Control Chart Formulas

The following abbreviations are used:

• CL: center line

• D4: statistics variable D4, d4 = 3.267
• LCL: lower control limit
• LCL(N): lower control limit for row N
• LCL_VAR: lower control limit multiplier
• LCL_VAR_MR: LCL_VAR with a suggested value of E2 = 2.660
• N: number of data rows
• pbar: subgroup unit mean for P Chart

• SIZE: subgroup size

• sgN: subgroup column value for row N
• SQRT: square root
• UCL: upper control limit

• UCL(N): upper control limit for row N

• UCL_VAR: upper control limit multiplier

• UCL_VAR_MR: UCL_VAR with a suggested value of E2 = 2.660

• vN: displayed row value for row N
• xN: data values for row N

Step 1: Calculate the mean of the data column. These are the graph display points.

Step 2: Calculate the standard deviation for each subgroup:

STD_DEV = SQRT [(01 - mean)² +(02 - mean)² + ... +(0i - mean)² / i - 1 ]

Step 3: Calculate the mean for the subgroup means (Mean_Bar) and the standard deviations (StdDev_Bar).

Step 4: Calculate the CL, UCL, and LCL:

CL = Mean_Bar

UCL = CL + A3 * StdDev_Bar

LCL = CL - A3 * StdDev_Bar

NOTE: A3 is a variable that is constant for any subgroup size, which means that display points of the same subgroup size will have the same value for A3, and thus the same UCL, LCL, and CL.

A3 = 3.0 / (c4 * SQRT(i)) where i = subgroup size, and c4 = (4 * (i - 1)) / ((4 * i) - 3)

vN = xN / sgN

CL = (v1) + (v2) + ... + (vN) / N

UCL(N) = CL + (UCL_VAR * SQRT (CL * (1 - CL) / sgN))

LCL(N) = CL - (LCL_VAR * SQRT (CL * (1 - CL) / sgN))

pbar = (x1 + x2 + ... + xN) / (N * SIZE)

CL = pbar * SIZE

UCL = CL + (UCL_VAR * SQRT(SIZE * pbar * (1 - pbar)))

LCL = CL - (LCL_VAR * SQRT(SIZE * pbar * (1 - pbar)))

CL = (x1 + x2 + x3+ .... + xN) / N

UCL = CL + (UCL_VAR * SQRT(CL))

LCL = CL - (LCL_VAR * SQRT(CL))

vN = xN / sgN

CL = (v1 + v2 + ... + vN) / (sg1 + sg2 + ... + sgN)

UCL(N) = CL + (UCL_VAR * SQRT(CL / sgN))

LCL(N) = CL - (LCL_VAR * SQRT(CL / sgN))

vN = xN - x(N - 1)

CL = v1 + v2 + ... + vN / N

UCL = CL + (D4 * CL)

LCL = 0

vN = xN - x(N - 1)

CL = v1 + v2 + ... + vN / N

UCL = CL + (UCL_VAR_MR * CL)

LCL = CL - (LCL_VAR_MR * CL)