# Process Capability Index Calculator

Created by Mariamy Chrdileli
Reviewed by Anna Szczepanek, PhD and Rijk de Wet
Last updated: Feb 02, 2023

Welcome to Omni's process capability index calculator, aka the Cpk calculator! With the process capability index calculator, you can quickly determine whether the products of your process meet specification limits (target values, usually set by customers, in which products or services are relative to customer requirements). Do you want to determine whether your process is going smoothly, or whether you need to make some improvements for the future? Read on to learn more about:

• How to calculate process capability index (a.k.a. process capability ratio);
• What the meaning of Cpk is; and
• What values of Cpk statistics are satisfactory.

As a bonus, you will learn why people are so excited when their Cpk is 1.33 or larger! 💡

## What is process capability index?

To better explain the process capability index, let's imagine a scenario. You are craving a delicious cappuccino on a Monday morning ☕. You decide to walk to the coffee shop at the corner of your street. That coffee shop is your favorite, because the way they make cappuccino meets your requirements, which have certain specifications. You know that if the cappuccino's milk temperature exceeds your specification limits of 55–65 °C, the fluffy foam will not be stable, and it will most definitely ruin your Monday morning. Unlike other coffee shops that are even closer to your house, the one at the corner of your street is your favorite because you can get a cappuccino with a perfect milk temperature within your specification limits every time.

Now that you have the scenario in mind, let's define the process capability index. Process capability index, process capability ratio, or Cpk, is a measure of the process's capability to produce desired results that meet the customer's specification limits (values within which a product should perform). In the scenario described, the Cpk of preparing milk for cappuccinos would indicate that the coffee shop is consistent with its average performance of producing cappuccinos for customers (i.e., baristas consistently deliver cappuccinos with milk temperature that is 60 °C on average 🥛, and rarely exceeds 55–65 °C).

## How to calculate process capability index? Process capability index formula

Before presenting the process capability index (or process capability ratio) formula, it is essential to note that you can apply different process capability indices (Cpk, Cp, Cpm, Cpkm) to different situations.

For example, you can successfully utilize the process capability index formula when you want to observe variation in your process relative to specification limits and determine how the process can perform in the future. Cpk can be used when your process average is not perfectly centered. The formula for Cpk is as follows:

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

where:

• $C_{pk}$ is the process capability index;
• $\text{USL}$ is the upper specification limit;
• $\text{LSL}$ is lower specification limit; and
• $\sigma$ is the standard deviation.

If you want to see the potential capability of the process when the process average is centered, you can use the following formula of Cp:

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

where :

• $C_p$ is the potential process capability;
• $USL$ is the upper specification limit;
• $LSL$ is the lower specification limit; and
• $\sigma$ is the standard deviation. If you're unsure how to calculate standard deviation, feel free to use our standard deviation calculator.

If you want to see the capability of your process around a specific target, you can use the following formula for Cpm below. However, remember that Cpm does not account for the process average that is off-centered.

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

where:

• $C_{pm}$ is the process capability to be relative to the target process mean. This is also called the Taguchi capability index.
• $C_p$ is the potential process capability;
• $μ$ is the mean;
• ${\displaystyle {{\sigma }}}$ is the standard deviation; and
• $T$ is the target process mean.

If you're determining the process capability around a target, and you don't have a process that is centered around the target value, you can use the Cpkm formula below:

$C_{pkm} = \frac{C_{pk}}{\sqrt{1 + \left( \frac{\mu-T}{\sigma} \right)^2}}$
• $C_{pkm}$ the "third generation" process capability index;
• $C_{pk}$ is the process capability index;
• $μ$ is the mean;
• $\sigma$ is the standard deviation; and
• $T$ is the target process mean.

🙋 Before calculating process capability indices, check whether you have normally distributed data and a stable process. To learn more about normal distribution, and the consistency of the process, use our normal distribution calculator and upper control limit calculator, respectively. Also, don't forget our calculator's Advanced mode 🧠 feature, to calculate Cpm and Cpkm!

## Interpretation of process capability index and the meaning of Cpk statistics

Now that you're familiar with the process capability index formula, how to calculate the process capability index, and what Cpk is, let's talk about its interpretation. As mentioned before, the higher your Cpk value, the better your process produces. Experts agree that you want your Cpk index to be at least 1.33. Here is the explanation why you want to reach this magic number:

Rare events in your process become detectable when the mean moves off-center by 1.5 sigmas (i.e. when normal distribution shifts by 1.5 standard deviations on either side). When Cpk is 1.33, upper and lower specification limits are four standard deviations from the process mean. In this case, there is some (one standard deviation) room for variability within specification limits, and you can consider the process capable. However, a Cpk of 1.33 is not ideal since you want larger variability before defects are displayed.

The process with Cpk of 2 corresponds to a 6-sigma level process; therefore, it can bear a larger process mean shift while having most process-related output within specifications (has desired room for variability). Therefore, a Cpk of 2 would indicate excellent process capability.

## Examples of process capability index calculation

Is it still confusing? Let's discuss an example. Soups at your buffet must be served between 64–80 °C, as per your customer's specifications 🍲. The mean temperature of the soups is usually 72 °C, and the process of reheating has a standard deviation of 1.5 °C. What is the Cpk value of reheating soups?

$\begin{split} C_{pk} &= \min \left[\left(\frac{80-72}{3\times1.5}\right), \left(\frac{72-64}{3\times1.5}\right)\right] \\ C_{pk} &= \frac{8}{4.5} \approx 1.778 \end{split}$

Since the smallest acceptable Cpk is 1.33, your value of 1.778 is a great result and we can conclude that your process is capable!

Mariamy Chrdileli
Upper specification limit
Lower specification limit
Standard deviation
Mean
Results
Potential process capability (Cₚ)
Process capability index (Cₚₖ)
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