Control chart constants are the multipliers that turn an average range or an average standard deviation into 3-sigma control limits. They depend only on the subgroup size n. A2, D3 and D4 build the X-bar and R chart. A3, B3 and B4 build the X-bar and s chart. E2 and D4 (for n = 2) build the I-MR chart. d2 and c4 turn R-bar or s-bar into an estimate of the process standard deviation. The table below gives every one of them for n = 2 to 25.
Download: control-chart-constants.csv (one row per subgroup size, every constant on this page, three decimals). Exact values to six decimals: control-chart-constants-exact.csv.
Which constant goes with which chart
| Chart | Center line | Upper control limit | Lower control limit | Constants |
|---|---|---|---|---|
| X-bar (with R) | X-double-bar | X-double-bar + A2 R-bar | X-double-bar minus A2 R-bar | A2 |
| R | R-bar | D4 R-bar | D3 R-bar | D3, D4 |
| X-bar (with s) | X-double-bar | X-double-bar + A3 s-bar | X-double-bar minus A3 s-bar | A3 |
| s | s-bar | B4 s-bar | B3 s-bar | B3, B4 |
| Individuals (I) | X-bar | X-bar + E2 MR-bar | X-bar minus E2 MR-bar | E2 (n = 2 row: 2.660) |
| Moving range (MR) | MR-bar | D4 MR-bar | D3 MR-bar | D3 = 0, D4 = 3.267 (n = 2 row) |
| Process sigma from ranges | sigma-hat = R-bar / d2 | d2 | ||
| Process sigma from standard deviations | sigma-hat = s-bar / c4 | c4 |
Every one of these limits is the center line plus or minus three standard deviations of the plotted statistic. The constant is just that arithmetic, done once, for a given n.
The table: control chart constants for n = 2 to 25
| n | A2 | d2 | D3 | D4 | A3 | c4 | B3 | B4 | E2 | d3 |
|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 1.880 | 1.128 | 0 | 3.267 | 2.659 | 0.7979 | 0 | 3.267 | 2.660 | 0.853 |
| 3 | 1.023 | 1.693 | 0 | 2.574 | 1.954 | 0.8862 | 0 | 2.568 | 1.772 | 0.888 |
| 4 | 0.729 | 2.059 | 0 | 2.282 | 1.628 | 0.9213 | 0 | 2.266 | 1.457 | 0.880 |
| 5 | 0.577 | 2.326 | 0 | 2.114 | 1.427 | 0.9400 | 0 | 2.089 | 1.290 | 0.864 |
| 6 | 0.483 | 2.534 | 0 | 2.004 | 1.287 | 0.9515 | 0.030 | 1.970 | 1.184 | 0.848 |
| 7 | 0.419 | 2.704 | 0.076 | 1.924 | 1.182 | 0.9594 | 0.118 | 1.882 | 1.109 | 0.833 |
| 8 | 0.373 | 2.847 | 0.136 | 1.864 | 1.099 | 0.9650 | 0.185 | 1.815 | 1.054 | 0.820 |
| 9 | 0.337 | 2.970 | 0.184 | 1.816 | 1.032 | 0.9693 | 0.239 | 1.761 | 1.010 | 0.808 |
| 10 | 0.308 | 3.078 | 0.223 | 1.777 | 0.975 | 0.9727 | 0.284 | 1.716 | 0.975 | 0.797 |
| 11 | 0.285 | 3.173 | 0.256 | 1.744 | 0.927 | 0.9754 | 0.321 | 1.679 | 0.946 | 0.787 |
| 12 | 0.266 | 3.258 | 0.283 | 1.717 | 0.886 | 0.9776 | 0.354 | 1.646 | 0.921 | 0.778 |
| 13 | 0.249 | 3.336 | 0.307 | 1.693 | 0.850 | 0.9794 | 0.382 | 1.618 | 0.899 | 0.770 |
| 14 | 0.235 | 3.407 | 0.328 | 1.672 | 0.817 | 0.9810 | 0.406 | 1.594 | 0.881 | 0.763 |
| 15 | 0.223 | 3.472 | 0.347 | 1.653 | 0.789 | 0.9823 | 0.428 | 1.572 | 0.864 | 0.756 |
| 16 | 0.212 | 3.532 | 0.363 | 1.637 | 0.763 | 0.9835 | 0.448 | 1.552 | 0.849 | 0.750 |
| 17 | 0.203 | 3.588 | 0.378 | 1.622 | 0.739 | 0.9845 | 0.466 | 1.534 | 0.836 | 0.744 |
| 18 | 0.194 | 3.640 | 0.391 | 1.608 | 0.718 | 0.9854 | 0.482 | 1.518 | 0.824 | 0.739 |
