Cpk
Cpk is a statistical process capability index that measures how reliably a process produces output within customer-defined specification limits. Unlike potential capability (Cp), which only evaluates total process spread against tolerance width, Cpk accounts for both process variation and centering by measuring the distance from the process mean to the nearest specification limit in units of three standard deviations. A Cpk value of 1.33 or higher typically represents an industry standard for capable, stable production processes.
- Specification limits from the customer
- The curve is the spread of shaft diameters from one lathe. The customer specifies the shaft at 10.00 ± 0.10 mm. The two crimson bands are the limits: 9.90 mm below and 10.10 mm above. They come from the drawing, not from the process. The control chart for this shaft uses different lines. Its control limits, 9.94 and 10.06 mm, are calculated from the process itself.
- The process mean against the target
- The target is 10.00 mm, the middle of the specification. The mean is where the process actually centers. At mean 10.04 mm and σ 0.02 mm, the nearest limit is 0.06 mm away. Cpk falls to 1.00 while Cp stays at 1.67. Moving the center changes Cpk and leaves Cp alone.
- The spread of the process
- The width of the curve is the spread, measured by the standard deviation σ. Six σ is the full width the process needs. At σ 0.04 mm, six σ is 0.24 mm, wider than the 0.20 mm specification. Cp drops to 0.83 and Cpk drops with it. The mean still sits on the target.
- The Cpk readout
- Cpk compares the distance from the mean to the nearest limit with three σ. Centered at 10.00 mm with σ 0.02 mm, both distances are 0.10 mm. Cpk = 0.10 / 0.06 = 1.67, the same as Cp. Blue means 1.33 or above, amber means 1.00 to 1.33, and crimson means below 1.00.
- Parts outside the specification
- The crimson area is the part of the curve beyond the upper limit. Every unit of that area is a shaft too large to ship. At mean 10.06 mm and σ 0.02 mm, the upper limit is two σ away. Cpk reads 0.67, and the tail past 10.10 mm is real scrap.
Key facts
- Origin
- 1970s Japanese industry
- Standard mature process minimum
- Cpk >= 1.33
- Six Sigma short-term target
- Cpk = 2.00 (3.4 DPMO)
- Key mathematical divisor
- 3 standard deviations to nearest limit
- Prerequisite assumptions
- Process stability and normal distribution
By Matthew Savas — Founder of Kaizumi. Reviewed 1 September 2026.
Cpk is a process capability index that measures how well a process fits inside its specification limits, accounting for both its spread and where it is centered. It evaluates whether a manufacturing or business process can reliably produce output within customer-defined tolerances. The index divides the distance from the process mean to the nearest specification limit by three standard deviations. By incorporating process centering alongside variability, Cpk provides an accurate reflection of the actual yield of non-defective product produced during stable operating conditions.
History and background
Process capability indices were developed in Japanese industry during the 1970s as part of broader statistical quality control and total quality management frameworks. Japanese manufacturing engineers recognized that monitoring whether a process was merely statistically stable was insufficient to guarantee that parts would meet engineering requirements. They required standardized numerical metrics to quantify how comfortably a process could operate within design tolerances.
These techniques spread rapidly to Western manufacturing throughout the 1980s, driven by the competitive pressures of global trade and the widespread adoption of statistical process control across the automotive, aerospace, and electronics sectors. Capability indices became standard deliverables in supplier qualification procedures, production part approval processes, and continuous improvement initiatives. Over time, Cpk became an integral analytical metric within the Six Sigma methodology, serving as a primary benchmark for baseline assessment and post-improvement verification.
Conceptual framework and mechanics
A process generates output that naturally varies due to common cause variation. When graphed on a histogram, this variation often forms a normal distribution characterized by its central tendency (the process mean) and its dispersion (the standard deviation). Customers define acceptable product parameters using an upper specification limit and a lower specification limit. The space between these two boundaries is the tolerance band or specification width.
Capability evaluation depends on two distinct questions:
- What is the total spread of the process relative to the total width of the specification?
- Where is the distribution located relative to the specification limits?
The first question relates to potential capability, designated as Cp. The second question relates to actual capability, designated as Cpk.
