Control Chart Constants

Chart for Averages					Chart for Standard Deviations
Observations in Sample,?n	Factors for Control Limits		Factors for Central Line		Factors for Control Limits
Observations in Sample,?n	A	A₂	A₃	c₄	1 / c₄	B₃	B₄	B₅	B₆
2	2.121	1.880	2.659	0.7979	1.2533	0	3.267	0	2.606
3	1.732	1.023	1.954	0.8862	1.1284	0	2.568	0	2.276
4	1.500	0.729	1.628	0.9213	1.0854	0	2.266	0	2.088
5	1.342	0.577	1.427	0.9400	1.0638	0	2.089	0	1.964
6	1.225	0.483	1.287	0.9515	1.0510	0.030	1.970	0.029	1.874
7	1.134	0.419	1.182	0.9594	1.0423	0.118	1.882	0.113	1.806
8	1.061	0.373	1.099	0.9650	1.0363	0.185	1.815	0.179	1.751
9	1.000	0.337	1.032	0.9693	1.0317	0.239	1.761	0.232	1.707
10	0.949	0.308	0.975	0.9727	1.0281	0.284	1.716	0.276	1.669
11	0.905	0.285	0.927	0.9754	1.0252	0.321	1.679	0.313	1.637
12	0.866	0.266	0.886	0.9776	1.0229	0.354	1.646	0.346	1.610
13	0.832	0.249	0.850	0.9794	1.0210	0.382	1.618	0.374	1.585
14	0.802	0.235	0.817	0.9810	1.0194	0.406	1.594	0.399	1.563
15	0.775	0.223	0.789	0.9823	1.0180	0.428	1.572	0.421	1.544
16	0.750	0.212	0.763	0.9835	1.0168	0.448	1.552	0.440	1.526
17	0.728	0.203	0.739	0.9845	1.0157	0.466	1.534	0.458	1.511
18	0.707	0.194	0.718	0.9854	1.0148	0.482	1.518	0.475	1.496
19	0.688	0.187	0.698	0.9862	1.0140	0.497	1.503	0.490	1.483
20	0.671	0.180	0.680	0.9869	1.0133	0.510	1.490	0.504	1.470
21	0.655	0.173	0.663	0.9876	1.0126	0.523	1.477	0.516	1.459
22	0.640	0.167	0.647	0.9882	1.0119	0.534	1.466	0.528	1.448
23	0.626	0.162	0.633	0.9887	1.0114	0.545	1.455	0.539	1.438
24	0.612	0.157	0.619	0.9892	1.0109	0.555	1.445	0.549	1.429
25	0.600	0.153	0.606	0.9896	1.0105	0.565	1.435	0.559	1.420

	Centerline	Control Limits		$$$ \sigma_x $$$
X bar and R Charts	$$$ CL_\overline{X} =\overline{\overline{X}} $$$	$$$ UCL_\overline{X} =\overline{\overline{X}}+A_2 \overline{R} $$$	$$$ LCL_\overline{X} =\overline{\overline{X}}-A_2 \overline{R} $$$	$$$ \frac{\overline{R}}{d_{2}} $$$
X bar and R Charts	$$$ CL_R =\overline{R} $$$	$$$ UCL_R =D_4 \overline{R} $$$	$$$ LCL_R =D_3 \overline{R} $$$	$$$ \frac{\overline{R}}{d_{2}} $$$
X bar and s Charts	$$$ CL_\overline{X} =\overline{\overline{X}} $$$	$$$ UCL_\overline{X} =\overline{\overline{X}}+A_3 \overline{S} $$$	$$$ LCL_\overline{X} =\overline{\overline{X}}-A_3 \overline{S} $$$	$$$ \frac{\overline{s}}{c_{4}} $$$
X bar and s Charts	$$$ CL_R =\overline{s} $$$	$$$ UCL_R =B_4 \overline{s} $$$	$$$ LCL_R =B_3 \overline{s} $$$	$$$ \frac{\overline{s}}{c_{4}} $$$
Median Charts	$$$ CL_\overline{X} =\overline{\widetilde{X}} $$$	$$$ UCL_\overline{X} =\overline{\widetilde{X}}+\overline{\widetilde{A}}_2 \overline{R} $$$	$$$ LCL_\overline{X} =\overline{\widetilde{X}}-\overline{\widetilde{A}}_2 \overline{R} $$$	-
Median Charts	$$$ CL_R =\overline{R} $$$	$$$ UCL_R =D_4 \overline{R} $$$	$$$ LCL_R =D_3 \overline{R} $$$	-
Charts for Individuals	$$$ CL_X =\overline{X} $$$	$$$ UCL_X =\overline{X}+ E_2 \overline{R} $$$	$$$ LCL_X =\overline{X}- E_2 \overline{R} $$$	-
Charts for Individuals	$$$ CL_R =\overline{R} $$$	$$$ UCL_R =D_4 \overline{R} $$$	$$$ LCL_R =D_3 \overline{R} $$$	-

Centerline	Control Limits
$$$ CL_{p} = \overline{P} $$$	Samples not necessarily of constant size $$$ UCL_{pi} = \overline{P}+3 \frac{\sqrt{\overline{p} (1-\overline{p}})}{\sqrt{n_{i}}} $$$ $$$ LCL_{pi} = \overline{P}-3 \frac{\sqrt{\overline{p} (1-\overline{p}})}{\sqrt{n_{i}}} $$$
$$$ CL_{p} = \overline{P} $$$	If the Sample size is constant (n) $$$ UCL_{pi} = \overline{P}+3 \frac{\sqrt{\overline{p} (1-\overline{p}})}{\sqrt{n}} $$$ $$$ LCL_{pi} = \overline{P}-3 \frac{\sqrt{\overline{p} (1-\overline{p}})}{\sqrt{n}} $$$

Centerline	Control Limits
$$$ CL_{np} = \overline{P} $$$	$$$ UCL_{np} = \overline{nP}+3 {\sqrt{\overline{np} (1-\overline{p}})} $$$ $$$ LCL_{np} = \overline{nP}-3 {\sqrt{\overline{np} (1-\overline{p}})} $$$

Centerline	Control Limits
$$$ CL_{c} = \overline{c} $$$	$$$ UCL_{c} = \overline{c}+3 {\sqrt {\overline{c}}} $$$ $$$ LCL_{c} = \overline{c}-3 {\sqrt {\overline{c}}} $$$

Centerline	Control Limits
$$$ CL_{u} = \overline{u} $$$	Samples not necessarily of constant size $$$ UCL_{u} = \overline{u}+3 \sqrt{\frac{\overline{u}}{n_i}} $$$ $$$ LCL_{u} = \overline{u}-3 \sqrt{\frac{\overline{u}}{n_i}} $$$
	using average sample size $$$ UCL_{u} = \overline{u}+3 \sqrt{\frac{\overline{u}}{\overline{n}}} $$$ $$$ LCL_{u} = \overline{u}-3 \sqrt{\frac{\overline{u}}{\overline{n}}} $$$
	If the sample size is constant (n) $$$ UCL_{u} = \overline{u}+3 \sqrt{\frac{\overline{u}}{n}} $$$ $$$ LCL_{u} = \overline{u}-3 \sqrt{\frac{\overline{u}}{n}} $$$