| 19 | 0.187 | 3.689 | 0.403 | 1.597 | 0.698 | 0.9862 | 0.497 | 1.503 | 0.813 | 0.733 |
| 20 | 0.180 | 3.735 | 0.415 | 1.585 | 0.680 | 0.9869 | 0.510 | 1.490 | 0.803 | 0.729 |
| 21 | 0.173 | 3.778 | 0.425 | 1.575 | 0.663 | 0.9876 | 0.523 | 1.477 | 0.794 | 0.724 |
| 22 | 0.167 | 3.819 | 0.434 | 1.566 | 0.647 | 0.9882 | 0.534 | 1.466 | 0.785 | 0.720 |
| 23 | 0.162 | 3.858 | 0.443 | 1.557 | 0.633 | 0.9887 | 0.545 | 1.455 | 0.778 | 0.716 |
| 24 | 0.157 | 3.895 | 0.451 | 1.548 | 0.619 | 0.9892 | 0.555 | 1.445 | 0.770 | 0.712 |
| 25 | 0.153 | 3.931 | 0.459 | 1.541 | 0.606 | 0.9896 | 0.565 | 1.435 | 0.763 | 0.708 |
A dash or a zero in the D3 and B3 columns means the same thing: the lower limit of the range or s chart is zero for that subgroup size. For n of 6 or less the R chart has no lower limit. For n of 5 or less the s chart has no lower limit.
Where the numbers come from. We recomputed every value on this page from the sampling distribution of the range and of the sample standard deviation. The values match the NIST/SEMATECH e-Handbook and the AIAG SPC manual tables at three decimals, with one family of exceptions: a few cells (D4 at n = 3, 18, 19 and 22, D3 at n = 19, 22 and 24, and E2 at n = 2) differ by one in the last digit depending on whether a table computes D3, D4 and E2 from rounded or unrounded d2 and d3. The table above prints the conventional values, the ones in the AIAG-referenced table and on most exam reference sheets. The exact values are in the six-decimal CSV. The difference moves a control limit by less than 0.1 percent. It will never change a decision.
Constants for charts with a known standard (A, B5, B6, D1, D2)
If the process standard deviation sigma is given (a standard, or a long history) rather than estimated from the data, the limits use sigma directly. These constants are less common, but exams and SPC software both use them.
| n | A | D1 | D2 | B5 | B6 |
|---|---|---|---|---|---|
| 2 | 2.121 | 0 | 3.686 | 0 | 2.606 |
| 3 | 1.732 | 0 | 4.358 | 0 | 2.276 |
| 4 | 1.500 | 0 | 4.698 | 0 | 2.088 |
| 5 | 1.342 | 0 | 4.918 | 0 | 1.964 |
| 6 | 1.225 | 0 | 5.079 | 0.029 | 1.874 |
| 7 | 1.134 | 0.205 | 5.204 | 0.113 | 1.806 |
| 8 | 1.061 | 0.388 | 5.307 | 0.179 | 1.751 |
| 9 | 1.000 | 0.547 | 5.394 | 0.232 | 1.707 |
| 10 | 0.949 | 0.686 | 5.469 | 0.276 | 1.669 |
| 11 | 0.905 | 0.811 | 5.535 | 0.313 | 1.637 |
| 12 | 0.866 | 0.923 | 5.594 | 0.346 | 1.610 |
| 13 | 0.832 | 1.025 | 5.647 | 0.374 | 1.585 |
| 14 | 0.802 | 1.118 | 5.696 | 0.399 | 1.563 |
| 15 | 0.775 | 1.203 | 5.740 | 0.421 | 1.544 |
| 16 | 0.750 | 1.282 | 5.782 | 0.440 | 1.526 |
| 17 | 0.728 | 1.356 | 5.820 | 0.458 | 1.511 |
| 18 | 0.707 | 1.424 | 5.856 | 0.475 | 1.496 |
| 19 | 0.688 | 1.489 | 5.889 | 0.490 | 1.483 |
| 20 | 0.671 | 1.549 | 5.921 | 0.504 | 1.470 |
| 21 | 0.655 | 1.606 | 5.951 | 0.516 | 1.459 |
| 22 | 0.640 | 1.660 | 5.979 | 0.528 | 1.448 |
| 23 | 0.626 | 1.711 | 6.006 | 0.539 | 1.438 |
| 24 | 0.612 | 1.759 | 6.032 | 0.549 | 1.429 |
| 25 | 0.600 | 1.805 | 6.056 | 0.559 | 1.420 |
With a known sigma: X-bar limits are the target plus or minus A sigma; R chart limits are D1 sigma and D2 sigma; s chart limits are B5 sigma and B6 sigma.
