General FAQs

How are control limits calculated for the different Shewhart control charts?

Mean and Range Charts:
You can compute the limits in the following ways:

• as a specified multiple (k) of the standard errors of X-bar_i and R_i above and below the central line. The default limits are computed with k=3 (these are referred to as 3σ limits).
• as probability limits defined in terms of α, a specified probability that X-bar_i or R_i exceeds the limits.

The following notation is used for the formulas:

μ	process mean (expected value of the population of measurements)
σ	process standard deviation (standard deviation of the population of measurements)
Ri	range of measurements in ith subgroup
ni	sample size of ith subgroup
N	number of subgroups
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
zp	100pth percentile of the standard normal distribution
Dp(n)	100pth percentile of the distribution of the range of n independent observations from a normal population with unit standard deviation
	mean of measurements in ith subgroup
	weighted average of subgroup means

The following tables provide the formulas for the limits:

Mean and Standard Deviation Charts:
You can compute the limits in the following ways:

• as a specified multiple (k) of the standard errors of X-bar_i and s_i above and below the central line. The default limits are computed with k=3 (these are referred to as 3σ limits).
• as probability limits defined in terms of α, a specified probability that X-bar_i or s_i exceeds the limits.

The following notation is used for the formulas:

μ	process mean (expected value of the population of measurements)
σ	process standard deviation (standard deviation of the population of measurements)
si	standard deviation of the measurements xi1,...,xini in the ith subgroup
ni	sample size of ith subgroup
N	number of subgroups
zp	100pth percentile of the standard normal distribution
c4(n)	expected value of the standard deviation of n independent normally distributed variables with unit standard deviation
c5(n)	standard error of the standard deviation of n independent observations from a normal population with unit standard deviation
	mean of measurements in ith subgroup
	weighted average of subgroup means
	100pth percentile (0<p<1) of the Χ2 distribution with n degrees of freedom

The following tables provide the formulas for the limits:

IR Charts:
You can compute the limits in the following ways:

• as a specified multiple (k) of the standard errors of X_i and R_i above and below the central line. The default limits are computed with k=3 (these are referred to as 3σ limits).
• as probability limits defined in terms of α, a specified probability that X_i or R_i exceeds the limits.

The following notation is used for the formula:

μ	process mean (expected value of the population of measurements)
σ	process standard deviation (standard deviation of the population of measurements)
Xi	the ith individual measurement
n	number of consecutive measurements used to calculate the moving ranges (by default, n=2)
Ri	moving range computed for the ith subgroup (corresponding to the ith individual measurement). If i<n, then Ri is assigned a missing value. Otherwise Ri = max(Xi,Xi-1,...,Xi-n+1) - min(Xi,Xi-1,...,Xi-n+1) This formula assumes that Xi,Xi-1,...Xi-n+1 are nonmissing.
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
zp	100pth percentile (0<p<1) of the standard normal distribution
Dp(n)	100pth percentile (0<p<1) of the distribution of the range of n independent observations from a normal population with unit standard deviation
	mean of the individual measurements, computed as (X1+...+XN)/N, where N is the number of individual measurements
	average of the nonmissing moving ranges, computed as [(Rn+Rn+1...+RN)/(N+1-n)]

The following tables provide the formulas for the limits:

C Charts:
You can compute the limits in the following ways:

• as a specified multiple (k) of the standard error of c_i above and below the central line. The default limits are computed with k=3 (these are referred to as 3σ limits).
• as probability limits defined in terms of α, a specified probability that c_i exceeds the limits.

The following notation is used for the formula:

u	expected number of nonconformities per unit produced by the process
ui	number of nonconformities per unit in the ith subgroup
ci	total number of nonconformities in the ith subgroup
ni	number of inspection units in the ith subgroup. Typically ni = 1 and ui = ci for c charts. In general, ui = ci/ni.
N	number of subgroups
	average number of nonconformities per unit taken across subgroups. The quantity is computed as a weighted average:
	has a central Χ2 distribution with ν degrees of freedom

The lower and upper control limits, LCLC and UCLC respectively, are given by

The upper probaility limit UCLC is calculated by setting

and solving for UCLC.

The lower probability limit LCLC is calculated by setting

and solving for LCLC.

NP Charts:
You can compute the limits in the following ways:

• as a specified multiple (k) of the standard error of X_i above and below the central line. The default limits are computed with k=3 (these are referred to as 3σ limits).
• as probability limits defined in terms of α, a specified probability that X_i exceeds the limits.

The following notation is used for the formula:

The lower and upper control limits, LCLC and UCLC respectively, are given by

The upper probaility limit UCLC is calculated by setting

and solving for UCLC.

The lower probability limit LCLC is calculated by setting

and solving for LCLC.

P Charts:
You can compute the limits in the following ways:

• as a specified multiple (k) of the standard error of p_i above and below the central line. The default limits are computed with k=3 (these are referred to as 3σ limits).
• as probability limits defined in terms of α, a specified probability that p_i exceeds the limits.

The notation for the p chart is the same as the notation for the np chart.

The lower and upper control limits, LCLC and UCLC respectively, are given by

The upper probaility limit UCLC is calculated by setting

and solving for UCLC.

The lower probability limit LCLC is calculated by setting

and solving for LCLC.

U Charts:
You can compute the limits in the following ways:

• as a specified multiple (k) of the standard error of u_i above and below the central line. The default limits are computed with k=3 (these are referred to as 3σ limits).
• as probability limits defined in terms of α, a specified probability that u_i exceeds the limits.

The following notation is used for the formula:

u	expected number of nonconformities per unit produced by the process
ui	number of nonconformities per unit in the ith subgroup. In general, ui=ci/ni.
ci	total number of nonconformities in the ith subgroup
ni	number of inspection units in the ith subgroup.
N	number of subgroups
	average number of nonconformities per unit taken across subgroups. The quantity is computed as a weighted average:
	has a central Χ2 distribution with ν degrees of freedom

The lower and upper control limits, LCLC and UCLC respectively, are given by

The upper probaility limit UCLC is calculated by setting

and solving for UCLC.

The lower probability limit LCLC is calculated by setting

and solving for LCLC.

FAQ # 2075
Last Updated: 2005 May 16

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