## What is an I-MR Chart?

I-MR chart was introduced by Walter Shewart hence control charts are also called as Shewart Charts. I-MR chart also called X-MR chart is a combination of two charts (Individual and Moving Range) is to track the process variability based on the samples taken from a process over the period of time.

An Individual moving range (I-MR ) chart is used when data is continuous and not collected in subgroups. In other words collect the single observation at a time. An I-MR chart provides process variation over time in graphical method. Graphical I-MR chart helps to identify the when the process goes out of control and also indicates where to focus for source of assignable cause.

## I-MR Definition

I-MR chart is basically two separate charts – Individuals (I) chart and Moving Range (MR) chart, the combination of two charts provides the complete picture of process behavior.

**I-Chart: **Individual chart displays the individual data points and monitors mean and shifts in the process when the data points collected at regular intervals of time. This chart will help to identify the common and assignable causes in the process, if any.

**MR Chart:** While Individual chart monitors the process mean, the Moving Range chart monitors the process variation when the data points collected at regular intervals of time. In other words the moving range chart tracks the absolute difference of each measurement to its previous measurement.

## An I-MR used for:

- To check any indications of assignable causes in the process which may leads to process out of control
- To monitor the process performance before and after implementation of process improvements
- When the production rate is very slow in the shop floor and difficult to wait for more samples
- The measurement systems is too expensive (like destructive testing)
- Mostly used for batch process

## How is an I-MR chart different than X̅-R chart ?

Individual chart displays the individual data points and monitors shifts in the process, where as X̅ monitors mean change over the time.

Moving Range chart monitors the absolute difference of each measurement to its previous measurement. Range chart monitors the subgroups range change over the time.

### When would you use one over the other?

X̅ – R chart to be used when the large data is readily available, where as I-MR chart to be used in case of limited data. I-MR chart to be used when subgroup size is one while X̅ – R chart to be consider when subgroup size is between two & ten, and use X̅ -S chart if the subgroup size is eleven or more.

## How to use an I-MR Chart

To effectively monitor any process, the initial phase the values are used to evaluate the process mean and standard deviation, these will establish the control limits for Individual and ranges. During this phase, monitor are any points are moving out of control and identify the special cause for variation and try to eliminate those causes to make the process in control. Once the process is in control and control limits for Individual and moving range charts are in place, use these control limits and collect the data points at regular interval of time to monitor the process variation. Identify the special cause if the process is out of control.

## How to build an I-MR Chart

- First of all, calculate the Moving range (MR) that is absolute value of the difference between consecutive measurements
- Calculate the process average (x̅) and average of moving range (M̅R̅)
- For I-MR chart n is always equal to 2 ( because we are calculating the moving average of two consecutive points, in other words it is basically comparing the current sate with previous state)
- Finally, construct the control charts for Individual and Moving range

- Where x
_{i }= value at point i - k= number of subgroups

The formula for center line, Lower and Upper control limits for Individuals are:

The formula for center line, Lower and Upper control limits for Moving range are:

Where,
D_{3}, D_{4} and E_{2} are control chart constants

E2 being the constant for I-MR Chart: http://web.mit.edu/2.810/www/files/readings/ControlChartConstantsAndFormulae.pdf

## Important notes on I-MR chart

- I-MR data points always should be in time sequence order, in other words capture the data in the same order when it happened.
- The cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) charts are monitors the mean of the process but they consider the cumulative information whereas Individual chart considers just current values. Hence they are more sensitive when compared to Individual chart for minor shifts in the process.
- Individual charts can be interpret like X̅ charts. While Moving range charts cannot be create like X̅ – R charts.
- I-MR charts are to monitor individual observation rather than sub group averages.

## Example of an I-MR Chart

A sales person travels various shops in the city to deliver the sample products. Below are the distance traveled data (in miles) for the last 11 months. Calculate the control limits for I-MR chart.

First calculate Moving Range

Calculate the X̅ and M̅R̅

- k=11
- X̅ = 880/11= 80
- M̅R̅ =32/10 =3.2
- n=2 (comparing the current state with previous state)

E_{2} = A_{2 }√n = 1.88* √2 = 2.66

Calculate the Control limits for the Individual Chart

Calculate the Control limits for the Moving Average Chart

## Example of Using an I-MR Chart in a DMAIC Project

I-MR control charts are used when subgroup size is equal to 1. Example Monthly reporting data (like customer complaints, Inventory, monthly sales data etc.,). Usually for any process assignable causes will be takes care during Analyze and Improve phases of DMAIC using different types of statistical tools like Regression, Design of Experiments etc.,.

The variation of the process due to common causes still exists while the process is under control. And I-MR control chart will help to identify the process abnormality due to assignable causes in timely manner in Control Phase of DMAIC. These charts will only signals that there is a problem in the process.

Let’s say that the Project was chartered to run a DMAIC process with the stated goal of increasing customer satisfaction via Net Promoter Scores. Through out the measure phase customer complaints was measured to get a baseline. An I-MR Chart could be used to show trends of customer complaints.

Later, the analyze phase could have determined ways to remove several causes of variation. In the Improve phase, changes could have then been made to the process and piloted. You could use an I-MR chart then to see what the effects of the pilot were. Also, in the Control Phase, you might make continual monitoring of the I-MR Chart part of your control plan.

The Six Sigma team has to identify the cause of the problem and appropriate corrective action to be initiated. This is an iterative procedure for continuous improvement of the process.

## What You Need to Know for Certification

### Six Sigma Green Belts

The IASSC Six Sigma Green Belt BOK lists knowledge of I-MR charts under it’s control phase

The ASQ Six Sigma Green Belt BOK requires:

Selection and application of control chartsIdentify, select, construct, and apply the following types of control charts: X Bar−R, X Bar−s, individuals and moving range (ImR / XmR), median (), p, np, c, and u. (Apply)

### Six Sigma Black Belts

The ASQ Six Sigma Black Belt BOK Requires:

Select and use the following control charts in various situations: X¯ – R, X¯ – s, individual and moving range (ImR), attribute charts (p, np, c, u), short-run SPC and moving average. (Apply)

The IASSC Six Sigma Black Belt BOK lists knowledge of I-MR charts under it’s control phase.

The Villanova Six Sigma Black Belt BOK requires the following

Advanced statistical process controlUnderstand appropriate uses of short-run SPC, EWMA, CuSum, and moving average.

**Six Sigma Black Belt Certification I-MR Chart Test Questions: **

**Question**: Which of the following control chart to be used when the rational sub group size is one?

(A) X̅ -R Chart

(B) p Chart

(C) I-MR Chart

(D) c Chart

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## Comments (2)

Where did you mention about E2 being the constant for ImR Chart?

Here’s a good reference, Henry: http://web.mit.edu/2.810/www/files/readings/ControlChartConstantsAndFormulae.pdf