Methods and formulas for Moving Range Chart

Select the method or formula of your choice.

Plotted points

when i v
when i > v

Notation

TermDescription
vnumber of sample means that are averaged
μprocess mean
kparameter for Test 1. The default is 3.
σprocess standard deviation
nisize of subgroup i

Center line

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

center line = MR * d2(w)

Notation

MR
the estimate of the average moving range for the method that you use to estimate the standard deviation
d2(w)
the unbiasing constant given in this table
w
the number of observations that are used in the moving range

Control limits

Lower control limit (LCL)

The LCL is the greater of the following:

or

Upper control limit (UCL)

Notation

TermDescription
d2() a constant used to estimate the standard deviation
wnumber of observations in the moving range. By default, w = 2.
σ process standard deviation
k parameter for Test 1 (default is 3)
d3() A constant used to estimate LCL and UCL.

Average moving range method

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

where MRi is the moving range for observation i, calculated as follows:

Minitab uses to calculate Smr, which is an unbiased estimate of σ:

Notation

TermDescription
nnumber of observations
w length of the moving range. The default is 2.
d2() value of unbiasing constant d2 that corresponds to the value specified in parentheses.

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

d2(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) = d2(N)σ.

d3(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) = d3(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.)

For values of N from 51 to 100, use the following approximation for d2(N):
For values of N from 26 to 100, use the following approximations for d3(N) and d4(N):
For more information on these constants, see the following:
  • D. J. Wheeler and D. S. Chambers. (1992). Understanding Statistical Process Control, Second Edition, SPC Press, Inc.
  • H. Leon Harter (1960). "Tables of Range and Studentized Range". The Annals of Mathematical Statistics, Vol. 31, No. 4, Institute of Mathematical Statistics, 1122−1147.
Table 1. Table of values
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 0.8480 2.472
7 2.704 0.8332 2.645
8 2.847 0.8198 2.791
9 2.970 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 0.7630 3.356
15 3.472 0.7562 3.422
16 3.532 0.7499 3.482
17 3.588 0.7441 3.538
18 3.640 0.7386 3.591
19 3.689 0.7335 3.640
20 3.735 0.7287 3.686
21 3.778 0.7242 3.730
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
N d2(N)
26 3.964
27 3.997
28 4.027
29 4.057
30 4.086
31 4.113
32 4.139
33 4.165
34 4.189
35 4.213
36 4.236
37 4.259
38 4.280
39 4.301
40 4.322
41 4.341
42 4.361
43 4.379
44 4.398
45 4.415
46 4.433
47 4.450
48 4.466
49 4.482
50 4.498
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