21.1 Process Capability Indices and Defect Levels

FE Industrial and Systems Study Guide Exam

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20 Chapter 20: Statistical Process Control

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21 Chapter 21: Process Capability & Acceptance Sampling

21.1 Process Capability Indices and Defect Levels 21.2 Acceptance Sampling Concepts and Operating Characteristic Curves 21.3 Sampling Risks, AQL, LTPD, and Standard Sampling Plans

22 Chapter 22: Systems Engineering & Life-Cycle Management

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23 Chapter 23: Reliability Engineering & Risk Analysis

23.1 Reliability Mathematics, Hazard Rates, and the Bathtub Curve 23.2 System Reliability Configurations: Series, Parallel, and k-out-of-n Systems 23.3 Failure Modes and Effects Analysis and Fault Tree Analysis

21.1 Process Capability Indices and Defect Levels

Key Takeaways Explain This Topic AI

Statistical control (absence of assignable causes) and verified distribution normality are absolute prerequisites before evaluating process capability; indices calculated on an unstable process are mathematically meaningless.

Potential capability Cp = (USL - LSL) / (6*sigma) reflects the inherent precision of the process relative to specification tolerance assuming perfect centering; actual capability Cpk = min(Cpu, Cpl) penalizes for process mean off-centering.

The mathematical relationship Cpk = Cp * (1 - k), where k = |USL + LSL - 2*mu| / (USL - LSL), dictates that Cpk <= Cp always, with equality holding strictly when the process mean is centered at the nominal specification midpoint (k = 0).

Benchmark capability values correspond directly to defect rates: Cpk = 1.0 represents a barely capable 3-sigma process (2,700 PPM total defects), Cpk = 1.33 is the standard 4-sigma industrial minimum (~63 to 66 PPM), and Cpk = 2.0 represents a Six Sigma process (3.4 DPMO under a 1.5-sigma shift).

Short-term capability (Cp, Cpk) uses within-subgroup dispersion sigma_hat = R_bar / d2 (or s_bar / c4), while long-term performance (Pp, Ppk) uses overall sample standard deviation s, capturing machine drift, tool wear, and batch-to-batch variations.

Last updated: September 2026

Process capability analysis bridges statistical quality control and design engineering. While statistical process control (SPC) charts monitor an active process to maintain stability over time, capability analysis evaluates whether that stable process can consistently produce parts that conform to customer engineering specifications. On the FE Industrial and Systems exam, mastery of capability formulas ( C p C_p C p ​ , C p k C_{pk} C p k ​ , P p P_p P p ​ , P p k P_{pk} P p k ​ ), the centering parameter k k k , standard normal Z Z Z -score defect conversions, and the distinction between short-term within-subgroup variation and long-term overall variation is essential.

1. Preconditions for Process Capability Analysis

Before computing any process capability index, an industrial engineer must satisfy two non-negotiable statistical preconditions: Prerequisites for Valid Capability Analysis

• ┌───────────────────────────┐ ┌───────────────────────────┐
• │ Statistical Stability │ │ Normality of Data │
• │ (In Statistical Control)│ │ X ~ N(mu, sigma^2) │
• └─────────────┬─────────────┘ └─────────────┬─────────────┘
• │ │
• ▼ ▼
• No Assignable Causes Goodness-of-Fit / Normality
• (Common Causes Only) (Anderson-Darling / W-Test)
• │ │
• └────────────────┬────────────────┘
• ▼
• Proceed to Calculate Cp, Cpk, PPM
• 1. The Process Must Be in Statistical Control

A process is in a state of statistical control when it exhibits only natural, random, common-cause variation and is free from assignable (special) causes. Stability must be verified using Shewhart control charts (such as X ˉ − R \bar{X}-R X ˉ − R or X ˉ − s \bar{X}-s X ˉ − s charts) over an extended baseline period (typically at least 20 to 25 rational subgroups):

All subgroup sample points must fall strictly within the 3-sigma control limits ( U C L UCL U C L and L C L LCL L C L ).

The plotted points must exhibit random scatter without non-random patterns, runs, trends, cycles, or stratification.

Exam Rule : If a process is not in statistical control, calculating C p C_p C p ​ or C p k C_{pk} C p k ​ is completely invalid. An unstable process has a shifting mean μ ( t ) \mu(t) μ ( t ) and/or time-varying variance σ 2 ( t ) \sigma^2(t) σ 2 ( t ) ; predicting future fraction defective or conformance percentage from an unstable distribution is mathematically meaningless.

2. Normality of the Quality Characteristic

Standard capability indices assume that the underlying continuous quality characteristic X X X is normally distributed: X ∼ N ( μ , σ 2 ) X \sim N(\mu, \sigma^2) X ∼ N ( μ , σ 2 ) .

Normality should be evaluated using normal probability plots and statistical goodness-of-fit tests (such as the Anderson-Darling, Shapiro-Wilk, or Kolmogorov-Smirnov tests).

