Control charts for variables: X-bar and R

Why it matters

Every process varies. Control charts tell an operator whether today's variation is the normal noise of the process or a sign that something has changed (a worn tool, a new batch of material, a wrong setting), so that the process is adjusted only when it needs to be. X-bar and R charts are the most widely used charts for measured characteristics such as diameters, lengths and weights, and they are a staple of GATE and university papers.

Key ideas

Common and special causes. Common (chance) causes are the many small, always-present sources of variation built into the process. Special (assignable) causes are occasional identifiable events such as tool breakage, a new operator or a material change. A process affected only by common causes is in statistical control: its output is predictable. Control charts detect special causes; they do not say whether parts meet the specification.

Rational subgroups. Small samples (typically n = 4 or 5) of consecutive parts are taken at regular intervals. Parts within a subgroup are made under nearly identical conditions, so the within-subgroup range reflects only common-cause variation; differences between subgroups reveal special causes.

The two charts.

• X-bar chart: plots the subgroup mean X̄; it detects shifts in the process mean (setting).
• R chart: plots the subgroup range R = largest − smallest; it detects changes in process spread (dispersion). They are always used as a pair, and the R chart is read first: the X-bar limits are calculated from R̄, so they are meaningless if the spread is out of control.

Where the limits come from. Limits are set at ±3 standard deviations of the plotted statistic. The process standard deviation is estimated from the average range, σ̂ = R̄/d₂, and the standard deviation of a subgroup mean is σ/√n. Combining these gives the tabulated factor A₂ = 3/(d₂√n). With 3σ limits, a process in control gives a point outside the limits only about 0.27 % of the time (a false alarm, Type I error), so on average about one false alarm per 370 subgroups.

Setting up the charts (Phase I). Collect 20–25 subgroups, compute trial limits, look for points outside them, find and remove the assignable causes, drop those subgroups and recompute. The revised limits are then used for ongoing monitoring (Phase II).

Out-of-control signals. Besides a point beyond the 3σ limits, common run rules (Western Electric) flag: 2 of 3 consecutive points beyond 2σ on the same side; 4 of 5 beyond 1σ on the same side; 8 consecutive points on one side of the centre line (some texts use 7); trends of 6–7 points steadily rising or falling; cycles or hugging the centre line. Run rules make the chart more sensitive to small shifts but increase false alarms.

Control limits are not specification limits. Control limits come from the process; specification (tolerance) limits come from the designer. A process can be in control and still make out-of-tolerance parts (capability problem), or have wide tolerances and drift without producing scrap. Comparing the two is the subject of process capability.

When to use other charts. For subgroups larger than about 10 the range wastes information; an X̄–s chart is used instead. For one reading at a time, an individuals and moving-range (I-MR) chart is used. For counts of defectives or defects, attribute charts are used.

Formulas

X̿ = ΣX̄ᵢ / k and R̄ = ΣRᵢ / k

• X̄ᵢ = mean of subgroup i, Rᵢ = range of subgroup i (unit of the measurement, e.g. mm); k = number of subgroups.

UCL_X̄ = X̿ + A₂·R̄, CL = X̿, LCL_X̄ = X̿ − A₂·R̄

• X-bar chart limits (same unit as the data).

UCL_R = D₄·R̄, CL = R̄, LCL_R = D₃·R̄

• R chart limits.

σ̂ = R̄ / d₂ and σ_X̄ = σ / √n

• Estimated process standard deviation and standard deviation of subgroup means; n = subgroup size.

A₂ = 3 / (d₂·√n)

Factors (from the standard SQC table; take other n from your data book):

n	A₂	D₃	D₄	d₂
2	1.880	0	3.267	1.128
3	1.023	0	2.574	1.693
4	0.729	0	2.282	2.059
5	0.577	0	2.114	2.326
6	0.483	0	2.004	2.534

ARL = 1 / p

• Average run length: mean number of subgroups until a signal, where p = probability that one point falls outside the limits.

Worked examples

Example 1 (standard): trial limits Given: 20 subgroups of n = 5 shaft diameters give ΣX̄ = 500.40 mm and ΣR = 1.60 mm. Find the limits for both charts and estimate σ.

1. X̿ = 500.40 / 20 = 25.020 mm; R̄ = 1.60 / 20 = 0.080 mm.
2. For n = 5: A₂ = 0.577, D₃ = 0, D₄ = 2.114, d₂ = 2.326.
3. UCL_X̄ = 25.020 + 0.577 × 0.080 = 25.020 + 0.0462 = 25.066 mm.
4. LCL_X̄ = 25.020 − 0.0462 = 24.974 mm.
5. UCL_R = 2.114 × 0.080 = 0.169 mm; LCL_R = 0 × 0.080 = 0.
6. σ̂ = R̄ / d₂ = 0.080 / 2.326 = 0.0344 mm.

X-bar chart: 24.974 to 25.066 mm (CL 25.020 mm); R chart: 0 to 0.169 mm (CL 0.080 mm); σ̂ = 0.034 mm.

Example 2 (GATE level): detecting a shift Given: an X-bar chart with n = 4 has X̿ = 50.00 mm and R̄ = 2.059 mm. The process mean suddenly shifts to 51.00 mm with the spread unchanged. Find the probability that the first subgroup after the shift falls outside the limits, and the average run length to detect it.

1. σ̂ = R̄ / d₂ = 2.059 / 2.059 = 1.000 mm; σ_X̄ = 1.000 / √4 = 0.500 mm.
2. Limits: A₂·R̄ = 0.729 × 2.059 = 1.501 mm, so UCL = 51.501 mm and LCL = 48.499 mm.
3. Upper tail: z = (51.501 − 51.00) / 0.500 = 1.00; P(X̄ > UCL) = 1 − Φ(1.00) = 0.159.
4. Lower tail: z = (48.499 − 51.00) / 0.500 = −5.00; P ≈ 0.
5. p = 0.159; ARL = 1 / 0.159 = 6.3 subgroups.

Probability of detection on the first subgroup ≈ 0.159; about 6.3 subgroups on average to detect a 1σ shift. This is why run rules or larger subgroups are used to catch small shifts quickly.

Common mistakes

• Using a sample standard deviation formula in place of R̄/d₂, or using A₂ from the wrong subgroup size.
• Interpreting the X-bar chart before checking that the R chart is in control.
• Drawing specification limits on an X-bar chart; individual parts and subgroup means have different spreads.
• Forgetting that LCL of the R chart is zero (not negative) for n ≤ 6.
• Forgetting to drop out-of-control subgroups and recompute trial limits.
• Treating every point near a limit as a signal and adjusting the process (over-adjustment, or tampering, increases variation).

For GATE PI

Expect numericals giving subgroup means and ranges, or their sums, with the factor table supplied: compute the centre line, UCL and LCL of X-bar and R charts, and estimate σ from R̄/d₂. Conceptual questions cover common versus special causes, what each chart detects, Type I error with 3σ limits, and the difference between control and specification limits. Practise one probability-of-detection problem using the normal table.

Quick check

1. Which chart detects a change in process spread?
2. For n = 5, X̿ = 40.0 mm and R̄ = 2.0 mm. What is the UCL of the X-bar chart?
3. What fraction of points falls outside 3σ limits for an in-control normal process?
4. For n = 4 and R̄ = 4.118 mm, estimate σ.

Answers: 1. The R chart. 2. 41.154 mm. 3. About 0.27 %. 4. 2.0 mm.