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In this article we will explore in brief about IMR control charts used in Statistical Process Control, and the formulas used in statistical software.

## What is IMR Control Chart?

IMR, or Individual moving range control charts are used to monitor the mean and variation of a process when the following conditions exists:

- Continuous (numeric / decimal) data is collected over a period.
- The data is for an individual observation (not for a subgroup of size > 1)
- The date is moderately normally distributed
- Each data point is independent of the other date points (non-corelated data)

IMR chart includes two charts, i.e individual chart & moving range chart.

## Formula for calculation of I Chart control limits

Each data point, *x _{i}*, is an observation.

### I Chart center line

The center line represents the process mean, *μ*. If you do not specify a historical value for the process mean, we use the mean of the observations.

### I Chart control limits

If historical value for the process standard deviation, *σ*, is not specified, then *σ* is estimated from the data using the specified method.

**Notation**

Term |
Description |

μ |
process mean |

k |
parameter for Test 1. The default is 3. Usually, additional limits are specified at k = 1 and K =2 |

## Estimation of sigma

### Average moving range method

The average moving range, , of length *w* is given by the following formula:

where *MR _{i}* is the moving range for observation

*i*, calculated as follows:

is used to calculate *S _{mr}*, which is an unbiased estimate of

*σ*:

**Notation**

Term |
Description |

N |
number of observations |

w |
length of the moving range. The default is 2. |

d_{2}() |
value of unbiasing constant d_{2} that corresponds to the value specified in parentheses. |

**Note:** Zometric Qtools and Zometric IntelliQS products uses w =2 by default. Therefore, d_{2}(2) = 1.128 is used by default.

## Moving Range Chart Control limit formula

### Moving Range plotted points

Each data point, MR* _{i}*, is the moving range of the x values in each group. MR

*is not plotted for*

_{i}*i*<

*w*because it is undefined.

Term |
Description |

MR | moving range |

w |
Number of observations in the moving range. By default, w = 2. |

d_{2}(w) |
Unbiasing constant given in this table later |

### Moving Range center line formula

The center line is the unbiased estimate of the average of the moving range.

center line = MR * d_{2}(w)

### Moving Range Lower control limit (LCL) formula

The LCL is the greater of the following:

or

### Moving Range Upper control limit (UCL) formula

**Notation**

Term |
Description |

d_{2}() |
a constant used to estimate the standard deviation |

w |
number of observations in the moving range. By default, w = 2. |

σ |
process standard deviation |

k |
parameter for Test 1 (default is 3) |

d_{3}() |
A constant used to estimate LCL and UCL. |

Zometric Qtools and Zometric IntelliQS products uses w =2 by default. Therefore, d_{2}(2) = 1.128 & d3(2) = 0.8525 is used by default.

### Unbiasing constants d2(), d3(), and d4()

- d
_{2}(*N*) is the expected value of the range of*N*observations from a normal population with standard deviation = 1. Thus, if*r*is the range of a sample of*N*observations from a normal distribution with standard deviation =*σ*, then E(*r*) = d_{2}(*N*)*σ*. - d
_{3}(*N*) is the standard deviation of the range of*N*observations from a normal population with*σ*= 1. Thus, if*r*is the range of a sample of*N*observations from a normal distribution with standard deviation =*σ*, then stdev(*r*) = d_{3}(*N*)*σ*. - Use the following table to find an unbiasing constant for a given value,
*N*. (To determine the value of*N*, consult the formula for the statistic of interest.)

N | d2(N) | d3(N) | d4(N) |

2 | 1.128 | 0.8525 | 0.954 |

3 | 1.693 | 0.8884 | 1.588 |

4 | 2.059 | 0.8798 | 1.978 |

5 | 2.326 | 0.8641 | 2.257 |

6 | 2.534 | 0.848 | 2.472 |

7 | 2.704 | 0.8332 | 2.645 |

8 | 2.847 | 0.8198 | 2.791 |

9 | 2.97 | 0.8078 | 2.915 |

10 | 3.078 | 0.7971 | 3.024 |

11 | 3.173 | 0.7873 | 3.121 |

12 | 3.258 | 0.7785 | 3.207 |

13 | 3.336 | 0.7704 | 3.285 |

14 | 3.407 | 0.763 | 3.356 |

15 | 3.472 | 0.7562 | 3.422 |

16 | 3.532 | 0.7499 | 3.482 |

17 | 3.588 | 0.7441 | 3.538 |

18 | 3.64 | 0.7386 | 3.591 |

19 | 3.689 | 0.7335 | 3.64 |

20 | 3.735 | 0.7287 | 3.686 |

21 | 3.778 | 0.7242 | 3.73 |

22 | 3.819 | 0.7199 | 3.771 |

23 | 3.858 | 0.7159 | 3.811 |

24 | 3.895 | 0.7121 | 3.847 |

25 | 3.931 | 0.7084 | 3.883 |

For values of *N* from 26 to 100, use the following approximations for d_{3}(*N*) and d_{4}(*N*):

For values of *N* from 51 to 100, use the following approximation for d_{2}(*N*):