To calculate potential capability, subtract the lower specification limit from the upper specification limit, and divide that total specification width by six standard deviations. Six standard deviations represent the total natural spread of a normally distributed process, encompassing 99.73 percent of all individual measurements. Cp indicates whether the process could fit inside the specification limits if it were centered between them. However, Cp does not account for the process location; a process can have a high Cp value even if its mean is located entirely outside the customer specification limits.
To calculate actual capability, evaluate both sides of the distribution independently. First, calculate the upper capability by subtracting the process mean from the upper specification limit and dividing the result by three standard deviations. Second, calculate the lower capability by subtracting the lower specification limit from the process mean and dividing the result by three standard deviations. Cpk is defined as the minimum of these two values. By selecting the smaller number, Cpk evaluates the worst-case side of the distribution, reflecting the distance between the process center and the closest specification limit.
When a process is centered between the upper and lower specification limits, the upper capability equals the lower capability, and Cpk equals Cp. When the process mean drifts toward either limit, the distance to the nearer limit decreases, causing Cpk to drop below Cp.
Worked example: a turned steel shaft
The interaction between process spread, process centering, and capability indices can be demonstrated using a CNC lathe turning a steel shaft.
A customer specifies the shaft diameter at 10.00 plus or minus 0.10 millimeters. The lower specification limit is 9.90 millimeters and the upper specification limit is 10.10 millimeters. The total specification width is 0.20 millimeters.
Assume the lathe operates with a standard deviation of 0.02 millimeters under stable conditions. Six standard deviations of the process equal 0.12 millimeters. Dividing the specification width of 0.20 millimeters by the six-sigma spread of 0.12 millimeters yields a Cp of 1.67.
When the lathe operates with its mean centered at 10.00 millimeters, the distance from the mean to each specification limit is 0.10 millimeters. Three standard deviations equal 0.06 millimeters. Dividing the distance of 0.10 millimeters by 0.06 millimeters yields a Cpk of 1.67. In this centered state, the actual capability matches the potential capability.
Suppose tool wear causes the process mean to shift to 10.04 millimeters while the standard deviation remains constant at 0.02 millimeters. The distance from the mean to the lower specification limit increases to 0.14 millimeters, giving a lower capability of 0.14 divided by 0.06, which equals 2.33. However, the distance from the mean to the upper specification limit decreases to 0.06 millimeters. Dividing this 0.06 millimeter margin by three standard deviations (0.06 millimeters) yields a Cpk of 1.00. The potential capability Cp remains 1.67 because the variation has not changed, but Cpk reflects the reduced margin on the upper side.
If the cutting tool wears further, shifting the mean to 10.06 millimeters, the distance to the upper specification limit shrinks to 0.04 millimeters. Dividing 0.04 millimeters by 0.06 millimeters produces a Cpk of 0.67. The upper specification limit now sits two standard deviations from the process mean. In a normal distribution, approximately 2.3 percent of the turned shafts will exceed the upper specification limit of 10.10 millimeters and will be defective.
Consider another scenario where the lathe mean is restored to the center at 10.00 millimeters, but mechanical looseness doubles the process standard deviation to 0.04 millimeters. Six standard deviations now equal 0.24 millimeters, which is wider than the 0.20 millimeter customer specification. Dividing 0.20 millimeters by 0.24 millimeters yields a Cp of 0.83. Because the process is centered, the distance to each limit is 0.10 millimeters, and three standard deviations equal 0.12 millimeters. Dividing 0.10 millimeters by 0.12 millimeters gives a Cpk of 0.83.
Specification limits versus control limits
Process capability indices must not be confused with statistical process control limits. Specification limits and control limits serve distinct functions and are calculated independently.
Specification limits represent the voice of the customer. They are determined by product designers, engineering specifications, or contractual requirements. They define what dimensions or performance values are acceptable for the product to function. Specification limits do not depend on the manufacturing process used to make the part.
Control limits represent the voice of the process. They are calculated from process data collected over time and appear on a control chart to show whether a process is behaving consistently. Control limits are established at three standard errors above and below the centerline of subgroup averages.
