To calculate the Cpk value, we need to find the mean, standard deviation, upper specification limit (USL) and lower specification limit (LSL) of the data. The mean is the average of all the observations, which can be found by adding them up and dividing by the total number of observations. The standard deviation is a measure of how much the data varies from the mean, which can be found by using the formula:
$$\\sigma = \\sqrt{\\frac{\\sum (x-\\bar{x})^2}{n}}$$
where $x$ is an observation, $\bar{x}$ is the mean, and $n$ is the number of observations. The USL and LSL are given by the specifications, which are 8.000 millimeters. The Cpk value is then calculated by using the formula:
$$Cpk = min \\left( \\frac{USL-\\bar{x}}{3\\sigma}, \\frac{\\bar{x}-LSL}{3\\sigma} \\right)$$
where $min$ means taking the smaller of the two values. The Cpk value measures how well the process can produce within the specifications, with a higher value indicating a better capability.
Using a spreadsheet or a calculator, we can find that the mean of the data is 8.325 millimeters, and the standard deviation is 0.829 millimeters. Plugging these values into the Cpk formula, we get:
$$Cpk = min \\left( \\frac{8.000-8.325}{3(0.829)}, \\frac{8.325-8.000}{3(0.829)} \\right)$$
$$Cpk = min \\left( -0.131, 0.131 \\right)$$
$$Cpk = -0.131$$
The Cpk value is negative, which means that the process is not capable of producing within the specifications. The mean is too far away from the target value of 8.000 millimeters, and the variation is too high. The process needs to be improved to reduce the bias and variation, and increase the Cpk value. A Cpk value of at least 1.33 is usually considered acceptable for most processes¹.
¹: [Process Capability (Cp & Cpk)](^2^)