If the distribution is significantly skewed or non-normal (such as flatness, roundness, surface roughness, or cycle times), standard formulas produce severe errors in estimated defect levels. In non-normal cases, the industrial engineer must either:

Transform the raw data into an approximately normal distribution using mathematical transformations (e.g., Box-Cox power transformation or Johnson system of distributions).

Apply non-parametric percentile-based capability models, where natural process width is defined by the 0.135th and 99.865th percentiles ( P 0.135 P_{0.135} P 0.135 ​ and P 99.865 P_{99.865} P 99.865 ​ ): C p k = min ⁡ [ USL − P 50 P 99.865 − P 50 , P 50 − LSL P 50 − P 0.135 ] C_{pk} = \min\left[ \frac{\text{USL} - P_{50}}{P_{99.865} - P_{50}}, \, \frac{P_{50} - \text{LSL}}{P_{50} - P_{0.135}} \right] C p k ​ = min [ P 99.865 ​ − P 50 ​ USL − P 50 ​ ​ , P 50 ​ − P 0.135 ​ P 50 ​ − LSL ​ ]

2. Potential Process Capability Index ( C p C_p C p ​ )

The potential capability index ( C p C_p C p ​ ) evaluates the inherent precision of a manufacturing process. It compares the allowable engineering tolerance band against the natural dispersion of the process, completely disregarding where the process distribution is centered: C p = USL − LSL 6 σ C_p = \frac{\text{USL} - \text{LSL}}{6\sigma} C p ​ = 6 σ USL − LSL ​

Where:

• USL \text{USL} USL = Upper Specification Limit (customer-defined maximum allowable dimension)
• LSL \text{LSL} LSL = Lower Specification Limit (customer-defined minimum allowable dimension)
• USL − LSL \text{USL} - \text{LSL} USL − LSL = Total specification tolerance band ("Voice of the Customer")

6 σ 6\sigma 6 σ = Natural process spread ("Voice of the Process"), containing 99.73 % 99.73\% 99.73% of output under a normal distribution

σ \sigma σ = Inherent process standard deviation Potential Capability (Cp) Anatomy

• LSL Target USL
• │ │ │
• ├─────────────────────┼──────────────────────┤
• │ Tolerance = USL - LSL │
• │ │
• ┌───┐
• ┌─┘ └─┐
• ┌─┘ └─┐ Process Spread = 6*sigma
• ┌┘ └┐ (Voice of the Process)
• ────┴─────────────┴────────────────────────────
• 3sigma mu +3sigma
• Properties and Limitations of C p C_p C p ​

Theoretical Upper Bound : C p C_p C p ​ represents the maximum capability that the process could achieve if its operating mean μ \mu μ were adjusted to sit perfectly at the nominal specification midpoint M = USL + LSL 2 M = \frac{\text{USL} + \text{LSL}}{2} M = 2 USL + LSL ​ .

Insensitivity to Location : C p C_p C p ​ does not depend on the process mean μ \mu μ . A machining center could have an outstanding C p = 2.50 C_p = 2.50 C p ​ = 2.50 , but if its tool offset is improperly calibrated such that μ > USL \mu > \text{USL} μ > USL , every single manufactured part will be oversized scrap. Thus, C p C_p C p ​ measures potential , never actual conformance.

Bilateral Specifications Only : C p C_p C p ​ can only be calculated when both an upper and lower specification limit are defined. For unilateral (one-sided) specifications, C p C_p C p ​ is undefined.

3. Actual Process Capability Index ( C p k C_{pk} C p k ​ ) and Centering Factor ( k k k )

To account for process centering relative to specification limits, industrial engineers compute the actual capability index ( C p k C_{pk} C p k ​ ). C p k C_{pk} C p k ​ measures the clearance between the operating mean μ \mu μ and the nearest specification limit, scaled against half the natural process spread ( 3 σ 3\sigma 3 σ ).

Unilateral Capability Indices

First, define capability relative to each individual specification boundary:

Upper Capability Index ( C p u C_{pu} C p u ​ ) : C p u = USL − μ 3 σ C_{pu} = \frac{\text{USL} - \mu}{3\sigma} C p u ​ = 3 σ USL − μ ​

Lower Capability Index ( C p l C_{pl} C pl ​ ) : C p l = μ − LSL 3 σ C_{pl} = \frac{\mu - \text{LSL}}{3\sigma} C pl ​ = 3 σ μ − LSL ​

Defining Actual Capability ( C p k C_{pk} C p k ​ )

The overall actual capability index is the minimum of the unilateral indices: C p k = min ⁡ ( C p u , C p l ) = min ⁡ ( USL − μ 3 σ , μ − LSL 3 σ ) C_{pk} = \min(C_{pu}, \, C_{pl}) = \min\left( \frac{\text{USL} - \mu}{3\sigma}, \, \frac{\mu - \text{LSL}}{3\sigma} \right) C p k ​ = min ( C p u ​ , C pl ​ ) = min ( 3 σ USL − μ ​ , 3 σ μ − LSL ​ )

Because C p k C_{pk} C p k ​ is bounded by the specification limit closest to the process mean, it directly reflects the worst-case tail probability of producing defective parts.

