Quality Assurance Lines

Samples are taken at regular intervals. A statistic, such as the sample mean, median or range, is calculated for each sample. The mean and median help monitor the typical value; the range helps monitor how much individual items differ.

A control chart plots these statistics in the order the samples were taken. Its horizontal axis shows sample number or time, and its vertical axis shows the statistic being monitored, with its units. Horizontal lines show the target and the limits used to decide whether a result needs attention.

Warning lines and action lines have different jobs

For a sample-mean chart, the central line represents the target mean. Above and below it are two warning lines, with two action lines further out.

Both directions matter. An unusually low sample mean might indicate underfilled packets; an unusually high one might indicate overfilling. The chart detects departures from the target rather than checking only whether products contain enough.

Calculating limits for sample means

Repeated samples produce a collection of sample means. These means have their own distribution and standard deviation: this describes how much the means vary, rather than how much individual items vary.

Let $\mu$ be the target mean and $\sigma$ be the standard deviation of the sample means. The limits are:

The minus sign gives the lower limit and the plus sign gives the upper limit.

When sample means follow a Normal distribution, approximately 95% lie within two standard deviations of the mean. Consequently, even a correctly functioning process produces about one sample mean in twenty outside the warning limits. A warning is therefore a reason to check, not proof that the machinery is faulty.

Values more than three standard deviations from the mean are highly unusual. Almost all sample means should lie within the action limits when the process is working correctly, so a result beyond them calls for a stronger response.

Reading a control chart in context

Suppose a packet-filling process has a target sample mean of 500 g. When it is working correctly, sample means are Normally distributed with a standard deviation of 2 g.

The warning limits are $500 - 2\times2 = 496$ g and $500 + 2\times2 = 504$ g. The action limits are $500 - 3\times2 = 494$ g and $500 + 3\times2 = 506$ g. Draw each limit as a horizontal line across the chart.

The first three sample means, 500 g, 501 g and 503 g, are within the warning limits. They require no change to the routine sampling schedule.

Sample 4 has a mean of 505 g. It lies above the upper warning line but below the upper action line, so another sample should be taken immediately. If that next sample has a mean of 507 g, as sample 5 does here, it is above the upper action line. Production should be stopped and the filling machinery checked and corrected.

Notice that the decision depends on the sample mean, not on the sample number or the steepness of the line joining the points.

Why the range matters too

A mean close to the target can conceal large differences between items. For example, two packets weighing 498 g and 502 g have a mean of 500 g and a range of 4 g. Two packets weighing 490 g and 510 g also have a mean of 500 g, but their range is 20 g. The low and high measurements balance in the mean, while the range reveals their spread.

A sample-range chart therefore provides a different check. Usually only upper warning and action lines are shown: the concern is that the variation becomes too large. An ideal range of zero would mean every item had the same measurement.

For example, if a question gives an upper range action limit of 12 g, the 20 g range is beyond it. Production should be stopped and corrected even though the sample mean is exactly on target.

For charts monitoring sample medians or ranges, use the limits supplied in the question. Apply the same warning-versus-action distinction, and consider all the charts before deciding what the manufacturer should do.