Constructing Charts for Numbers of Nonconformities (c Charts)

CCHART Statement

Constructing Charts for Numbers of Nonconformities (c Charts)

The following notation is used in this section:

u expected number of nonconformities per unit produced by the process

u_i number of nonconformities per unit in the i^th subgroup

c_i total number of nonconformities in the i^th subgroup

n_i number of inspection units in the i^th subgroup. Typically, n_i = 1 and u_i=c_i for c charts. In general, u_i=c_i/n_i.

$\bar{u}$ average number of nonconformities per unit taken across subgroups. The quantity $\bar{u}$ is computed as a weighted average:
$\bar{u} = \frac{n_{1}u_{1} + ... + n_{N}u_{N}} {n_{1} + ... + n_{N}} = \frac{c_{1} + ... + c_{N}} {n_{1} + ... + n_{N}}$

N number of subgroups

$\chi^2_{\nu}$ has a central $\chi^2$ distribution with $\nu$ degrees of freedom

Plotted Points

Each point on a c chart represents the total number of nonconformities (c_i) in a subgroup. For example, Figure 33.10 displays three sections of pipeline that are inspected for defective welds (indicated by an X). Each section represents a subgroup composed of a number of inspection units, which are 1000-foot-long sections. The number of units in the i^th subgroup is denoted by n_i, which is the subgroup sample size. The value of n_i can be fractional; Figure 33.10 shows n₃=2.5 units in the third subgroup.

Figure 33.10: Terminology for c Charts and u Charts

The number of nonconformities in the i^th subgroup is denoted by c_i. The number of nonconformities per unit in the i^th subgroup is denoted by u_i=c_i/n_i. In Figure 33.10, the number of welds per inspection unit in the third subgroup is u₃=2/2.5=0.8.

A u chart created with the UCHART statement plots the quantity u_i for the i^th subgroup (see Chapter 41). An advantage of a u chart is that the value of the central line at the i^th subgroup does not depend on n_i. This is not the case for a c chart, and consequently, a u chart is often preferred when the number of units n_i is not constant across subgroups.

Central Line

On a c chart, the central line indicates an estimate for n_iu, which is computed as $n_{i}\bar{u}$ .If you specify a known value (u₀) for u, the central line indicates the value of n_iu₀.

Note that the central line varies with subgroup sample size n_i. When n_i=1 for all subgroups, the central line has the constant value $\bar{c} = (c_{1} + ... + c_{N})/N$ .

Control Limits

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 $3\sigma$ limits).
as probability limits defined in terms of $\alpha$ , a specified probability that c_i exceeds the limits

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

${LCLC} & = & {max}(n_{i}\bar{u} - k\sqrt{n_{i}\bar{u}}\; ,0 ) \ {UCLC} & = & n_{i}\bar{u}+ k\sqrt{n_{i}\bar{u}}$

The upper and lower control limits vary with the number of inspection units per subgroup n_i. If n_i=1 for all subgroups, the control limits have constant values.

${LCLC} & = & {max}(\bar{c} - k\sqrt{\bar{c}}\; ,0 ) \ {UCLC} & = & \bar{c}+ k\sqrt{\bar{c}}$

An upper probability limit UCLC for c_i can be determined using the fact that

$P\{c_{i} \gt {UCLC}\} & = 1 - P\{c_{i} \leq {UCLC} \} \ & = 1 - P\{\chi^2_{2(\!{{\scriptsize UCLC}}+1)} \geq 2n_{i}\bar{u}\}$

The upper probability limit UCLC is then calculated by setting

$1 - P\{\chi^2_{2(\!{{\scriptsize UCLC}}+1)} \geq 2n_{i}\bar{u}\} = \alpha/2$

and solving for UCLC.

A similar approach is used to calculate the lower probability limit LCLC, using the fact that

$P\{c_{i} \lt {LCLC}\} = P\{\chi^2_{2(\!{{\scriptsize LCLC}}+1)} \gt 2n_{i}\bar{u}\}$

The lower probability limit LCLC is then calculated by setting

$P\{\chi^2_{2(\!{{\scriptsize LCLC}}+1)} \gt 2n_{i}\bar{u}\} = \alpha/2$

and solving for LCLC. This assumes that the process is in statistical control and that c_i has a Poisson distribution. For more information, refer to Johnson, Kotz, and Kemp (1992). Note that the probability limits vary with the number of inspection units per subgroup (n_i) and are asymmetric about the central line.

If a standard value u₀ is available for u, replace $\bar{u}$ with u₀ in the formulas for the control limits. You can specify parameters for the limits as follows:

Specify k with the SIGMAS= option or with the variable _SIGMAS_ in a LIMITS= data set.
Specify $\alpha$ with the ALPHA= option or with the variable _ALPHA_ in a LIMITS= data set.
Specify a constant nominal sample size $n_{i} \equiv n$ for the control limits with the LIMITN= option or with the variable _LIMITN_ in a LIMITS= data set.
Specify u₀ with the U0= option or with the variable _U_ in a LIMITS= data set.

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u	expected number of nonconformities per unit produced by the process
u_i	number of nonconformities per unit in the i^th subgroup
c_i	total number of nonconformities in the i^th subgroup
n_i	number of inspection units in the i^th subgroup. Typically, n_i = 1 and u_i=c_i for c charts. In general, u_i=c_i/n_i.
$\bar{u}$	average number of nonconformities per unit taken across subgroups. The quantity $\bar{u}$ is computed as a weighted average: $\bar{u} = \frac{n_{1}u_{1} + ... + n_{N}u_{N}} {n_{1} + ... + n_{N}} = \frac{c_{1} + ... + c_{N}} {n_{1} + ... + n_{N}}$
N	number of subgroups
$\chi^2_{\nu}$	has a central $\chi^2$ distribution with $\nu$ degrees of freedom