The Centering Factor ( k k k )

The degree of process off-centering is quantified by the non-dimensional parameter k k k (also called the bias factor). Let M M M be the nominal specification midpoint: M = USL + LSL 2 M = \frac{\text{USL} + \text{LSL}}{2} M = 2 USL + LSL ​

The centering factor k k k expresses the distance between the process mean μ \mu μ and midpoint M M M as a fraction of the allowable half-tolerance band: k = ∣ M − μ ∣ USL − LSL 2 = ∣ USL + LSL − 2 μ ∣ USL − LSL k = \frac{|M - \mu|}{\frac{\text{USL} - \text{LSL}}{2}} = \frac{|\text{USL} + \text{LSL} - 2\mu|}{\text{USL} - \text{LSL}} k = 2 USL − LSL ​ ∣ M − μ ∣ ​ = USL − LSL ∣ USL + LSL − 2 μ ∣ ​

The Fundamental Identity Linking C p C_p C p ​ and C p k C_{pk} C p k ​

Using k k k , the relationship between potential and actual capability is formulated as: C p k = C p ( 1 − k ) C_{pk} = C_p (1 - k) C p k ​ = C p ​ ( 1 − k ) Process Centering Regimes and k Values

• k = 0 0 < k < 1 k = 1 k > 1
• (Centered) (Off-Center) (On Spec Limit) (Outside Spec)
• │ │ │ │
• Cpk = Cp 0 < Cpk < Cp Cpk = 0 Cpk < 0
• Zero Bias Partial Shift 50% Defective >50% Defective

Diagnostic Interpretation of k k k :

k = 0 k = 0 k = 0 ( C p k = C p C_{pk} = C_p C p k ​ = C p ​ ) : The process mean is perfectly centered at M M M . The process operates at its maximum theoretical capability.

0 < k < 1 0 < k < 1 0 < k < 1 ( 0 < C p k < C p 0 < C_{pk} < C_p 0 < C p k ​ < C p ​ ) : The process mean is shifted away from target, but remains within the specification boundaries. As off-centering ∣ μ − M ∣ |\mu - M| ∣ μ − M ∣ increases, C p k C_{pk} C p k ​ drops linearly.

k = 1 k = 1 k = 1 ( C p k = 0 C_{pk} = 0 C p k ​ = 0 ) : The process mean falls exactly on one of the specification limits ( μ = USL \mu = \text{USL} μ = USL or μ = LSL \mu = \text{LSL} μ = LSL ). In this condition, exactly 50 % 50\% 50% of production violates that specification limit under a normal distribution.

k > 1 k > 1 k > 1 ( C p k < 0 C_{pk} < 0 C p k ​ < 0 ) : The process mean lies entirely outside the specification limits. More than half of total production is defective.

Exam Rule : C p k C_{pk} C p k ​ can never exceed C p C_p C p ​ ( C p k ≤ C p C_{pk} \le C_p C p k ​ ≤ C p ​ always). If your calculation yields C p k > C p C_{pk} > C_p C p k ​ > C p ​ , an arithmetic error has occurred.

4. Interpretation of Capability Values and Defect Benchmarks

Industrial manufacturing systems classify process capability according to established thresholds. Conformance is traditionally quantified in parts per million (PPM) or defects per million opportunities (DPMO) .

Capability Value Process Classification Equivalent Half-Spread Conformance Rate (Centered) Expected Defect Level (Centered)

C p k < 1.0 C_{pk} < 1.0 C p k ​ < 1.0 Incapable < 3 σ < 3\sigma < 3 σ to spec < 99.73 % < 99.73\% < 99.73% > 2 , 700 PPM > 2,700\text{ PPM} > 2 , 700 PPM (severe scrap/rework)

C p k = 1.00 C_{pk} = 1.00 C p k ​ = 1.00 Barely Capable (3-Sigma) 3.0 σ 3.0\sigma 3.0 σ to spec 99.7300 % 99.7300\% 99.7300% 2 , 700 PPM 2,700\text{ PPM} 2 , 700 PPM ( 1 , 350 PPM 1,350\text{ PPM} 1 , 350 PPM per side)

C p k = 1.33 C_{pk} = 1.33 C p k ​ = 1.33 Industry Minimum (4-Sigma) 4.0 σ 4.0\sigma 4.0 σ to spec 99.9937 % 99.9937\% 99.9937% 63.3 PPM 63.3\text{ PPM} 63.3 PPM two-sided ( 31.7 PPM 31.7\text{ PPM} 31.7 PPM per side)

C p k = 1.50 C_{pk} = 1.50 C p k ​ = 1.50 Highly Capable (4.5-Sigma) 4.5 σ 4.5\sigma 4.5 σ to spec 99.99932 % 99.99932\% 99.99932% 6.8 PPM 6.8\text{ PPM} 6.8 PPM

C p k = 1.67 C_{pk} = 1.67 C p k ​ = 1.67 Critical / Safety (5-Sigma) 5.0 σ 5.0\sigma 5.0 σ to spec 99.999943 % 99.999943\% 99.999943% 0.57 PPM 0.57\text{ PPM} 0.57 PPM ( 570 PPB 570\text{ PPB} 570 PPB )

