13.3S-Control Charts 555
where the next to last equality follows from Equation 13.2.2 and the fact that the expected
value of a chi-square random variable is equal to its degrees of freedom parameter.
On using the fact that, when in control,Sihas the distribution of a constant (equal
toσ/
√
n−1) times the square root of a chi-square random variable withn−1 degrees
of freedom, it can be shown thatSiwill, with probability near to 1, be within 3 standard
deviations of its mean. That is,
P{E[Si]− 3√
Var(Si)<Si<E[Si]+ 3√
Var(Si)}≈.99Thus, using the formulas 13.3.1 and 13.3.2 forE[Si]and Var(Si), it is natural to set the
upper and lower control limits for theSchart by
UCL=σ[c(n)+ 3√
1 −c^2 (n)] (13.3.3)LCL=σ[c(n)− 3√
1 −c^2 (n)]The successive values ofSishould be plotted to make certain they fall within the upper
and lower control limits. When a value falls outside, the process should be stopped and
declared to be out of control.
When one is just starting up a control chart andσis unknown, it can be estimated
fromS/c(n). Using the foregoing, the estimated control limits would then be
UCL=S[ 1 + 3√
1/c^2 (n)− 1 ] (13.3.4)LCL=S[ 1 − 3√
1/c^2 (n)− 1 ]As in the case of starting up anX-control chart, it should then be checked that thek
subgroup standard deviationsS 1 ,S 2 ,...,Skall fall within these control limits. If any of
them falls outside, then those subgroups should be discarded andSrecomputed.
EXAMPLE 13.3a The following are theX andSvalues for 20 subgroups of size 5 for
a recently started process.
Subgroup XSSubgroup XSSubgroup XSSubgroup XS
1 35.1 4.2 6 36.4 4.5 11 38.1 4.2 16 41.3 8.2
2 33.2 4.4 7 35.9 3.4 12 37.6 3.9 17 35.7 8.1
3 31.7 2.5 8 38.4 5.1 13 38.8 3.2 18 36.3 4.2
4 35.4 3.2 9 35.7 3.8 14 34.3 4.0 19 35.4 4.1
5 34.5 2.6 10 27.2 6.2 15 43.2 3.5 20 34.6 3.7