Data Analysis with Microsoft Excel: Updated for Office 2007

494 Statistical Methods

with subgroup 2 occurring after subgroup 1 and before subgroup 3. As an

example, consider a clothing store in which the owner monitors the length

of time customers wait to be served. He decides to calculate the average wait

time in half hour increments. The fi rst half hour of customers who were

served between 9 and 9:30 a.m. forms the fi rst subgroup, and the owner re-

cords the average wait time during this interval. The second subgroup cov-

ers customers served from 9:30 to 10:00 a.m., and so forth.

The x chart is based on the standard normal distribution. The standard

normal distribution underlies the mean chart, because the Central Limit

Theorem (see Chapter 5) states that the subgroup averages approximately

follow the normal distribution even when the underlying observations are

not normally distributed.

The applicability of the normal distribution allows the control limits

to be calculated very easily when the standard deviation of the process is

known. You might recall from Chapter 5 that 99.74% of the observations

in a normal distribution fall within 3 standard deviations of the mean (μ).

In SPC, this means that points that fall more than 3 standard deviations

from the mean occur only 0.26% of the time. Because this probability is so

small, points outside the control limits are assumed to be the result of un-

controlled special causes. Why not narrow the control limits to ±2 standard

deviations? The problem with this approach is that you might increase the

false-alarm rate, that is, the number of times you stop a process that you in-

correctly believed was out of control. Stopping a process can be expensive,

and adjusting a process that doesn’t need adjusting might increase the vari-

ability through tampering. For this reason, a 3-standard-deviation control

limit was chosen as a balance between running an out-of-control process

and incorrectly stopping a process when it doesn’t need to be stopped.

You might also recall that the statistical tests you learned earlier in the

book differed slightly depending on whether the population standard devia-

tion was known or unknown. An analogous situation occurs with control

charts. The two possibilities are considered in the following sections.

Calculating Control Limits When s Is Known

If the true standard deviation of the process (s) is known, then the control

limits are

LCL5m2

3 s

!n

UCL5m1

3 s

!n

and 99.74% of the points should lie between the control limits if the process

is in control. If s is known, it usually derives from historical values. Here,