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Control Chart Constants Table: A2, D3, D4, d2 and More

Every X-bar, R and S chart relies on a handful of constants that turn average ranges or standard deviations into control limits. This table puts them all in one searchable place so you do not have to dig through a textbook appendix.

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Subgroup size nA2D3D4d2d3A3c4B3B4
2
1.880
0
3.267
1.128
0.853
2.659
0.7979
0
3.267
3
1.023
0
2.574
1.693
0.888
1.954
0.8862
0
2.568
4
0.729
0
2.282
2.059
0.880
1.628
0.9213
0
2.266
5
0.577
0
2.114
2.326
0.864
1.427
0.9400
0
2.089
6
0.483
0
2.004
2.534
0.848
1.287
0.9515
0.030
1.970
7
0.419
0.076
1.924
2.704
0.833
1.182
0.9594
0.118
1.882
8
0.373
0.136
1.864
2.847
0.820
1.099
0.9650
0.185
1.815
9
0.337
0.184
1.816
2.970
0.808
1.032
0.9693
0.239
1.761
10
0.308
0.223
1.777
3.078
0.797
0.975
0.9727
0.284
1.716
11
0.285
0.256
1.744
3.173
0.787
0.927
0.9754
0.321
1.679
12
0.266
0.283
1.717
3.258
0.778
0.886
0.9776
0.354
1.646
13
0.249
0.307
1.693
3.336
0.770
0.850
0.9794
0.382
1.618
14
0.235
0.328
1.672
3.407
0.763
0.817
0.9810
0.406
1.594
15
0.223
0.347
1.653
3.472
0.756
0.789
0.9823
0.428
1.572
16
0.212
0.363
1.637
3.532
0.750
0.763
0.9835
0.448
1.552
17
0.203
0.378
1.622
3.588
0.744
0.739
0.9845
0.466
1.534
18
0.194
0.391
1.608
3.640
0.739
0.718
0.9854
0.482
1.518
19
0.187
0.403
1.597
3.689
0.734
0.698
0.9862
0.497
1.503
20
0.180
0.415
1.585
3.735
0.729
0.680
0.9869
0.510
1.490

Values are the standard published factors for normally distributed data, rounded to three or four decimals. Use A2, D3, D4 and d2 for X-bar and R charts, and A3, B3, B4 and c4 for X-bar and S charts. D3 and B3 of 0 mean the lower limit is set to zero.

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How to use the table

Find the row for your subgroup size n, the number of readings in each sample. For an X-bar and R chart, use A2 for the X-bar limits, D3 and D4 for the R limits, and d2 to convert R-bar to a sigma estimate. For an X-bar and S chart, use A3 for the X-bar limits, B3 and B4 for the S limits, and c4 to convert S-bar to sigma. Search by n to jump to a row, then export the table to Excel to use it in your own spreadsheets.

  • X-bar chart limits: grand mean plus or minus A2 times R-bar.
  • R chart limits: LCL is D3 times R-bar, UCL is D4 times R-bar.
  • Sigma estimate: R-bar divided by d2, or S-bar divided by c4.

Where the constants come from

The constants come from the sampling distribution of the range and of the standard deviation for normal data. The value d2 is the expected range of n standard normal observations, and d3 is the standard deviation of that range. A2 equals 3 divided by d2 times the square root of n. D3 and D4 are 1 minus and 1 plus 3 d3 over d2, with the lower value floored at zero. Similarly c4 corrects the bias in the sample standard deviation, and A3, B3 and B4 follow from it.

Worked example

A shaft process is sampled with n of 5. The grand mean is 25.004 and R-bar is 0.012. The X-bar chart UCL is 25.004 plus 0.577 times 0.012, which is 25.011, and the LCL is 24.997. The R chart UCL is 2.114 times 0.012, which is 0.0254, and the LCL is zero because D3 is 0 for n below 7. The within sigma is 0.012 divided by 2.326, about 0.0052. This sigma then goes into Cp and Cpk.

Common mistakes

The most frequent mistake is using the constants for the wrong subgroup size, particularly when the sample size changes over time. Another is using R chart factors for an S chart or the reverse. Remember that these constants assume normal data and subgroups of constant size. For individual readings use an I-MR chart with a moving range of two, where the relevant d2 is 1.128 and the limits are the mean plus or minus 2.66 times the average moving range.

Frequently Asked Questions

What does A2 mean in an X-bar chart?

A2 is a factor that converts the average range into three-sigma limits for subgroup means. The limits are the grand mean plus and minus A2 times R-bar. It depends on subgroup size and decreases as the subgroup gets larger.

Why is D3 zero for small subgroups?

For subgroup sizes of 6 or fewer, the calculated lower limit for the range would be negative, which is impossible. By convention the lower control limit of the R chart is then set to zero.

When should I use an S chart instead of an R chart?

Use X-bar and S when subgroups are larger than about 8 to 10, because the range wastes information as n grows. The standard deviation uses all the readings, so it is more efficient for larger samples.

What constants do I use for individual measurements?

For an I-MR chart use d2 of 1.128 for moving ranges of two. Individual limits are the mean plus or minus 2.660 times average moving range, and the moving range UCL is 3.267 times the average moving range.

If you want control limits computed automatically from inspection results in your ERP, we can help you design a private, read-only SPC report.

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