C p k = 2.00 C_{pk} = 2.00 C p k ​ = 2.00 Six Sigma Quality (6-Sigma) 6.0 σ 6.0\sigma 6.0 σ to spec 99.9999998 % 99.9999998\% 99.9999998% 0.002 PPM 0.002\text{ PPM} 0.002 PPM ( 2 PPB 2\text{ PPB} 2 PPB ) without shift Process Tolerance vs. Specification Limits

• Cpk = 1.00 (3-sigma): Spec width = 6*sigma
• LSL Nominal USL
• ├───────────────┬───────────────┼───────────────┬───────────────┤
• │◄───────────── 3*sigma ───────►│◄───────────── 3*sigma ───────►│
• Defects: 1350 PPM Defects: 1350 PPM

• Cpk = 1.33 (4-sigma): Spec width = 8*sigma (Safety Margin = 1*sigma per side)
• LSL Nominal USL
• ├─────────┬─────┬───────────────┼───────────────┬─────┬─────────┤
• │ Margin │◄─── 3*sigma ───────►│◄───────────── 3*sigma ───────►│ Margin │
• Defects: 31.7 PPM Defects: 31.7 PPM

• Cpk = 2.00 (6-sigma): Spec width = 12*sigma (Safety Margin = 3*sigma per side)
• LSL Nominal USL
• ├─── Margin ────┬───────────────┼───────────────┬──── Margin ───┤
• │◄── 3*sigma ──►│◄─── 3*sigma ─►│◄─── 3*sigma ─►│◄── 3*sigma ──►│
• The Motorola 1.5-Sigma Shift and 3.4 DPMO

In standard Six Sigma methodology (developed at Motorola), real-world manufacturing processes are observed to experience non-random long-term drifts due to temperature cycles, tool wear, bearing play, and raw material batch variance. This drift is empirically modeled as a 1.5 σ 1.5\sigma 1.5 σ shift in the process mean over the long run:

If a process has short-term capability of C p = 2.00 C_p = 2.00 C p ​ = 2.00 , the specification limits lie at ± 6 σ \pm 6\sigma ± 6 σ from nominal.

When the process mean drifts by 1.5 σ 1.5\sigma 1.5 σ toward the upper limit, the clearance to the nearest limit decreases from 6 σ 6\sigma 6 σ to 6 σ − 1.5 σ = 4.5 σ 6\sigma - 1.5\sigma = 4.5\sigma 6 σ − 1.5 σ = 4.5 σ .

The resulting capability drops to: C p k = 4.5 σ 3 σ = 1.50 C_{pk} = \frac{4.5\sigma}{3\sigma} = 1.50 C p k ​ = 3 σ 4.5 σ ​ = 1.50

The probability of exceeding the near specification limit is: P ( Z > 4.5 ) = 1 − Φ ( 4.5 ) = 3.39767 × 10 − 6 ≈ 3.4 × 10 − 6 P(Z > 4.5) = 1 - \Phi(4.5) = 3.39767 \times 10^{-6} \approx 3.4 \times 10^{-6} P ( Z > 4.5 ) = 1 − Φ ( 4.5 ) = 3.39767 × 1 0 − 6 ≈ 3.4 × 1 0 − 6

This corresponds to exactly 3.4 DPMO 3.4\text{ DPMO} 3.4 DPMO (defects per million opportunities).

Exam Trap : A centered 6-sigma process produces 0.002 PPM (2 parts per billion). The famous 3.4 DPMO figure applies only when assuming the standard 1.5 σ 1.5\sigma 1.5 σ off-center drift!

5. Short-Term Capability ( C p , C p k C_p, C_{pk} C p ​ , C p k ​ ) vs. Long-Term Performance ( P p , P p k P_p, P_{pk} P p ​ , P p k ​ )

A frequent source of confusion on the FE exam is the operational distinction between capability ( C p , C p k C_p, C_{pk} C p ​ , C p k ​ ) and performance ( P p , P p k P_p, P_{pk} P p ​ , P p k ​ ). The difference lies entirely in how process variability ( σ \sigma σ ) is estimated.

Attribute Capability Indices ( C p , C p k C_p, C_{pk} C p ​ , C p k ​ ) Performance Indices ( P p , P p k P_p, P_{pk} P p ​ , P p k ​ )

Time Horizon Short-term (hours, days, single batch) Long-term (weeks, months, multiple batches)

Variation Captured Within-subgroup variation only (inherent machine precision) Overall variation (within-subgroup + between-subgroup drift + tool wear)

Dispersion Estimator σ ^ within = R ˉ d 2 or s ˉ c 4 \hat{\sigma}_{\text{within}} = \frac{\bar{R}}{d_2} \quad \text{or} \quad \frac{\bar{s}}{c_4} σ ^ within ​ = d 2 ​ R ˉ ​ or c 4 ​ s ˉ ​ s = ∑ i = 1 N ( X i − X ˉ ) 2 N − 1 s = \sqrt{\frac{\sum_{i=1}^N (X_i - \bar{X})^2}{N - 1}} s = N − 1 ∑ i = 1 N ​ ( X i ​ − X ˉ ) 2 ​ ​ (sample standard deviation)