What each constant means
Each constant is a piece of the 3-sigma formula, worked out once for each subgroup size so you do not have to.
- d2 is the average range of n values drawn from a normal process, in units of sigma. For n = 2 the average range is 1.128 sigma. For n = 5 it is 2.326 sigma. That is why R-bar / d2 estimates sigma.
- d3 is the standard deviation of that range, again in sigma units. You rarely use it directly, but D3 and D4 are built from it.
- c4 is the average sample standard deviation in sigma units. A sample standard deviation runs a little low on average (0.94 sigma for n = 5), so s-bar / c4 corrects it.
- A2 = 3 / (d2 times the square root of n). Three sigma of the subgroup mean, with sigma estimated from R-bar.
- A3 = 3 / (c4 times the square root of n). The same thing, with sigma estimated from s-bar.
- D3 = 1 minus 3 d3 / d2 and D4 = 1 + 3 d3 / d2. R-bar plus or minus three standard deviations of the range. When the arithmetic goes negative, D3 is set to zero.
- B3 = 1 minus 3 (the square root of 1 minus c4 squared) / c4 and B4 = 1 + the same term. s-bar plus or minus three standard deviations of s. B3 is floored at zero.
- E2 = 3 / d2. Three sigma of an individual value, with sigma estimated from the average moving range. With the usual moving range of two, E2 = 3 / 1.128 = 2.660.
Check one yourself: for n = 4, A2 = 3 / (2.059 times 2) = 0.7285, which the table rounds to 0.729.
Which constant do I use?
X-bar and R chart (subgroups of 2 to 10)
Read the row for your subgroup size. Use A2 for the X-bar chart limits, D3 and D4 for the R chart limits, and d2 if you want a sigma estimate for a capability study. The range works well up to about n = 10. Past that, the range throws away too much information and the s chart is the better choice. Full method: X-bar R control charts.
X-bar and s chart (subgroups larger than 10, or software)
Same row, different columns. A3 for the X-bar chart, B3 and B4 for the s chart, c4 for the sigma estimate. Minitab and most SPC software offer it as its own chart, and it is the better choice once subgroups pass about ten. Full method: X-bar S control chart.
I-MR chart (one value at a time)
Always the n = 2 row, because each moving range is the difference between two consecutive points. E2 = 2.660 for the individuals chart, D4 = 3.267 for the moving range chart, D3 = 0, and d2 = 1.128 for the sigma estimate. Some books write the individuals limits as X-bar plus or minus 3 MR-bar / 1.128. That is the same number. Full method: I-MR chart and the XmR chart page.
Attribute charts
The p, np, c and u charts do not use this table. Their limits come from the binomial and Poisson distributions and depend on the proportion or count itself. See attribute charts.
Worked example: X-bar R with n = 5
A bottling line fills 500 mL bottles. The operator pulls five bottles every half hour. Ten subgroups, in mL:
| Subgroup | Values | Mean | Range |
|---|---|---|---|
| 1 | 501.2, 499.8, 500.5, 500.1, 499.4 | 500.20 | 1.8 |
| 2 | 500.3, 500.9, 499.1, 500.6, 499.7 | 500.12 | 1.8 |
| 3 | 499.5, 500.2, 501.0, 500.4, 499.9 | 500.20 | 1.5 |
| 4 | 500.8, 499.6, 500.0, 501.3, 500.2 | 500.38 | 1.7 |
| 5 | 499.9, 500.4, 499.3, 500.7, 500.1 | 500.08 | 1.4 |
| 6 | 500.6, 501.1, 500.2, 499.8, 500.5 | 500.44 | 1.3 |
| 7 | 499.4, 500.0, 500.9, 499.7, 500.3 | 500.06 | 1.5 |
| 8 | 500.1, 499.5, 500.6, 500.8, 499.6 | 500.12 | 1.3 |
| 9 | 500.7, 500.3, 499.2, 500.0, 500.4 | 500.12 | 1.5 |
| 10 | 499.8, 500.5, 500.1, 499.9, 501.2 | 500.30 | 1.4 |
X-double-bar (the average of the ten means) = 500.202 mL. R-bar (the average of the ten ranges) = 1.52 mL.
From the n = 5 row: A2 = 0.577, D3 = 0, D4 = 2.114, d2 = 2.326.