For the shaft example described above, the specification limits sit at 9.90 millimeters and 10.10 millimeters. On the corresponding control chart for subgroup averages, the upper and lower control limits sit at 9.94 millimeters and 10.06 millimeters. A process can operate in complete statistical control within its control limits while still failing to meet customer specification limits, just as an out-of-control process might temporarily produce parts that fall within engineering tolerances.
Interpreting Cpk values
The numerical value of Cpk indicates the operating safety margin of the process relative to its specification limits:
- Cpk greater than or equal to 1.33: This is the standard industry minimum for an established, mature production process. A Cpk of 1.33 means the distance between the process mean and the nearest specification limit equals four standard deviations. This creates a safety buffer of one full standard deviation between the natural three-sigma edge of the distribution and the specification limit.
- Cpk between 1.00 and 1.33: The process meets the specification requirements but operates with little or no safety margin. When Cpk equals 1.00, the natural three-sigma boundary of the process touches the specification limit. Under a normal distribution, roughly 0.13 percent of output (1,350 parts per million) will fall outside the limit on that side. Minor process shifts will immediately create non-conforming output.
- Cpk less than 1.00: The process is incapable of meeting the specification. The process mean is located within three standard deviations of a specification limit, meaning defective parts are produced during routine, normal operation.
- Cpk equal to 2.00: This represents the baseline target for Six Sigma quality on short-term data. For a centered process, a Cpk of 2.00 places the nearest specification limit six standard deviations away from the mean. This wide margin ensures that even if the process mean drifts by 1.5 standard deviations over the long term, the process will generate no more than 3.4 defects per million opportunities.
Corrective actions based on capability data
Comparing Cp and Cpk allows engineers and operational leaders to identify the root cause of capability deficits and select appropriate corrective interventions.
When Cp is high (such as 1.67) and Cpk is low (such as 0.67), the issue is process centering, not process variability. The process is well-behaved and capable of high yields, but it is running in the wrong place. The appropriate corrective action involves an offset adjustment:
- For the turned shaft running with a mean of 10.04 millimeters, the machine tool requires a physical work offset adjustment of negative 0.04 millimeters to move the mean back to 10.00 millimeters.
- Centering adjustments typically require low capital expenditure, involving changes to tooling offsets, machine settings, chemical concentrations, temperature setpoints, or operator positioning.
When both Cp and Cpk are low (such as 0.83), the issue is excess variability. The natural spread of the process is too wide for the specification window. Adjusting the mean will not resolve the problem because the six-sigma spread exceeds the tolerance band. Resolving low Cp values requires fundamental variation reduction:
- For the shaft process operating with a standard deviation of 0.04 millimeters, adjusting tool offsets will not prevent defects.
- The operation requires a more rigid machine tool, a redesigned clamping fixture, improved raw material consistency, enhanced temperature stabilization, or a different manufacturing method altogether.
Common misconceptions
A frequent misunderstanding in quality management is the belief that achieving a high Cpk guarantees zero defects. Process capability calculations rely on statistical probability models. A Cpk of 2.00 still allows for an expected non-conformance rate of 3.4 parts per million when accounting for standard long-term mean drift. No finite capability index denotes absolute zero defects.
Another common error is calculating Cpk on unstable processes. Capability indices are only mathematically valid when the underlying process is in a state of statistical control. If special causes of variation are present, the process mean and standard deviation will change over time, rendering capability calculations inaccurate and unrepresentative of future production.
Similarly, standard Cpk calculations assume that the underlying data follow a normal distribution. If the process data are skewed, bimodal, or non-normal, such as flatness, roundness, or cycle time measurements, standard Cpk equations will miscalculate the defect rate. In such situations, data transformation or non-parametric capability techniques must be applied.
Examples across industries
Process capability analysis applies across manufacturing, clinical, and administrative processes where measurable specifications exist.
Manufacturing
A high-speed bottling line fills beverages with a target volume of 500 milliliters. The specification limits are defined as 495 milliliters and 505 milliliters. Quality audits determine that the filling process runs with an average fill volume of 502 milliliters and a standard deviation of 1.2 milliliters.