Sensitivity to Instability Insensitive to shifts between subgroups Highly sensitive to shifts between subgroups

Formulas C p = USL − LSL 6 σ ^ within C_p = \frac{\text{USL}-\text{LSL}}{6\hat{\sigma}_{\text{within}}} C p ​ = 6 σ ^ within ​ USL − LSL ​ ; C p k = min ⁡ ( USL − μ 3 σ ^ within , μ − LSL 3 σ ^ within ) C_{pk} = \min\left(\frac{\text{USL}-\mu}{3\hat{\sigma}_{\text{within}}}, \frac{\mu-\text{LSL}}{3\hat{\sigma}_{\text{within}}}\right) C p k ​ = min ( 3 σ ^ within ​ USL − μ ​ , 3 σ ^ within ​ μ − LSL ​ ) P p = USL − LSL 6 s P_p = \frac{\text{USL}-\text{LSL}}{6s} P p ​ = 6 s USL − LSL ​ ; P p k = min ⁡ ( USL − X ˉ 3 s , X ˉ − LSL 3 s ) P_{pk} = \min\left(\frac{\text{USL}-\bar{X}}{3s}, \frac{\bar{X}-\text{LSL}}{3s}\right) P p k ​ = min ( 3 s USL − X ˉ ​ , 3 s X ˉ − LSL ​ )

Engineering Diagnostics Using C p k C_{pk} C p k ​ and P p k P_{pk} P p k ​

Because total variation incorporates all sources of disturbance, the sample standard deviation is virtually always larger than within-subgroup variation: s ≥ σ ^ within s \ge \hat{\sigma}_{\text{within}} s ≥ σ ^ within ​ . Consequently: P p ≤ C p and P p k ≤ C p k P_p \le C_p \quad \text{and} \quad P_{pk} \le C_{pk} P p ​ ≤ C p ​ and P p k ​ ≤ C p k ​

Case 1 ( C p k ≈ P p k C_{pk} \approx P_{pk} C p k ​ ≈ P p k ​ ) : The process is operating with negligible between-subgroup drift. Total variation is dominated by pure white-noise within-subgroup variation. The process is stable and predictable.

Case 2 ( C p k ≫ P p k C_{pk} \gg P_{pk} C p k ​ ≫ P p k ​ ) : Inherent machine capability is excellent, but significant assignable causes, day-to-day temperature fluctuations, raw material batch shifts, or setup differences exist between subgroups. Improvement efforts must focus on process standardization, setup controls, and environmental regulation rather than purchasing a higher-precision machine.

6. Calculating Expected Defect Rates (PPM) Using Z-Scores

To compute the expected fraction defective ( p p p ) or defect rate in parts per million (PPM) for a normally distributed process, calculate the standard normal Z Z Z -scores relative to each specification limit:

Upper and Lower Z-Scores Z USL = USL − μ σ = 3 C p u Z_{\text{USL}} = \frac{\text{USL} - \mu}{\sigma} = 3 C_{pu} Z USL ​ = σ USL − μ ​ = 3 C p u ​ Z LSL = μ − LSL σ = 3 C p l Z_{\text{LSL}} = \frac{\mu - \text{LSL}}{\sigma} = 3 C_{pl} Z LSL ​ = σ μ − LSL ​ = 3 C pl ​

Notice the direct relationship between Z Z Z -scores and unilateral capability indices: the Z Z Z -score distance to a specification limit is identically three times the corresponding unilateral capability index ( Z = 3 C Z = 3C Z = 3 C ).

Computing Tail Probabilities

Using the cumulative standard normal distribution function Φ ( z ) = P ( Z ≤ z ) \Phi(z) = P(Z \le z) Φ ( z ) = P ( Z ≤ z ) :

Fraction Defective Above USL ( p upper p_{\text{upper}} p upper ​ ) : p upper = P ( X > USL ) = 1 − Φ ( Z USL ) p_{\text{upper}} = P(X > \text{USL}) = 1 - \Phi(Z_{\text{USL}}) p upper ​ = P ( X > USL ) = 1 − Φ ( Z USL ​ )

Fraction Defective Below LSL ( p lower p_{\text{lower}} p lower ​ ) : p lower = P ( X < LSL ) = P ( Z < − Z LSL ) = 1 − Φ ( Z LSL ) p_{\text{lower}} = P(X < \text{LSL}) = P(Z < -Z_{\text{LSL}}) = 1 - \Phi(Z_{\text{LSL}}) p lower ​ = P ( X < LSL ) = P ( Z < − Z LSL ​ ) = 1 − Φ ( Z LSL ​ )

Total Fraction Defective ( p total p_{\text{total}} p total ​ ) : p total = p upper + p lower = [ 1 − Φ ( Z USL ) ] + [ 1 − Φ ( Z LSL ) ] p_{\text{total}} = p_{\text{upper}} + p_{\text{lower}} = [1 - \Phi(Z_{\text{USL}})] + [1 - \Phi(Z_{\text{LSL}})] p total ​ = p upper ​ + p lower ​ = [ 1 − Φ ( Z USL ​ )] + [ 1 − Φ ( Z LSL ​ )]

