X-bar R chart

Each point on the top chart is the average of one subgroup: a handful of items measured at about the same time. Each point on the bottom chart is that subgroup's range, its largest value minus its smallest. The averages show whether the process has moved; the ranges show whether it has become more or less consistent.

Averaging makes the chart sensitive. An average of five moves less from chance than a single value does, so a real shift stands out sooner than it would on an individuals chart.

Use it when you can take small samples of two to nine items at regular times: five calls a day, four forms a shift, three deliveries a round. The items in a subgroup should be made or done under the same conditions, so that the range measures only the short term variation, and the differences between subgroups carry any change.

Not when your data arrives one value at a time; use an I-MR chart. Not for subgroups of about ten or more, where an X-bar S chart, using the standard deviation, makes better use of the data. And not when the items in a subgroup come from different conditions, such as one call from each of five teams; the range then measures the difference between teams and the limits become meaninglessly wide.

Average each subgroup and find its range. Average the averages to get the centre line, and average the ranges to get the average range. The averages chart has limits at the centre line plus and minus A2 times the average range. The range chart has an upper limit of D4 times the average range and, for subgroups under seven, a lower limit of zero.

A2 and D4 depend on the subgroup size and come from a standard table. For subgroups of five they are 0.577 and 2.114.

An advice line sampled five calls at random each working day for 20 days and recorded how long each took to handle. The figures are illustrative, built to show the method.

The average of the daily averages is 6.32 minutes and the average range is 1.05 minutes, which puts the averages limits at 5.71 and 6.93 minutes and the range limit at 2.22 minutes. Day 14 averaged 7.84 minutes, above the upper limit. Its range, 1.2 minutes, is ordinary.

Read together, the two charts say something specific: on day 14 every call took longer, by about the same amount, rather than a few calls going badly. That points to a cause that touched the whole day, such as a system running slowly or a new script, rather than to one difficult caller.

There is a second signal, and it teaches something. With day 14 included, the other 19 days all sit below the centre line, a run far longer than the eight the rules allow. That run is not a second change in the process: the one high day has pulled the centre line up above everything else. Set day 14 aside once its cause is found and recalculate: the centre line drops to 6.24 minutes, the limits become 5.64 and 6.84, every remaining day sits inside them with no run, and the process is stable. Always look for the cause of the loudest signal first; it can create the others.

Four steps: the average and the range of each day, then the average of each, then the limits. Each is stated in words first, then written out, then worked with the 20 days above.

One row per subgroup, one column per item. Add a column for the average and one for the range. Average those two columns, apply the constants for your subgroup size, and draw two line charts, averages above and ranges below, each with its centre line and limits. Read the range chart first: if the ranges are not stable, the averages limits, which are built from them, cannot be trusted.

In Measure it shows whether the process is stable before you compare it with what is needed. In Control it holds the gain, and its sensitivity to small shifts makes it the better choice than an individuals chart wherever sampling in groups is natural.