Methods and formulas for Individuals Chart

The center line represents the process mean, μ. If you do not specify a historical value for the process mean, Minitab uses the mean of the observations.

If you do not specify a historical value for the process standard deviation, σ, then Minitab estimates σ from your data using the specified method.

Lower control limit (LCL)

Upper control limit (UCL)

Notation

Extended control limits: Between/within approach

Use the between/within approach to calculate extended control limits from within-subgroup and between-subgroup standard variation.

Subgroup size = 1

Where u_I and u_O are quantiles of the standardized normal distribution.

For the expected mean square, calculate σ̂_O as follows:

Notation for between/within approach

Extended control limits: Min/max mean approach

Use the min/max mean approach to calculate extended control limits from the minimum and maximum group means.

Subgroup size = 1

Where u_I is the quantile of the standardized normal distribution.

For the m largest subgroup means:

For the m smallest subgroup means:

Notation for min/max approach

Standard deviation for extended limits

When extended limits are selected, use the following standard deviation for stored variation values and additional sigma limits.

The average moving range, , of length w is given by the following formula:

where MR_i is the moving range for observation i, calculated as follows:

Minitab uses to calculate S_mr, which is an unbiased estimate of σ:

Notation

Use the Nelson estimate to correct for unusually large moving range values in the calculation of the control limits. The procedure is similar to the procedure proposed by Nelson¹ Minitab eliminates any moving range values that are more than 3σ larger than the average moving range, then recalculates the average moving range and control limits.

The median moving range, , of length w is given by the following formula:

where w is the number of observations used in the moving range and MR_i is the moving range for observation i, calculated as follows:

Minitab uses to calculate S_mr, which is an unbiased estimate of σ:

Notation

MSSD stands for the mean of squared successive differences. The square root of the MSSD (SRMSSD) is calculated as follows:

With unbiasing constant

Without unbiasing constant

Notation

d₂(N) is the expected value of the range of N observations from a normal population with standard deviation = 1. Thus, if r is the range of a sample of N observations from a normal distribution with standard deviation = σ, then E(r) = d₂(N)σ.

d₃(N) is the standard deviation of the range of N observations from a normal population with σ = 1. Thus, if r is the range of a sample of N observations from a normal distribution with standard deviation = σ, then stdev(r) = d₃(N)σ.

Use the following table to find an unbiasing constant for a given value, N. (To determine the value of N, consult the formula for the statistic of interest.)

For values of N from 51 to 100, use the following approximation for d₂(N):

For values of N from 26 to 100, use the following approximations for d₃(N) and d₄(N):

Use the following tables to find values for the unbiasing constant, c₄'(), which is used in the formulas for the square root of MSSD method of estimating sigma.

Box-Cox formula

If you use a Box-Cox transformation, Minitab transforms the original data values (Y_i) according to the following formula:

where λ is the parameter for the transformation. Minitab then creates a control chart of the transformed data values (W_i). To learn how Minitab chooses the optimal value for λ, go to Methods and formulas for Box-Cox Transformation.

Common λ values

The following table shows some commonly used λ values and their transformations.

λ	Transformation
2
0.5
0
−0.5
−1