same insert, same offset, give a range that reflects the machine's short-term repeatability. If you instead pull five parts from a tote that spans a morning, the range swallows the drift and the limits come out too wide to detect anything.
Write down, with each subgroup, the time, the machine, the operator, and any event: an insert change, an offset adjustment, a new bar, a restart after a break. Those notes are what turn a signal on the chart into a cause you can fix. Also make sure the gauge is good enough: if the gauge repeatability is a large share of the tolerance, the range chart measures the gauge, not the process. The gage R&R guide covers that check.
The example below is a hypothetical but typical case: an OD of Ø12.000 mm with limits of 11.975 to 12.025 mm on a turned pin, five consecutive parts measured every hour, 25 subgroups in the baseline. The first five subgroups are shown so you can follow the arithmetic.
| Subgroup | Readings (mm) | Average X-bar (mm) | Range R (mm) |
|---|---|---|---|
| 1 | 12.003, 12.006, 12.001, 12.008, 12.002 | 12.004 | 0.007 |
| 2 | 12.010, 12.000, 11.998, 12.005, 12.007 | 12.004 | 0.012 |
| 3 | 12.012, 12.004, 11.999, 12.006, 12.009 | 12.006 | 0.013 |
| 4 | 11.996, 12.008, 12.003, 12.001, 12.007 | 12.003 | 0.012 |
| 5 | 12.001, 12.013, 12.004, 11.999, 12.008 | 12.005 | 0.014 |
| All 25 (summary) | 125 readings | Grand average 12.004 | Average range 0.012 |
How do you calculate the control limits?
Average the subgroup averages to get the center line, average the ranges to get R-bar, then apply the A2, D3 and D4 factors for your subgroup size. For subgroups of 5 the factors are A2 = 0.577, D3 = 0 and D4 = 2.115, per the NIST/SEMATECH tables.
The formulas in the NIST/SEMATECH e-Handbook are: X-bar chart limits = grand average ± A2 × R-bar, and R chart limits from D3 × R-bar to D4 × R-bar (NIST/SEMATECH e-Handbook). The factor table in the same section gives A2 = 0.577, D3 = 0 and D4 = 2.115 for subgroups of five. Applied to the example:
| Chart | Center line (mm) | Upper control limit (mm) | Lower control limit (mm) |
|---|---|---|---|
| X-bar chart | 12.004 | 12.004 + 0.577 × 0.012 = 12.011 | 12.004 − 0.577 × 0.012 = 11.997 |
| R chart | 0.012 | 2.115 × 0.012 = 0.025 | 0 × 0.012 = 0 |
| Process sigma estimate | R-bar / d2 = 0.012 / 2.326 = 0.0052 | – | – |
| Capability check | Cp = 0.050 / (6 × 0.0052) ≈ 1.6 | Cpk = (12.025 − 12.004) / (3 × 0.0052) ≈ 1.36 | – |
Two cautions. First, the control limits come from the process data, not from the drawing tolerance, and the X-bar chart plots averages, so never draw the spec limits on it; averages are always tighter than individual parts. Second, check the R chart first: if the ranges are not in control, R-bar is not a trustworthy measure of spread and the X-bar limits are wrong too. The d2 value of 2.326 for n = 5 follows from the NIST definition A2 = 3 / (d2 × √n), and the capability numbers are only meaningful once both charts are stable.
How do you read the chart once it is running?
React to defined signals: a point outside the limits, a run of points on one side of the center line, or a steady trend. Each signal on a lathe usually has a short list of likely causes.
The Western Electric rules quoted by NIST include any point beyond 3 sigma, 2 of the last 3 beyond 2 sigma on the same side, 4 of the last 5 beyond 1 sigma, and 8 consecutive points on one side of the center line; NIST also lists 6 in a row trending up or down as a trend signal (NIST/SEMATECH e-Handbook, section 6.3.2). Agree with the supplier which rules apply before the run starts; using all of them raises false alarms, using only the first misses slow drift.
| What you see | Likely cause on a CNC lathe | Typical action |
|---|---|---|
| X-bar trending upward over hours | Insert flank wear growing the OD | Planned offset or insert change before the limit, not after |
| Step change in X-bar after an event | Insert change, offset entry, new bar lot | Check the event log; confirm the offset was entered correctly |
| Run of points on one side early in a shift | Spindle and machine warm-up | Warm-up cycle, or first-part check after start |
| R chart point above its limit | Chip wrap, loose collet or chuck, burr on the gauging face, gauge error | Stop, inspect the subgroup, check workholding and the gauge |
| R chart creeping upward | Worn insert edge, built-up edge, coolant or chip control problems | Inspect insert and coolant; review cutting data |
| Points hugging the center line | Limits from bad data, or readings rounded or edited | Recheck the baseline and the gauge resolution |
When should the control limits be recalculated?
