Control Limits for X- and R-charts
LCL for X chart =
UCL for X chart =
LCL for R-chart =
UCL for R-chart =
Center line for R-chart: By default, the center line for the ith subgroup (where k is the sigma multiplier) indicates an estimate of the expected value of Ri. This value is computed as: , where is an estimate of σ. If you specify a known value (σ0) for σ, the central line indicates the value of . Note that the central line varies with ni.
The standard deviation of an X/R chart is estimated by:
σ = process standard deviation
ni = sample size of ith subgroup
d2(n) is the expected value of the range of n independent normally distributed variables with unit standard deviation
d3(n) is the standard error of the range of n independent observations from a normal population with unit standard deviation
N is the number of subgroups for which
Control Limits for X- and S-charts
LCL for X chart =
UCL for X chart =
LCL for S-chart =
UCL for S-chart =
Center line for S-chart: By default, the center line for the ith subgroup (where k is equal to the sigma multiplier) indicates an estimate of the expected value of si. This value is computed as , where is an estimate of σ. If you specify a known value (σ0) for σ, the central line indicates the value of . Note that the central line varies with ni.
σ = process standard deviation
ni = sample size of ith subgroup
c4(n) is the expected value of the standard deviation of n independent normally distributed variables with unit standard deviation
c5(n) is the standard error of the standard deviation of n independent observations from a normal population with unit standard deviation
N is the number of subgroups for which
si is the sample standard deviation of the ith subgroup
X = the mean of the individual measurements
MR = the mean of the nonmissing moving ranges computed as (MRn+MRn+1+...+MRN)/N
σ = the process standard deviation
k = the number of standard deviations
d2(n) = expected value of the range of n independent normally distributed variables with unit standard deviation.
d3(n) = standard error of the range of n independent observations from a normal population with unit standard deviation.
d4(n) = expected value of the range of a normally distributed sample of size n.
p is the average proportion of nonconforming items taken across subgroups
ni is the number of items in the ith subgroup
k is the number of standard deviations
u is the expected number of nonconformities per unit produced by process
ui is the number of nonconformities per unit in the ith subgroup. In general, ui = ci/ni.
ci is the total number of nonconformities in the ith subgroup
ni is the number of inspection units in the ith subgroup
u is the average number of nonconformities per unit taken across subgroups. The quantity u is computed as a weighted average
N is the number of subgroups
u is the expected number of nonconformities per unit produced by process
ui is the number of nonconformities per unit in the ith subgroup. In general, ui = ci/ni.
ci is the total number of nonconformities in the ith subgroup
ni is the number of inspection units in the ith subgroup
u is the average number of nonconformities per unit taken across subgroups. The quantity u is computed as a weighted average
N is the number of subgroups
The standard deviation, s, for the Levey-Jennings chart is calculated the same way standard deviation is in the Distribution platform.
P(X r) ~ P(X2v < )
is a chi-square variate with v = 2u/(1+uk) degrees of freedom.
Based on this approximation, approximate upper and lower control limits can be determined. For a nominal level αType 1 error probability in one direction, an approximate upper control limit is a limit UCL such that:
P(X > UCL) = 1 - P(X2v < ) = α
P(X < LCL) = 1 - P(X2v > ) = α
is the upper (lower) percentile of the chi-square distribution with
v = 2u/(1+uk) degrees of freedom. Negative lower control limits can be set to zero.