The capability procedures that follow infer a distribution of individuals based on information about sample means. However, capability procedures rely on normality of the data. A capability ratio based on small samples or non-normal processes is likely to misleading. The requirement that a process be in statistical control before capability can be measured attempts to guarantee that the process is at least free of nonrandom variation. If the original observations (the individual x's) are independent and identically distributed, the central limit theorem tells us the sample means ( 's) are normally distributed. A warning: the distribution of departs from normality when the sample size is small or the parent distribution is skewed.

By using the sample means and range to estimate 3 sigma limits for individual (x) measurements we can apply the empirical rule to individual measurements. To distinguish between the standard deviations for sample means and standard deviations of individuals, write for sample means and for individuals. Our goal is to calculate an estimate of the population standard deviation of individuals, . The formula for sigma hat uses the average range and a control chart constant which relates range, standard deviation and sample size. = .

For d2 values see Constants for an x-bar R chart .
Read across in the row for subgroup size 3 (n = 3), d2 = 1.693

The accuracy of will be strongly dependent on a normal distribution of individuals, as the following example shows.

Example: The capability of the booth #2 glue process.

Before determining capability, the process must be in statistical control, so that the individual observations will be stable and more likely to be normally distributed. We will artificially put the data into statistical control by eliminating the out of control data points. After having eliminated the offending points, the range is 2.2.

#35 From the booth #2 glue weight data, using = 2.2, calculate .

= =   The actual value is = 1.41

The difference probably comes from using a small sample size (n=3) and from non-normality of the data.

Use to calculate 3 standard deviation limits for individuals.

3= 3 1.30 = 3.90.

Because plus or minus 3 standard deviations include .997 of normally distributed measurements, over 99% of the production can be expected to fall within 3.9 of the center line represented by the grand average (27.8). The specification limits are 25 and 29. The 99% figure only holds if the measurements are normally distributed. Otherwise statistics based on will be inaccurate.

The z-score of the upper specification limit is = 0.92.

Note that the upper specification limit (29) is nearer to the process center (27.8) than is the lower specification limit (25).