Expected Defect Rate (PPM) : PPM = p total × 10 6 \text{PPM} = p_{\text{total}} \times 10^6 PPM = p total ​ × 1 0 6

• 7. Step-by-Step Worked Engineering Calculations
• Worked Example 21.1.1: Precision Shaft Grinding Capability Analysis

Problem : A precision automotive grinding cell manufactures cylindrical transmission shafts with an outside diameter specification of 25.000 ± 0.080 mm 25.000 \pm 0.080\text{ mm} 25.000 ± 0.080 mm (hence LSL = 24.920 mm \text{LSL} = 24.920\text{ mm} LSL = 24.920 mm and USL = 25.080 mm \text{USL} = 25.080\text{ mm} USL = 25.080 mm ). An industrial engineer gathers quality data from 25 rational subgroups of size n = 5 n = 5 n = 5 while the process is verified to be in statistical control. The grand mean and average range are:

• Grand average: X ˉ ˉ = 25.025 mm \bar{\bar{X}} = 25.025\text{ mm} X ˉ ˉ = 25.025 mm
• Average range: R ˉ = 0.04652 mm \bar{R} = 0.04652\text{ mm} R ˉ = 0.04652 mm
• Control chart factor for n = 5 n = 5 n = 5 : d 2 = 2.326 d_2 = 2.326 d 2 ​ = 2.326

An overall pooled calculation of all 125 parts yields a sample standard deviation of s = 0.0260 mm s = 0.0260\text{ mm} s = 0.0260 mm .

Required :

Estimate the within-subgroup process standard deviation σ ^ \hat{\sigma} σ ^ .

Compute the potential capability index C p C_p C p ​ and actual capability index C p k C_{pk} C p k ​ .

Compute the centering factor k k k and verify the identity C p k = C p ( 1 − k ) C_{pk} = C_p(1 - k) C p k ​ = C p ​ ( 1 − k ) .

Compute the expected defect rate in parts per million (PPM) for both upper and lower tails.

Compute the long-term performance indices P p P_p P p ​ and P p k P_{pk} P p k ​ , and interpret the process behavior.

Solution :

Step 1: Estimate Within-Subgroup Standard Deviation σ ^ = R ˉ d 2 = 0.04652 mm 2.326 = 0.0200 mm \hat{\sigma} = \frac{\bar{R}}{d_2} = \frac{0.04652\text{ mm}}{2.326} = 0.0200\text{ mm} σ ^ = d 2 ​ R ˉ ​ = 2.326 0.04652 mm ​ = 0.0200 mm

Step 2: Calculate Potential Capability ( C p C_p C p ​ )

Specification tolerance: USL − LSL = 25.080 − 24.920 = 0.160 mm \text{USL} - \text{LSL} = 25.080 - 24.920 = 0.160\text{ mm} USL − LSL = 25.080 − 24.920 = 0.160 mm

Process spread: 6 σ ^ = 6 × 0.0200 = 0.120 mm 6\hat{\sigma} = 6 \times 0.0200 = 0.120\text{ mm} 6 σ ^ = 6 × 0.0200 = 0.120 mm C p = USL − LSL 6 σ ^ = 0.160 0.120 = 1.333 C_p = \frac{\text{USL} - \text{LSL}}{6\hat{\sigma}} = \frac{0.160}{0.120} = 1.333 C p ​ = 6 σ ^ USL − LSL ​ = 0.120 0.160 ​ = 1.333

Interpretation : The inherent variability of the grinding machine is capable of achieving 4-sigma performance ( C p ≥ 1.33 C_p \ge 1.33 C p ​ ≥ 1.33 ).

Step 3: Calculate Actual Capability ( C p k C_{pk} C p k ​ )

Unilateral Upper Capability: C p u = USL − X ˉ ˉ 3 σ ^ = 25.080 − 25.025 3 ( 0.0200 ) = 0.055 0.060 = 0.9167 C_{pu} = \frac{\text{USL} - \bar{\bar{X}}}{3\hat{\sigma}} = \frac{25.080 - 25.025}{3(0.0200)} = \frac{0.055}{0.060} = 0.9167 C p u ​ = 3 σ ^ USL − X ˉ ˉ ​ = 3 ( 0.0200 ) 25.080 − 25.025 ​ = 0.060 0.055 ​ = 0.9167

Unilateral Lower Capability: C p l = X ˉ ˉ − LSL 3 σ ^ = 25.025 − 24.920 3 ( 0.0200 ) = 0.105 0.060 = 1.7500 C_{pl} = \frac{\bar{\bar{X}} - \text{LSL}}{3\hat{\sigma}} = \frac{25.025 - 24.920}{3(0.0200)} = \frac{0.105}{0.060} = 1.7500 C pl ​ = 3 σ ^ X ˉ ˉ − LSL ​ = 3 ( 0.0200 ) 25.025 − 24.920 ​ = 0.060 0.105 ​ = 1.7500

