I-MR chart

The top chart plots each value in order, with a centre line at the average and control limits either side. The bottom chart plots the moving range: the gap between each value and the one before it, which is how much the process jumps from one point to the next. It is also called an XmR chart.

The limits on both charts come from the average moving range rather than from the overall standard deviation. That matters: a slow drift or a sudden step in the data would inflate an overall standard deviation and hide itself, but it barely changes the point to point jumps.

Use it when each value stands alone: one figure a day, a week or a case. Waiting times for each patient, days to process each claim, cost of each job, a weekly total. In service work this is the chart you will use most, because data rarely arrives in neat samples.

If your data comes in groups taken at the same time, such as five calls sampled each day, use an X-bar R chart, which uses the information in the groups. If you are counting failures or defects, use a p or c chart. And if the values are heavily skewed, such as waiting times with a long tail, the limits can mislead; look at a histogram first, and consider charting a transformed measure or the median of each period.

Work out each moving range: the difference between each value and the one before, ignoring the sign. Average them to get the average moving range. The individuals limits are the average plus and minus 2.66 times the average moving range. The moving range chart has a lower limit of zero and an upper limit of 3.267 times the average moving range.

The 2.66 is three divided by 1.128, the constant that turns an average range of two points into an estimate of the standard deviation. You do not need to remember it; you need to know that the limits come from the process's own short term variation.

A skills provider wanted to know whether learners were starting reliably. For 24 learners in the order they enrolled, it recorded the days from enrolment to the first session. The figures are illustrative, built to show the method.

The average is 11.58 days and the average moving range is 3.52 days, which puts the individuals limits at 2.22 and 20.94 days. Learner 17 waited 24 days, above the upper limit: a special cause. On the moving range chart the jump back down after it, 12 days, is above the moving range limit of 11.50; the jump up into it, 11 days, falls just under. One clear signal on either chart is enough to look for a cause.

Every other learner sits inside the limits with no run or trend, so apart from that one case the process is stable: a new learner should expect to start somewhere between 3 and 20 days. The useful question is what happened to learner 17, not why one week was a little slower than another. In this kind of service the answer is often a course start date that was moved, or a referral that sat in an inbox.

In practice, once the cause of a point like that is found, recalculate the limits without it. Here they are calculated from all 24 values so that the arithmetic can be followed.

Three short formulas carry the whole chart. Each is stated in words first, then written out, then worked with the 24 learners above.

Put the values in time order in one column. In the next, from the second row, take the absolute difference from the row above. Average each column. Work out the three lines for the individuals chart and the upper limit for the moving range chart with the formulas above, and draw each chart as a line with the limits as flat lines.

Use at least twenty values for the limits. Fewer, and the limits move a lot as each new point arrives.

In Measure, an I-MR chart of the current process is the baseline: how it behaves before anyone changes it. In Control it is the chart that tells the team whether the improvement has held, and when to act.