The distance to the upper specification limit is 505 minus 502, which equals 3.0 milliliters. The distance to the lower limit is 502 minus 495, which equals 7.0 milliliters. Three standard deviations equal three multiplied by 1.2, which is 3.6 milliliters.
Evaluating the nearer limit gives a Cpk of 3.0 divided by 3.6, which equals 0.83. The process is incapable because overfilling pushes the distribution too close to the upper limit. By adjusting the filling valves to recenter the average at 500 milliliters without changing the filler mechanical variation, the distance to both limits becomes 5.0 milliliters. Dividing 5.0 milliliters by 3.6 milliliters raises the Cpk to 1.39 without requiring capital investment.
Healthcare
A diagnostic blood glucose meter must provide readings within plus or minus 10 milligrams per deciliter of a laboratory reference standard. Clinical testing reveals that a specific meter model exhibits a positive mean error of plus 3 milligrams per deciliter with a standard deviation of 2 milligrams per deciliter.
The upper specification limit is plus 10 milligrams per deciliter, and the lower specification limit is minus 10 milligrams per deciliter. Three standard deviations equal three multiplied by 2, which is 6 milligrams per deciliter.
The distance from the mean error of plus 3 to the nearest limit of plus 10 is 7 milligrams per deciliter. Dividing 7 milligrams per deciliter by 6 milligrams per deciliter results in a Cpk of 1.17. The positive measurement bias reduces the overall capability of the diagnostic device. Applying a calibration offset in the meter software to remove the 3 milligram bias centers the error at zero, increasing the distance to the nearest limit to 10 milligrams per deciliter and raising the Cpk to 10 divided by 6, which equals 1.67.
Administrative
A financial shared services center measures the performance of its accounts payable department using invoice accuracy. The operational specification requires that invoice accuracy must meet a lower specification limit of 97.0 percent, with no upper limit required.
Historical performance data show that the department averages an accuracy rate of 98.2 percent with a standard deviation of 0.5 percent.
Because this is a one-sided lower specification limit, capability is calculated using only the lower boundary. The distance from the mean of 98.2 percent to the lower specification limit of 97.0 percent is 1.2 percent. Three standard deviations equal three multiplied by 0.5 percent, which is 1.5 percent. Dividing 1.2 percent by 1.5 percent yields a Cpk of 0.80. Despite the average accuracy exceeding the 97.0 percent minimum target, the variability in invoice processing results in an incapable process that routinely produces batches falling below the required accuracy threshold.
Frequently asked questions
- Can Cpk be greater than Cp?
- Cpk can never exceed Cp. When a process is centered between specification limits, actual capability matches potential capability and Cpk equals Cp. As the process mean drifts toward either limit, the distance to the nearer limit decreases, causing Cpk to drop below Cp.
- What does a high Cp combined with a low Cpk indicate?
- A high Cp paired with a low Cpk indicates that the process has low variability but is running off-center relative to specification limits. The process spread is narrow enough to fit comfortably within tolerances, but the mean is located too close to one limit. Correcting this capability deficit requires an offset adjustment, such as updating tooling offsets or machine settings, rather than reducing process variation.
- What does a Cpk value of less than 1.00 mean for a process?
- A Cpk value below 1.00 means the process is incapable of consistently meeting customer specifications. In this condition, the process mean sits within three standard deviations of a specification limit, which produces non-conforming parts during routine operation. The process must either be recentered or fundamentally improved to reduce its natural spread.
- Does achieving a high Cpk guarantee zero defects?
- A high Cpk does not guarantee zero defects because capability calculations rely on statistical probability models rather than absolute limits. Even a target Cpk of 2.00 anticipates an expected non-conformance rate of 3.4 defects per million opportunities when accounting for standard long-term drift in the mean. No finite capability index denotes absolute zero defects.
- Why must a process be in statistical control before calculating Cpk?
- Cpk calculations assume that the process is predictable and subject only to common cause variation. When special causes of variation are present, the process mean and standard deviation shift over time. Calculating Cpk on an unstable process yields inaccurate capability figures that do not represent future production performance.