What is the difference between Cp and Cpk?

Cp (capability index) = (USL−LSL)/(6σ) measures the smallness of variation relative to the specification range. However, Cp does not account for the offset between the process center and specification center. Cpk does account for this offset: Cpk = min((USL−µ)/3σ, (µ−LSL)/3σ). If Cp is high but Cpk is low, the process center is offset from the specification center and centering adjustment is needed. Cpk ≥ 1.33 (4σ equivalent) is a common acceptance target.

How do Pp and Ppk differ from Cp and Cpk?

Cp and Cpk use the 'within-subgroup σ' derived from Shewhart's R̄/d2 or s̄/c4. Pp and Ppk use the sample standard deviation s (overall σ) of all data. If Pp ≤ Cp, it indicates drift or shift in the process mean. When Cp ≈ Pp, the process is considered stable in the long term.

What is Cpm (Taguchi capability index)?

Cpm incorporates the concept of Taguchi's loss function. Cpm = (USL−LSL)/(6τ), where τ² = σ² + (µ−T)² (T is the target value). When µ deviates from target T, τ > σ and Cpm < Cp. This means Cpm evaluates not only variation but also the deviation from target as a quality loss.

How is the PPM defect rate calculated?

The PPM (Parts Per Million) defect rate is calculated assuming a normal distribution. Upper defect probability = 1−Φ((USL−µ)/σ); lower defect probability = Φ((LSL−µ)/σ). The total defect rate is converted to PPM (×1,000,000). For example, at Cpk=1.33 (4σ), one-sided defects ≈ 31.7 PPM (two-sided ≈ 63.4 PPM). At Cpk=2.0 (6σ), less than 0.001 PPM one-sided.

Process Capability Indices Cp/Cpk · Histogram Visualization — Free Online Calculator

Q: How is the PPM defect rate calculated?

The PPM (Parts Per Million) defect rate is calculated assuming a normal distribution. Upper defect probability = 1−Φ((USL−µ)/σ); lower defect probability = Φ((LSL−µ)/σ). The total defect rate is converted to PPM (×1,000,000). For example, at Cpk=1.33 (4σ), one-sided defects ≈ 31.7 PPM (two-sided ≈ 63.4 PPM). At Cpk=2.0 (6σ), less than 0.001 PPM one-sided.

Data Input

Measurements (up to 100 points, comma or newline separated)

USL (Upper Spec. Limit)

LSL (Lower Spec. Limit)

Results

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Cpk

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PPM Defects

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Sigma Level

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Mean μ

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Std Dev σ

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Cpm

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Judgment

Histogram + Normal Distribution Fit

Hist

Normal Probability Plot

Quality Engineering & Manufacturing Applications

Dimensional tolerance analysis of automotive parts / Process evaluation for injection-molded and stamped parts / Process capability reporting for ISO/IATF 16949 / Data for PPAP (Production Part Approval Process).

Theory & Key Formulas

$$C_p = \frac{USL-LSL}{6\sigma},\quad C_{pk}= \min\!\left(\frac{USL-\mu}{3\sigma},\frac{\mu-LSL}{3\sigma}\right)$$ $$C_{pm}= \frac{USL-LSL}{6\tau},\quad \tau = \sqrt{\sigma^2+(\mu-T)^2}$$

PPM defect rate: $\text{PPM}= \left[1-\Phi\!\left(\frac{USL-\mu}{\sigma}\right)+\Phi\!\left(\frac{LSL-\mu}{\sigma}\right)\right]\times 10^6$

What are Cp and Cpk?

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What exactly are Cp and Cpk? I see them on quality charts, but I'm not sure what the difference is.

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Basically, they're both numbers that tell you how well a manufacturing process fits within its allowed tolerance limits. Cp measures the potential capability—how wide the tolerance is compared to the natural spread of your process. Cpk is stricter; it also checks if your process is centered between the limits. Try typing in a USL and LSL in the simulator above and watch the histogram. If the data is off-center, you'll see Cp and Cpk diverge.

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Wait, really? So a high Cp but a low Cpk means my process is precise but inaccurate? How do I fix that?

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Exactly! That's a classic scenario. It means your machine's variation is small enough (good Cp), but the average output is drifting toward one of the specification limits (bad Cpk). In practice, you'd adjust the machine to re-center the process. For instance, if you're drilling holes, you'd recalibrate the drill's depth setting. Use the simulator's 'Measurements' input to shift the mean (µ) and see how Cpk reacts instantly.

