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SPC Control Chart Constants Table (A2, D3, D4, d2)

Control chart constants turn the average range of your subgroups into control limits and an estimate of process sigma. This table gives A2, D3, D4 and d2 for subgroup sizes 2 to 10, which covers most X-bar and R charts used in PPAP initial studies.

Download:
Subgroup size nA2D3D4d2
2
1.880
0
3.267
1.128
3
1.023
0
2.574
1.693
4
0.729
0
2.282
2.059
5
0.577
0
2.114
2.326
6
0.483
0
2.004
2.534
7
0.419
0.076
1.924
2.704
8
0.373
0.136
1.864
2.847
9
0.337
0.184
1.816
2.970
10
0.308
0.223
1.777
3.078

Formulas: X-bar chart UCL = grand mean + A2 x R-bar, LCL = grand mean - A2 x R-bar. R chart UCL = D4 x R-bar, LCL = D3 x R-bar. Estimated sigma = R-bar / d2. For D3 shown as 0 the lower limit is zero.

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How to use the table

Pick the row that matches your subgroup size n, then plug the constants into the formulas in the notes. Calculate the mean of each subgroup and its range, then the grand mean and the average range R-bar. For n of 5, UCL for X-bar is the grand mean plus 0.577 times R-bar. Export the table to Excel to drive limit formulas in your own sheet.

Where the constants come from

The constants are derived from the distribution of the range of samples from a normal population. d2 is the expected value of the range in units of sigma, so sigma is estimated as R-bar divided by d2. A2 equals 3 divided by d2 times the square root of n. D3 and D4 set three-sigma limits on the range. These values are standard in SPC references.

Worked example

Subgroups of 5 with a grand mean of 10.002 and an R-bar of 0.020 give X-bar UCL = 10.002 + 0.577 x 0.020 = 10.0135 and LCL = 9.9905. The R chart UCL is 2.114 x 0.020 = 0.0423 and LCL is zero because D3 is zero. Estimated sigma = 0.020 / 2.326 = 0.0086, which feeds capability indices.

Common mistakes

Do not use these constants for individuals charts or for subgroup sizes above 10, where S charts are preferred. Calculating Ppk from the within-subgroup sigma instead of overall sigma is another frequent error: Ppk uses overall standard deviation while Cpk uses R-bar over d2. Check that the process is stable before using limits.

Frequently Asked Questions

What is the difference between A2 and d2?

d2 converts the average range into an estimate of process sigma. A2 converts the average range directly into the half-width of three-sigma limits on the subgroup mean. They are related by A2 = 3 / (d2 x sqrt(n)).

Why is D3 zero for small subgroups?

For subgroup sizes of 6 or fewer, the calculated lower limit on the range would be negative, but a range cannot be negative, so the lower limit is set to zero. From n of 7 upward the lower limit becomes positive.

What subgroup size is typical for PPAP initial studies?

Many customers expect 25 or more subgroups of 5 consecutive parts, giving at least 100 readings. Always follow the customer's requirement, since some specify different sizes or frequencies.

Can I use these for an S chart?

No. S charts use a different set of constants (A3, B3, B4, c4). The values here apply only to X-bar and R charts.

If your measurement data sits in an ERP or MES and you copy it into spreadsheets for charts, we can help automate the pipeline. Contact us with your data source.

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