Actual Capability Index: C p k = min ⁡ ( C p u , C p l ) = min ⁡ ( 0.9167 , 1.7500 ) = 0.9167 ≈ 0.92 C_{pk} = \min(C_{pu}, \, C_{pl}) = \min(0.9167, \, 1.7500) = 0.9167 \approx 0.92 C p k ​ = min ( C p u ​ , C pl ​ ) = min ( 0.9167 , 1.7500 ) = 0.9167 ≈ 0.92

Conclusion : Although potential capability is satisfactory ( C p = 1.33 C_p = 1.33 C p ​ = 1.33 ), the actual process is incapable ( C p k = 0.92 < 1.00 C_{pk} = 0.92 < 1.00 C p k ​ = 0.92 < 1.00 ) because the process mean is operating off-center toward the upper specification limit!

Step 4: Compute Centering Factor ( k k k ) and Verify Identity

Specification midpoint: M = 25.080 + 24.920 2 = 25.000 mm M = \frac{25.080 + 24.920}{2} = 25.000\text{ mm} M = 2 25.080 + 24.920 ​ = 25.000 mm

Half-tolerance: USL − LSL 2 = 0.160 2 = 0.080 mm \frac{\text{USL} - \text{LSL}}{2} = \frac{0.160}{2} = 0.080\text{ mm} 2 USL − LSL ​ = 2 0.160 ​ = 0.080 mm

Off-centering distance: ∣ X ˉ ˉ − M ∣ = ∣ 25.025 − 25.000 ∣ = 0.025 mm |\bar{\bar{X}} - M| = |25.025 - 25.000| = 0.025\text{ mm} ∣ X ˉ ˉ − M ∣ = ∣25.025 − 25.000∣ = 0.025 mm k = 0.025 0.080 = 0.3125 k = \frac{0.025}{0.080} = 0.3125 k = 0.080 0.025 ​ = 0.3125

Verify identity: C p k = C p ( 1 − k ) = 1.3333 × ( 1 − 0.3125 ) = 1.3333 × 0.6875 = 0.9167 C_{pk} = C_p (1 - k) = 1.3333 \times (1 - 0.3125) = 1.3333 \times 0.6875 = 0.9167 C p k ​ = C p ​ ( 1 − k ) = 1.3333 × ( 1 − 0.3125 ) = 1.3333 × 0.6875 = 0.9167

The mathematical identity holds exactly.

Step 5: Compute Expected Defect Rate (PPM)

Upper specification Z Z Z -score: Z USL = USL − X ˉ ˉ σ ^ = 25.080 − 25.025 0.0200 = 2.75 Z_{\text{USL}} = \frac{\text{USL} - \bar{\bar{X}}}{\hat{\sigma}} = \frac{25.080 - 25.025}{0.0200} = 2.75 Z USL ​ = σ ^ USL − X ˉ ˉ ​ = 0.0200 25.080 − 25.025 ​ = 2.75 Using standard normal cumulative probability: Φ ( 2.75 ) = 0.99702 \Phi(2.75) = 0.99702 Φ ( 2.75 ) = 0.99702 p upper = 1 − 0.99702 = 0.00298 = 2 , 980 PPM p_{\text{upper}} = 1 - 0.99702 = 0.00298 = 2,980\text{ PPM} p upper ​ = 1 − 0.99702 = 0.00298 = 2 , 980 PPM

Lower specification Z Z Z -score: Z LSL = X ˉ ˉ − LSL σ ^ = 25.025 − 24.920 0.0200 = 5.25 Z_{\text{LSL}} = \frac{\bar{\bar{X}} - \text{LSL}}{\hat{\sigma}} = \frac{25.025 - 24.920}{0.0200} = 5.25 Z LSL ​ = σ ^ X ˉ ˉ − LSL ​ = 0.0200 25.025 − 24.920 ​ = 5.25 Because Z LSL = 5.25 > 4.0 Z_{\text{LSL}} = 5.25 > 4.0 Z LSL ​ = 5.25 > 4.0 , lower tail defects are negligible: p lower = 1 − Φ ( 5.25 ) ≈ 7.9 × 10 − 8 ≈ 0.08 PPM p_{\text{lower}} = 1 - \Phi(5.25) \approx 7.9 \times 10^{-8} \approx 0.08\text{ PPM} p lower ​ = 1 − Φ ( 5.25 ) ≈ 7.9 × 1 0 − 8 ≈ 0.08 PPM .

Total expected defects: PPM total ≈ 2 , 980 PPM \text{PPM}_{\text{total}} \approx 2,980\text{ PPM} PPM total ​ ≈ 2 , 980 PPM

Takeaway : Because the grinding wheel offset is off by only + 0.025 mm +0.025\text{ mm} + 0.025 mm , the line generates nearly 3 , 000 PPM 3,000\text{ PPM} 3 , 000 PPM oversized scrap. Adjusting the CNC wheel offset by − 0.025 mm -0.025\text{ mm} − 0.025 mm to center the process at 25.000 mm 25.000\text{ mm} 25.000 mm immediately restores C p k C_{pk} C p k ​ to 1.333 1.333 1.333 , cutting total scrap to just 63.3 PPM 63.3\text{ PPM} 63.3 PPM without spending any capital on tooling.

