6.14 Process Capability and Performance Indices

Evaluating baseline process capability is a central objective of the Measure phase. Process Capability compares the natural voice of the process (process variation, $6\sigma$) against the voice of the customer (specification limits, $\text{USL} - \text{LSL}$). Before conducting capability calculations, a Black Belt must verify two strict prerequisites:

1. Statistical Stability: The process must be in a state of statistical control, free from special cause variation, as verified by control charts.
2. Normality: Data must follow a standard Normal distribution (or be transformed to normality).

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Capability ($C_p, C_{pk}$) vs. Performance ($P_p, P_{pk}$) Indices

Six Sigma maintains a strict distinction between short-term potential capability and long-term actual performance:

1. Short-Term Process Capability ($C_p, C_{pk}$)

Short-term capability represents the inherent technical capability of a stable process operating under controlled conditions over a short timeframe. It utilizes within-subgroup standard deviation ($\sigma_{\text{within}}$ or $\sigma_{\text{st}}$), estimated from control chart ranges or standard deviations:

$$\hat{\sigma}_{\text{st}} = \frac{\bar{R}}{d_2} \quad \text{or} \quad \hat{\sigma}_{\text{st}} = \frac{\bar{S}}{c_4}$$

2. Long-Term Process Performance ($P_p, P_{pk}$)

Long-term performance represents actual historical performance over extended time horizons, incorporating environmental shifts, raw material lot changes, and tool wear. It utilizes total sample standard deviation ($s_{\text{total}}$ or $s_{\text{lt}}$), calculated across all individual observations:

$$s_{\text{lt}} = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}}$$

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Mathematical Formulations of Capability Indices

Potential Capability Index ($C_p$)

$C_p$ measures the maximum potential capability of the process assuming the process mean $\mu$ is perfectly centered between specification limits. It ignores process centering:

$$C_p = \frac{\text{USL} - \text{LSL}}{6\sigma_{\text{st}}}$$

Centered Capability Index ($C_{pk}$)

$C_{pk}$ accounts for actual process centering by evaluating capability relative to both Upper Specification Limit (USL) and Lower Specification Limit (LSL):

$$C_{pu} = \frac{\text{USL} - \mu}{3\sigma_{\text{st}}}, \quad C_{pl} = \frac{\mu - \text{LSL}}{3\sigma_{\text{st}}}$$ $$C_{pk} = \min \left( C_{pu}, C_{pl} \right) = \min \left( \frac{\text{USL} - \mu}{3\sigma_{\text{st}}}, \frac{\mu - \text{LSL}}{3\sigma_{\text{st}}} \right)$$

• Key Relationship: $C_{pk} \le C_p$ always. $C_{pk} = C_p$ only when the process mean is centered perfectly at the specification midpoint $\frac{\text{USL} + \text{LSL}}{2}$.

Taguchi Capability Index ($C_{pm}$)

Developed by Genichi Taguchi, $C_{pm}$ incorporates target centering loss by incorporating the squared deviation of the process mean from the target nominal value $T$:

$$C_{pm} = \frac{\text{USL} - \text{LSL}}{6 \sqrt{\sigma^2 + (\mu - T)^2}} = \frac{C_p}{\sqrt{1 + \left( \frac{\mu - T}{\sigma} \right)^2}}$$

Performance Indices ($P_p, P_{pk}$)

Calculated using identical formulas to $C_p$ and $C_{pk}$, but substituting overall long-term standard deviation $s_{\text{lt}}$ for $\sigma_{\text{st}}$:

$$P_p = \frac{\text{USL} - \text{LSL}}{6 s_{\text{lt}}}, \quad P_{pk} = \min \left( \frac{\text{USL} - \mu}{3 s_{\text{lt}}}, \frac{\mu - \text{LSL}}{3 s_{\text{lt}}} \right)$$

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Six Sigma Capability Benchmarks & Worked Numerical Example

Worked Capability Calculation

A machining process produces shaft diameters with specifications $\text{LSL} = 19.80\text{ mm}$ and $\text{USL} = 20.20\text{ mm}$. Subgroup sample data ($n = 5$, $k = 20$ subgroups) yields a sample mean $\bar{\bar{x}} = 20.08\text{ mm}$, an average range $\bar{R} = 0.092\text{ mm}$, and an overall sample standard deviation $s_{\text{lt}} = 0.045\text{ mm}$. Using $d_2 = 2.326$ for $n = 5$:

1. $\hat{\sigma}_{\text{st}} = \frac{\bar{R}}{d_2} = \frac{0.092}{2.326} = 0.03955\text{ mm}$.
2. $C_p = \frac{20.20 - 19.80}{6 \times 0.03955} = \frac{0.40}{0.2373} = 1.685$.
3. $C_{pu} = \frac{20.20 - 20.08}{3 \times 0.03955} = \frac{0.12}{0.11865} = 1.011$.
4. $C_{pl} = \frac{20.08 - 19.80}{3 \times 0.03955} = \frac{0.28}{0.11865} = 2.360$.
5. $C_{pk} = \min(1.011, 2.360) = 1.011$.

• Diagnosis: The potential capability is excellent ($C_p = 1.685 > 1.33$), but the process is off-center toward the USL, reducing $C_{pk}$ to $1.011$. Re-centering the process mean from $20.08$ to $20.00\text{ mm}$ will immediately raise $C_{pk}$ to $1.685$.

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$Z$-Score Transformation & The 1.5$\sigma$ Shift

Process capability can be expressed on the Standard Normal Distribution $Z$-scale:

$$Z_{\text{USL}} = \frac{\text{USL} - \mu}{\sigma}, \quad Z_{\text{LSL}} = \frac{\mu - \text{LSL}}{\sigma}$$ $$Z_{\text{min}} = \min(Z_{\text{USL}}, Z_{\text{LSL}}) = 3 \cdot C_{pk}$$

• The 1.5$\sigma$ Shift Assumption: Motorola empirical research established that over long-term operations, process means shift and drift by approximately $1.5\sigma$. Thus:

$$Z_{\text{long-term}} = Z_{\text{short-term}} - 1.5$$ $$\text{Long-Term Sigma Level} = 3 \cdot C_{pk} - 1.5$$

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Handling Non-Normal Data: Data Transformations

When data violates normality (e.g., Anderson-Darling test $p$-value $< 0.05$), standard $C_{pk}$ equations produce severe calculation errors. Black Belts apply two primary transformation methods to achieve normality:

1. Box-Cox Power Transformation

The Box-Cox Transformation transforms skewed non-normal data $y$ into normal data $y^{(\lambda)}$ using a power parameter $\lambda$:

$$y^{(\lambda)} = \begin{cases} \frac{y^\lambda - 1}{\lambda} & \text{if } \lambda \neq 0 \\ \ln(y) & \text{if } \lambda = 0 \end{cases}$$

• Restriction: Box-Cox can ONLY be applied to strictly positive data ($y > 0$).

2. Johnson Transformation System

The Johnson Transformation fits non-normal data into one of three distribution families (Unbounded $S_U$, Bounded $S_B$, Log-Normal $S_L$), mapping complex non-normal data (including negative values or bounded data) into a Standard Normal distribution.

3. Non-Parametric Percentile Capability Method

If transformations fail, capability is computed directly from empirical percentiles ($P_{0.135}$, $P_{50}$, $P_{99.865}$):

$$C_{pk} = \min \left( \frac{\text{USL} - P_{50}}{P_{99.865} - P_{50}}, \frac{P_{50} - \text{LSL}}{P_{50} - P_{0.135}} \right)$$