4. Sampling distributions and the central limit theorem

Every statistic computed from a sample — a subgroup mean, a subgroup range, a sample standard deviation — is itself a random variable. If you draw a different subgroup from the same process, you get a different value. The probability distribution of that statistic over repeated sampling is called its sampling distribution. Control charting and process capability analysis are both built on sampling distributions: a control chart asks whether an observed statistic is plausible under the sampling distribution of a stable process, and a capability index asks how much of the specification band the underlying process distribution occupies. Confusing the two distributions — the distribution of individual measurements and the distribution of subgroup statistics — is one of the most common sources of error in SPC practice.

From Individual Values to Subgroup Means

Let individual measurements from a stable process have mean μ and standard deviation σ. For a subgroup of size n drawn independently, the sample mean X̄ has:

• Mean: E[X̄] = μ (the estimator is unbiased)
• Standard deviation: σ_X̄ = σ / √n (the standard error of the mean)

The √n reduction is the entire economic argument for subgrouping. With n = 4, the standard error is half of σ; with n = 9, it is one third. This is why an X̄ chart can detect a process shift of, say, 1σ far sooner than a chart of individual values (an I-MR chart) can. The price is that a subgroup mean is insensitive to within-subgroup variation — which is exactly why the R or S chart is plotted alongside it.

If the individual values are normally distributed, X̄ is exactly normal for any n. If they are not normal, X̄ is still approximately normal for moderate n — this is the central limit theorem.

The Central Limit Theorem

The central limit theorem (CLT) states that for independent, identically distributed observations with finite variance, the standardized sample mean converges in distribution to the standard normal:

(X̄ − μ) / (σ / √n)  →  N(0, 1)   as n → ∞

Three practical qualifications matter more than the theorem itself:

1. Convergence is in distribution, not in the tails. The CLT says nothing about how fast the extreme tails of X̄ approach normality. Control chart limits are placed at ±3 standard errors, which is precisely a tail statement. For a strongly skewed individual distribution, n = 5 may give a visually normal histogram of X̄ while the 0.135% tail probability assumed by 3-sigma limits is still badly wrong.
2. Independence is required. If consecutive parts are correlated (autocorrelated process, drifting tool wear, batch effects), the variance of X̄ is not σ²/n. Positive autocorrelation inflates it, so conventional limits are too tight and the chart produces false alarms.
3. Finite variance is required. Heavy-tailed processes (some particle, wear-debris, or defect-size measurements) can violate this in principle, though in practice the more common problem is simply slow convergence.

A useful rule of thumb for mildly skewed data is n ≥ 5; for moderately skewed data, n ≥ 10 to 15; for strongly skewed or bounded data (cycle times, counts, proportions near 0 or 1), transform the data or use an appropriate attribute/EWMA chart rather than relying on the CLT.

Why the Range and Standard Deviation Are Not Normal

The R and S statistics have their own sampling distributions, which are right-skewed and strictly positive. For normally distributed individuals, the expected value of the sample range is E[R] = d₂σ and the expected value of the sample standard deviation is E[S] = c₄σ, where d₂ and c₄ depend only on n. These constants are the bridge between the plotted statistic and the process σ:

σ̂ = R̄ / d₂        (X̄–R chart)
σ̂ = S̄ / c₄        (X̄–S chart)

For n = 5: d₂ = 2.326, c₄ = 0.9400. For n = 4: d₂ = 2.059, c₄ = 0.9213. Because R̄ and S̄ are themselves random, σ̂ carries estimation error — with 25 subgroups of n = 5, the relative standard error of σ̂ is roughly 1/√(2·25) ≈ 14%. This uncertainty propagates directly into capability indices, which is why capability studies demand far more data than control charts do.

Implications for Control Charting

Control limits are quantiles of the sampling distribution of the plotted statistic under the assumption of a stable process. For an X̄ chart with known μ and σ:

UCL = μ + 3σ/√n
CL  = μ
LCL = μ − 3σ/√n

When μ and σ are estimated from the data, the limits become random variables and the actual false-alarm rate differs from the nominal 0.27%. With k = 25 subgroups, the average run length in control is typically around 300–400 rather than 370, and short runs of data give limits that are too tight on average. This is a sampling-distribution effect, not a defect in the chart.

The CLT also explains why X̄ charts are robust to non-normality of individuals while I-MR charts are not. An I-chart plots individual values, so its limits depend directly on the shape of the individual distribution; a skewed process will generate one-sided false alarms. If you must chart individuals, consider a Box-Cox or Johnson transformation, or use an EWMA chart on the transformed scale.

Implications for Process Capability Estimation

Capability indices compare the specification band to the process spread. The critical distinction is which spread:

Cp  = (USL − LSL) / (6σ̂_within)
Cpk = min(USL − X̄, X̄ − LSL) / (3σ̂_within)
Pp  = (USL − LSL) / (6s_overall)
Ppk = min(USL − X̄, X̄ − LSL) / (3s_overall)

Because σ̂_within ≤ s_overall whenever between-subgroup variation exists, C_pk ≥ P_pk in practice. A large gap between them is diagnostic: it signals that the process mean is drifting between subgroups, and the appropriate response is to find and eliminate that source of variation, not to report the more flattering C_pk.

Sampling error in σ̂ translates directly into uncertainty in the indices. For a stable normal process with m subgroups of size n, the approximate relative standard error of C_pk is:

SE(Cpk)/Cpk ≈ sqrt( 1/(2(m(n−1))) + 1/(2(m−1)) )

With m = 25, n = 5, this gives roughly 0.10 — a 10% relative uncertainty. A reported C_pk = 1.33 could plausibly correspond to a true value anywhere from about 1.10 to 1.56. This is why automotive and aerospace customers typically require 25 or more subgroups, and why a single capability number without a confidence interval is weak evidence.

Worked Example

Suppose a machining process produces a shaft diameter with USL = 25.05 mm and LSL = 24.95 mm. From 25 subgroups of n = 5, we obtain X̄̄ = 25.001 mm and R̄ = 0.018 mm. Using d₂ = 2.326 for n = 5:

σ̂_within = R̄ / d₂ = 0.018 / 2.326 = 0.00774 mm

Cp  = (25.05 − 24.95) / (6 × 0.00774) = 0.10 / 0.0464 = 2.15
Cpk = min(25.05 − 25.001, 25.001 − 24.95) / (3 × 0.00774)
    = min(0.049, 0.051) / 0.0232
    = 0.049 / 0.0232 = 2.11

If the overall standard deviation computed from all 125 individual values is s = 0.0091 mm, then P_p = 0.10 / (6 × 0.0091) = 1.83 and P_pk ≈ 1.79. The gap between C_pk = 2.11 and P_pk = 1.79 indicates meaningful between-subgroup variation — perhaps a temperature drift or a fixture change between subgroups. The correct action is to investigate that source, not to accept the higher C_pk as the process capability.

Key Takeaways

• Control limits are quantiles of the sampling distribution of the plotted statistic, not of the individual measurements.
• The standard error σ/√n governs X̄ chart sensitivity; larger n detects smaller shifts but costs more per decision.
• The CLT justifies normal-based X̄ limits for moderate n, but convergence in the extreme tails is slow — be cautious with strongly skewed data.
• R and S statistics are right-skewed; d₂ and c₄ convert them to σ estimates, and those estimates carry substantial sampling error.
• C_p/C_pk use within-subgroup σ; P_p/P_pk use overall s. A large gap between them is a diagnostic signal of between-subgroup variation.
• Capability indices require a stable, in-control process and enough data (typically ≥ 25 subgroups) to keep estimation uncertainty acceptable.