Appendix A: SPC Constants Tables

Appendix A

SPC Constants Tables

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A.1 Purpose of SPC Constants

SPC constants are statistically derived factors used to calculate control limits for different control charts. These constants depend on sample size (n) and are essential for constructing X̄–R, X̄–S, R, and S charts accurately.

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A.2 Control Chart Constants for X̄–R Charts

Table A.1: Constants A₂, D₃, and D₄

Sample Size (n)	A₂	D₃	D₄
2	1.880	0.000	3.267
3	1.023	0.000	2.574
4	0.729	0.000	2.282
5	0.577	0.000	2.114
6	0.483	0.000	2.004
7	0.419	0.076	1.924
8	0.373	0.136	1.864
9	0.337	0.184	1.816
10	0.308	0.223	1.777
15	0.223	0.347	1.653
20	0.180	0.414	1.585
25	0.153	0.459	1.541

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Usage

• X̄-chart limits:
  [
  UCL = \bar{X} + A_2 \bar{R}
  ]
  [
  LCL = \bar{X} - A_2 \bar{R}
  ]

• R-chart limits:
  [
  UCL = D_4 \bar{R}
  ]
  [
  LCL = D_3 \bar{R}
  ]

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A.3 Control Chart Constants for X̄–S Charts

Table A.2: Constants A₃, B₃, and B₄

Sample Size (n)	A₃	B₃	B₄
2	2.659	0.000	3.267
3	1.954	0.000	2.568
4	1.628	0.000	2.266
5	1.427	0.000	2.089
6	1.287	0.030	1.970
7	1.182	0.118	1.882
8	1.099	0.185	1.815
9	1.032	0.239	1.761
10	0.975	0.284	1.716
15	0.789	0.428	1.572
20	0.680	0.510	1.490
25	0.606	0.565	1.435

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Usage

• X̄-chart limits (using S):
  [
  UCL = \bar{X} + A_3 \bar{S}
  ]

• S-chart limits:
  [
  UCL = B_4 \bar{S}, \quad LCL = B_3 \bar{S}
  ]

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A.4 Constants for Estimating Process Standard Deviation

Table A.3: d₂ and c₄ Constants

Sample Size (n)	d₂	c₄
2	1.128	0.7979
3	1.693	0.8862
4	2.059	0.9213
5	2.326	0.9400
6	2.534	0.9515
7	2.704	0.9594
8	2.847	0.9650
9	2.970	0.9693
10	3.078	0.9727
15	3.472	0.9823
20	3.735	0.9860
25	3.931	0.9880

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Usage

• From R-chart:
  [
  \sigma = \frac{\bar{R}}{d_2}
  ]

• From S-chart:
  [
  \sigma = \frac{\bar{S}}{c_4}
  ]

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A.5 Constants for Individuals Chart (I–MR)

Table A.4: Constants for Moving Range

Moving Range Size	E₂	D₃	D₄
2	2.660	0.000	3.267

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Usage

• I-chart limits:
  [
  UCL = \bar{X} + E_2 \bar{MR}
  ]

• MR-chart limits:
  [
  UCL = D_4 \bar{MR}
  ]

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A.6 Attribute Chart Constants (Quick Reference)

Chart Type	Center Line	UCL	LCL
p	( \bar{p} )	( \bar{p} + 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}} )	( \bar{p} - 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}} )
np	( n\bar{p} )	( n\bar{p} + 3\sqrt{n\bar{p}(1-\bar{p})} )	( n\bar{p} - 3\sqrt{n\bar{p}(1-\bar{p})} )
c	( \bar{c} )	( \bar{c} + 3\sqrt{\bar{c}} )	( \bar{c} - 3\sqrt{\bar{c}} )
u	( \bar{u} )	( \bar{u} + 3\sqrt{\frac{\bar{u}}{n}} )	( \bar{u} - 3\sqrt{\frac{\bar{u}}{n}} )

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✅ Exam & Industry Tips

• GATE frequently asks A₂, D₄, and d₂ values

• Memorize n = 5 row (most common in exams)

• LCL may be zero if negative

• Use R-chart first before X̄-chart

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📌 Recommended Next Appendices

1. Appendix B: SPC Formula Sheet (2-page revision)

2. Appendix C: GATE / PSU quick tricks & common mistakes

3. Appendix D: Glossary of SPC & Quality terms

4. References & Bibliography (APA style)

5. Index (print-ready)

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