Only for a real reason: enough new stable data to improve the estimate, or a deliberate process change. Do not recalculate every lot, or the chart stops detecting anything.
NIST advises changing limits only for a valid, compelling reason and gives three examples: at least 30 additional points from an unchanged process, a major process change, or a known, preventable event that altered the process (NIST/SEMATECH e-Handbook). On a turned part, a new machine, a new insert grade or a change of bar supplier counts as a process change; a new shift does not.
Exclude subgroups with an identified, corrected special cause from the baseline and recompute, but do not drop points simply because they look inconvenient. Keep the excluded subgroups and the reason in the record so the customer can audit the limits.
What should the drawing and RFQ say about control charts?
Mark the characteristic, say which chart, subgroup size and frequency you expect, and state what gets reported: the chart, the limits, and any out-of-control events with the action taken. A note that just says SPC REQUIRED is too vague to quote.
A typical way is to flag the dimension as a critical or key characteristic on the drawing (the critical characteristics guide covers the symbols) and put the SPC terms in the quality agreement or PO. For example: DIM 4 (OD): X-BAR AND R CHART, SUBGROUP 5 CONSECUTIVE PARTS, ONE SUBGROUP PER [INTERVAL]. SUBMIT CHART AND LIMITS WITH EACH LOT. REPORT OUT-OF-CONTROL POINTS WITH CAUSE AND ACTION.
Ask for raw subgroup data, not only the chart image, so you can recompute the limits. If a capability target applies, state it and say whether it is Cpk from a stable chart or Ppk from a lot study. For orders of a few hundred pieces, accept that 25 subgroups may not exist in one lot, and agree whether data carries over from lot to lot.
How are charted characteristics handled on a Saola order?
Tell us at drawing review which dimensions need charting, the chart type and the reporting you want, and it is agreed before quoting. Every new part gets a first-article inspection report and every lot a dimensional inspection record.
Turned diameters are run on the lead plant's CNC turning lines, and the inspection records on the quality page apply to every order: a first-article report on new parts and a dimensional inspection record each lot. Charting, subgroup size and the report format are extra requirements, so list them in the RFQ with the characteristic numbers.
Send the drawing for a quote with the charted dimensions marked and the reporting terms written out.
Frequently asked
Why use the range instead of the standard deviation?
For small subgroups the range is easy to compute on the shop floor and loses little information. NIST notes that a typical subgroup is 4 or 5 parts and that the standard deviation chart is preferable for larger subgroups.
How many subgroups do I need before the limits are valid?
Common practice is 20 to 25 subgroups for a first set of limits, treated as trial limits until more data confirm them.
Can I put the drawing tolerance on the X-bar chart?
No. The chart plots subgroup averages, which vary less than individual parts, so a tolerance line on it is misleading. Compare tolerance with individual parts or with a capability index instead.
Is a point inside the limits but outside tolerance possible?
Yes, if the process is stable but not capable. That is a capability problem, not a control problem, and the fix is a process change, not tighter monitoring.
What if the R chart shows zero range often?
The gauge resolution is probably too coarse for the tolerance, so readings repeat. Use a gauge that resolves at least a tenth of the tolerance, as the gauge accuracy guide explains.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods, 6.3.2.1 Shewhart X-bar and R and S Control Charts, National Institute of Standards and Technology, 2012
- NIST/SEMATECH e-Handbook of Statistical Methods, 6.3.2 What are Variables Control Charts? (Western Electric rules), National Institute of Standards and Technology, 2012
- ASTM E2587-25, Standard Practice for Use of Control Charts in Statistical Process Control (record and scope), ASTM International via DIN Media, 2025