X-bar chart: UCL = 500.202 + 0.577 x 1.52 = 501.079. LCL = 500.202 minus 0.877 = 499.325. Center line 500.202.
R chart: UCL = 2.114 x 1.52 = 3.213. LCL = 0 x 1.52 = 0. Center line 1.52.
Process sigma estimate: 1.52 / 2.326 = 0.654 mL.
Every subgroup mean sits between 499.325 and 501.079, and every range is under 3.213. The chart shows no signal. Ten subgroups is enough to start; collect 20 to 25 before you treat the limits as final.
The same data as an X-bar s chart gives nearly the same answer. s-bar = 0.596. A3 = 1.427, so the X-bar limits are 500.202 plus or minus 0.850, which is 499.352 to 501.052. B4 = 2.089, so the s chart UCL is 1.245. The sigma estimate is 0.596 / 0.94 = 0.634 mL. The two estimates (0.654 and 0.634) differ because the range and the standard deviation are two different statistics with their own sampling error. Both are fine. For n = 5 the range keeps about 95 percent of the information in s, which is why the R chart survives in practice.
A shorter I-MR example
Eight daily yields, in percent: 92.4, 93.1, 91.8, 92.9, 93.6, 92.2, 92.8, 93.3. The seven moving ranges are 0.7, 1.3, 1.1, 0.7, 1.4, 0.6, 0.5. X-bar = 92.763, MR-bar = 0.900.
Individuals chart: 92.763 plus or minus 2.660 x 0.900, so 90.369 to 95.156. Moving range chart: UCL = 3.267 x 0.900 = 2.940, LCL = 0. Sigma estimate = 0.900 / 1.128 = 0.798.
Common mistakes
- Reading the wrong row. n is the number of measurements in each subgroup, not the number of subgroups. Ten subgroups of five use the n = 5 row.
- Using the n = 1 row for an I-MR chart. There is no n = 1 row. A moving range of two consecutive points is a range of n = 2, so d2 = 1.128 and D4 = 3.267. If your software uses a moving range of three, use the n = 3 row.
- Mixing the R route with the s route. A2 goes with R-bar. A3 goes with s-bar. A2 times s-bar is a wrong number, and so is A3 times R-bar.
- Dividing by the wrong constant for sigma. R-bar / d2 or s-bar / c4. Not R-bar / A2, and not R-bar / 3.
- Using the overall standard deviation of all the data as sigma. That is the Pp and Ppk approach. Control limits use within-subgroup variation, which is what d2 and c4 estimate. Overall sigma is usually larger and would widen the limits. See Process Capability Cp and Cpk for why the two differ.
- Expecting a lower limit on the R chart with n = 5. D3 is zero through n = 6. A range cannot go below zero, and three standard deviations below R-bar is negative for small n. Zero is the floor, not a mistake in the table.
- Treating the constants as 2-sigma or 3-sigma optional. Every constant on this page assumes 3-sigma limits. Warning limits at 2 sigma and 1 sigma use two thirds and one third of the distance, which is how the Western Electric and Nelson rules zones are drawn.
- Taking the limits from a spec. Control limits come from the data and these constants. Specification limits come from the customer. They are never the same thing.
Exam notes
ASQ’s Green Belt body of knowledge (VI.A.3) asks you to select, construct and use X-bar R, X-bar s and ImR charts, and the Black Belt body of knowledge (VIII.A.4 and VIII.A.5) asks you to select, use and interpret them. Expect to calculate limits by hand. ASQ exams are open book, so bring a printed copy of this table (see what to bring to an open-book ASQ exam). IASSC exams are closed book. IASSC gives you its own Reference Document with the formulas and tables when the exam starts, so you work from theirs, not yours. Either way, practise reading the right row and the right column under time pressure. The Control Charts Study Guide covers chart selection, and Statistical Process Control covers the thinking behind it. Subgroup size itself is a design decision; see rational subgrouping.
Sources and how we checked the numbers
Every value in both tables was recomputed from first principles in Python: d2 and d3 by numerical integration of the distribution of the range of n normal values, c4 from the gamma function, and the rest from the formulas above. The results were compared, at the published rounding, with the NIST/SEMATECH e-Handbook of Statistical Methods, section 6.3.2.1 (A2, D3, D4 for n = 2 to 10) and with the Institute of Quality and Reliability’s tables of constants (reference: AIAG SPC manual), as hosted by MIT. Of 301 cells compared, 10 differ by one in the last digit, all for the rounding reason explained under the main table. c4 agrees with the ten-decimal values on Wikipedia’s unbiased standard deviation page.