Page 291 - Lean six sigma demystified

Chapter 8 Su S taining i mpr ovement 269

understanding Standard Deviation and Control Charts

Many people ask “Why aren’t my upper and lower control limits (UCL, LCL)
calculated as the mean +/− 3 times the standard deviation?” A simple answer
to this question is that the Levey Jennings chart does this exact calculation, but
there are other factors to consider. To answer this question, you have to under-
stand some key principles and underlying statistics: variation, standard devia-
tion, sampling, and populations.

Variance. The average of the square of the distance between each point in a
total population (N) and the mean (i.e., average). If your data are spread over
a wider range, you have a higher variance and standard deviation. If the data
are centered around the average, you have a smaller variance and standard
deviation.
Standard deviation (s). The square root of the variance.
Sampling. Early users of SPC found that it cost too much to evaluate every
item in the total population. To reduce the cost of measuring everything, they
had to find a way to evaluate a small sample and make inferences from it
about the total population.
Understanding control chart limits. Ask yourself this question: “If a simple
formula using the mean and standard deviation would work, why are there
so many different control charts?” Short answer: to save money by measuring
small samples, not the entire population. Another short answer is to handle
different distributions: binomial, normal, and so on.

When using small samples or varying populations the simple formula using

the mean and standard deviation just doesn’t work, because you don’t know the
average or standard deviation of the total population, only your sample. So why
are there so many control charts? Because you have to estimate the average and
standard deviation using the average and range of your samples. The formulas
to do this vary depending on the type of data (variable or attribute) and the
sample size. Each control chart’s formulas are designed for these varying
conditions.
In variable charts, the XmR uses a sample size of 1, XbarR 2 to 5, and XbarS
5 to 50. These small samples may be taken from lots of 1,000 or more. In attri-
bute charts, the c and np charts use small samples and fixed populations; the u
and p charts use varying populations. So, you have to adjust the formulas to
compensate for the varying samples and populations.