I-MR Chart (X-MR Chart): Moving Range, Control Limits and Example

What is an I-MR chart (X-MR chart)?

An I-MR chart (also called an X-MR chart or individuals and moving range chart) is a control chart for checking whether a process is stable when data can be collected only one point at a time. You look at the following two charts, one above the other.

• X chart (upper): Plots the measured values themselves in time order. Detects shifts in the process mean
• MR chart (lower): Plots the size of the difference between adjacent values (the moving range). Detects changes in the process variation

The name I-MR comes from the use of individual values.

When to use it

An I-MR chart is used when the data cannot be grouped into subgroups. Examples include the following.

• Measurement is made only once a day (for example, a daily quality check)
• Only one value is obtained per lot or batch (liquid concentration, chemical analysis values, etc.)
• Measurement takes effort or cost, as in destructive testing, so many items cannot be measured
• Production volume is small, so many items under the same conditions cannot be gathered

What is a moving range?

The moving range (MR) is the size of the difference between two adjacent values (the absolute value of the difference). The difference from the previous value is found in order.

The first value has no previous value, so there is one fewer moving range than the number of data points.

When to use it instead of an X-bar and R chart

If many items can be measured in a row under the same conditions, an X-bar and R chart makes changes in the mean easier to detect. The X-bar and R chart is covered in X-bar and R Chart.

Formulas

Let k be the number of data points.

The coefficient 2.66 comes from 3 ÷ 1.128. Dividing MR by 1.128 gives an estimate of the process standard deviation, and limits are placed 3 times that above and below the center line. 3.267 is the coefficient used for the range of two values (D₄ for a subgroup size of 2).

Worked example

Let us calculate with 6 data points (when actually finding control limits, a guideline is to collect 20 or more points).

• Mean x = 60.6 ÷ 6 = 10.10
• Mean of the moving ranges MR = 1.1 ÷ 5 = 0.22

X chart

• Center line = 10.10
• UCL = 10.10 + 2.66 × 0.22 = 10.10 + 0.59 = 10.69
• LCL = 10.10 − 0.59 = 9.51

MR chart

• Center line = 0.22
• UCL = 3.267 × 0.22 = 0.72

How to read the charts

In a stable process, the points on both the X chart and the MR chart are scattered randomly within the control limits.

With an I-MR chart, looking at the X chart and the MR chart together makes it easier to notice changes in the process. For example, if the process mean shifts upward partway through, it looks like this.

Point Point flagged by a rule

• X chart: After the shift, the points continue above the center line (a run)
• MR chart: At the point of the shift, the difference between adjacent values becomes large, and one large moving range appears

Rules for detecting out-of-control signals

An out-of-control condition is possible when even 1 point falls outside the control limits, when points continue on one side of the center line (a run), or when points keep rising or keep falling (a trend). The eight typical rules given in standards such as ISO 7870-2 (JIS Z 9020-2 in Japan) are listed in X-bar and R Chart.

Each point on the MR chart is made from two adjacent values, so neighboring points tend to have similar values. For this reason, the run and trend rules are mainly applied to the X chart, and on the MR chart it is common to focus on points outside the control limits.

Which rules to use varies from workplace to workplace. If your company or customer has its own rules, follow them.

Points to note when using it

• Arrange the data in time order: The moving range is the difference between adjacent values, so the data must be arranged in the order measured. If the order is changed, the values become meaningless
• Watch the shape of the distribution: Because single values are used, if the distribution of the data departs greatly from a normal distribution, misjudgments become more likely; for example, more points fall outside the control limits even though nothing is out of control. It is reassuring to check the shape of the distribution with a histogram
• Control limits and specification limits are different: Control limits are lines calculated from the process data, and they are separate from the specification limits required of the product. Even if the process is stable, out-of-specification product will occur if the variation is large relative to the specification limits

Relationship with Cpk

When finding the process capability index Cpk from individual data, one method is to use the standard deviation estimated from the moving range instead of the within-subgroup standard deviation.

In the worked example, 0.22 ÷ 1.128 ≈ 0.195. The Process Capability (Cpk) Calculator on this site also finds the within-subgroup standard deviation this way when the subgroup size is set to 1.

The idea behind Cpk is explained in What Is Cpk? The difference between the within-subgroup and overall standard deviations is explained in Cp vs Cpk vs Pp vs Ppk.

Making one in Excel

This is an example where 25 data points are arranged in the order measured in B2 to B26.

Make columns B and C into line charts and add the center line and control limit lines, and you have an I-MR chart.

Calculating with the tool on this site

With the X-MR Control Chart (Individuals and Moving Range), just paste in your data and it calculates the moving ranges and control limits and draws an X chart and an MR chart. It also flags points outside the control limits. You can paste data from Excel together with labels such as dates, and you can also fix the control limits and use them. No registration is required, and it is free to use.