Step 6: Compute Long-Term Performance Indices ( P p , P p k P_p, P_{pk} P p ​ , P p k ​ ) Using overall sample standard deviation s = 0.0260 mm s = 0.0260\text{ mm} s = 0.0260 mm : P p = USL − LSL 6 s = 0.160 6 × 0.0260 = 0.160 0.156 = 1.026 P_p = \frac{\text{USL} - \text{LSL}}{6s} = \frac{0.160}{6 \times 0.0260} = \frac{0.160}{0.156} = 1.026 P p ​ = 6 s USL − LSL ​ = 6 × 0.0260 0.160 ​ = 0.156 0.160 ​ = 1.026 P p u = USL − X ˉ ˉ 3 s = 25.080 − 25.025 3 × 0.0260 = 0.055 0.078 = 0.705 P_{pu} = \frac{\text{USL} - \bar{\bar{X}}}{3s} = \frac{25.080 - 25.025}{3 \times 0.0260} = \frac{0.055}{0.078} = 0.705 P p u ​ = 3 s USL − X ˉ ˉ ​ = 3 × 0.0260 25.080 − 25.025 ​ = 0.078 0.055 ​ = 0.705 P p l = X ˉ ˉ − LSL 3 s = 25.025 − 24.920 3 × 0.0260 = 0.105 0.078 = 1.346 P_{pl} = \frac{\bar{\bar{X}} - \text{LSL}}{3s} = \frac{25.025 - 24.920}{3 \times 0.0260} = \frac{0.105}{0.078} = 1.346 P pl ​ = 3 s X ˉ ˉ − LSL ​ = 3 × 0.0260 25.025 − 24.920 ​ = 0.078 0.105 ​ = 1.346 P p k = min ⁡ ( 0.705 , 1.346 ) = 0.705 P_{pk} = \min(0.705, 1.346) = 0.705 P p k ​ = min ( 0.705 , 1.346 ) = 0.705

Diagnostic : Note that s = 0.0260 > σ ^ = 0.0200 s = 0.0260 > \hat{\sigma} = 0.0200 s = 0.0260 > σ ^ = 0.0200 , leading to P p k = 0.71 < C p k = 0.92 P_{pk} = 0.71 < C_{pk} = 0.92 P p k ​ = 0.71 < C p k ​ = 0.92 . This confirms that additional long-term between-batch variation (such as grinding wheel thermal expansion and dress cycles) further degrades process performance over time.

8. NCEES Reference Handbook Tips & Realistic Exam Traps

Trap: Confusing C p C_p C p ​ with C p k C_{pk} C p k ​ : Questions frequently ask: "Is the process capable of meeting customer specifications?" If the process mean is off-center, evaluate C p k C_{pk} C p k ​ , not C p C_p C p ​ . C p C_p C p ​ measures only potential spread under ideal centering. Conformance requires C p k ≥ 1.33 C_{pk} \ge 1.33 C p k ​ ≥ 1.33 .

Trap: Inverting Z-score Conversions : Remember that Z USL = 3 C p u Z_{\text{USL}} = 3 C_{pu} Z USL ​ = 3 C p u ​ and Z LSL = 3 C p l Z_{\text{LSL}} = 3 C_{pl} Z LSL ​ = 3 C pl ​ . A common mistake is multiplying by 6 instead of 3 because C p C_p C p ​ has 6 σ 6\sigma 6 σ in its denominator. Unilateral capability indices evaluate half the distribution ( 3 σ 3\sigma 3 σ ), so the multiplier is always 3.

Trap: Estimating σ \sigma σ from Subgroup Ranges : The NCEES handbook provides the divisor d 2 d_2 d 2 ​ for subgroup ranges ( σ ^ = R ˉ / d 2 \hat{\sigma} = \bar{R}/d_2 σ ^ = R ˉ / d 2 ​ ) and c 4 c_4 c 4 ​ for standard deviations ( σ ^ = s ˉ / c 4 \hat{\sigma} = \bar{s}/c_4 σ ^ = s ˉ / c 4 ​ ). Make sure you look up the factor corresponding to the subgroup sample size n n n (typically n = 4 n = 4 n = 4 or 5 5 5 ), not the number of subgroups m m m ( m = 20 m = 20 m = 20 or 25 25 25 ).

Trap: Negative C p k C_{pk} C p k ​ Values : Do not discard negative capability values as computational errors. If the process mean μ \mu μ drifts outside the tolerance band ( μ > USL \mu > \text{USL} μ > USL or μ < LSL \mu < \text{LSL} μ < LSL ), the numerator of either C p u C_{pu} C p u ​ or C p l C_{pl} C pl ​ is negative, resulting in C p k < 0 C_{pk} < 0 C p k ​ < 0 . This indicates that greater than 50 % 50\% 50% of production is defective.