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Okay, and what about Pp and Ppk? The FAQ says they use a different "sigma". Why have two sets of indices?

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Great question. Cp/Cpk use within-subgroup variation (short-term sigma), which tells you the process's inherent capability. Pp/Ppk use the overall standard deviation (long-term sigma) of all your data, which includes any shifts or drifts over time. If Pp is much lower than Cp, it signals your process isn't stable. It's a key check in automotive PPAP submissions. The simulator calculates all four indices side-by-side—play with the data to create a trend and watch the gap between Cp and Pp appear.

Physical Model & Key Equations

The fundamental capability index, Cp, compares the width of the specification "tunnel" to the natural width of the process variation, which is defined as 6 standard deviations (σ). This assumes the process data follows a normal distribution.

$$C_p = \frac{USL - LSL}{6\sigma}$$

USL/LSL: Upper and Lower Specification Limits (the tolerance).
σ (sigma): The standard deviation of the process, representing its natural spread.
A Cp = 1.0 means the process spread exactly fits the tolerance. Cp > 1.33 is generally considered capable.

Cpk introduces the process mean (µ) into the equation. It measures the capability relative to the *nearest* specification limit, thus accounting for how centered the process is.

$$C_{pk}= \min\!\left(\frac{USL-\mu}{3\sigma},\frac{\mu-LSL}{3\sigma}\right)$$

µ (mu): The mean (average) of the process data.
The term $(USL - µ)/3σ$ is the one-sided capability for the upper limit. Cpk is the worse of the two one-sided scores. If the mean is perfectly centered, Cpk equals Cp. If it shifts, Cpk drops, directly predicting a higher defect rate on one side.

Real-World Applications

Automotive Part Dimensional Control: A piston diameter must be 80.00 mm ± 0.05 mm. Engineers use this simulator to input measurements from the production line to calculate Cpk. A Cpk ≥ 1.67 is often required by car manufacturers to ensure less than a handful of defective parts per million, preventing engine failure.

Injection Molding Process Validation: For a plastic gear, the critical tooth thickness has tight tolerances. During the PPAP (Production Part Approval Process), manufacturers must report Pp and Ppk using initial production run data from this tool to prove long-term process stability before full-scale launch.

Semiconductor Wafer Fabrication: The thickness of a chemical layer on a silicon wafer is critical. Process engineers monitor Cp/Cpk in real-time. If Cpk falls below a threshold, the tool triggers an automatic maintenance cycle to recalibrate the deposition machine, minimizing costly scrap.

Medical Device Manufacturing: The burst pressure of a sterile IV bag seal must exceed a minimum specification (LSL) with high reliability. Quality teams use the simulator's defect rate (PPM) output, derived from Cpk, to provide statistical evidence for regulatory submissions to agencies like the FDA.

Common Misconceptions and Points to Note

First, understand that "high Cp/Cpk does not equal zero defects". For example, even with Cp=1.33 (equivalent to 4σ), there is a possibility of approximately 63 ppm (63 parts per million) defects on one side. This is a matter of probability, so a lot might coincidentally contain defects. The indices indicate the "degree of risk," not an absolute guarantee.

Next, the underlying assumptions of the data are often overlooked. Cp/Cpk calculations implicitly assume the data follows a "normal distribution." However, actual measured data might be bimodal or have peaks at the extremes. Always check the histogram in this simulator to see if it deviates significantly from a "nice bell curve." If it does, the process itself might be unstable.

Finally, the rationale behind the specification limits (USL/LSL) is crucial. Whether they are tolerances functionally required or just "arbitrarily decided values" completely changes how you interpret the indices. For instance, if a gap specification was set at ±0.5mm, but it can actually tolerate ±0.8mm, you don't need to worry about a worsened Cpk due to unnecessarily tight specs. Questioning the validity of the specifications themselves can be a good first step.

What is Process Capability Indices Cp/Cpk · Histogram Visualization?

Process Capability Indices Cp/Cpk · Histogram Visualization is a fundamental topic in engineering and applied physics. This interactive simulator lets you explore the key behaviors and relationships by directly manipulating parameters and observing real-time results.

By combining numerical computation with visual feedback, the simulator bridges the gap between abstract theory and physical intuition — making it an effective learning tool for students and a rapid-verification tool for practicing engineers.

Process Capability Indices Cp/Cpk · Histogram Visualization

What are Cp and Cpk?

Physical Model & Key Equations

Real-World Applications

Common Misconceptions and Points to Note

What is Process Capability Indices Cp/Cpk · Histogram Visualization?

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