Surfstat.australia: an online text in introductory Statistics

STATISTICAL INFERENCE

STATISTICAL CONTROL CHARTS

Individuals Charts

There are applications of control charts when the available data are not plentiful, occur at long or irregular time intervals, or simply do not lend themselves to form rational subgroups. For example, monthly sales or other accounting data, batch chemical reactions every 48 hours, absenteeism, lost time injury rate, bulk shipments of raw material. In these cases it is useful to have control charts for individual measurements, or individuals charts.

Recall that in order to have control limits we need an estimate of the standard deviation of the process which reflects only common cause variability. We did this for the chart by sampling output close together and forming the observations into rational subgroups.

When using individuals control charts, observations are scarce; if large subgroups were formed there would be few subgroups to start the control chart with; the long passage of time might mean more chance for special causes to inflate the variability within subgroups.

Moving Ranges

One way to estimate the process standard deviation is by taking the smallest possible subgroups, with subgroup size n = 2. In addition, to make maximum use of the data, the subgroups are overlapped. For example, in the data below the ranges are calculated for overlapping groups of two. 

Data:      21    22    22    20    20    18    23    23    24    22    18
Range (n=2):   1     0     2     0    2      5     0     1     2     4

We call these "moving ranges of 2". Then is calculated as the average of these moving ranges. We can use to obtain an unbiased estimate of s, using a constant d2, which theoretically links the range and the standard deviation of a normal distribution, = where d2 has the value 1.13 for moving ranges of size 2.

A control chart for individuals is constructed by calculating the average range, , based on moving ranges of size 2. The control chart has

CL =

UCL = + 3 = + 2.66

LCL = - 3 = - 2.66

(where the constant 2.66 is appropriate for moving ranges of n=2 because d2 takes the value 1.13)

To produce a control chart for individual observations using MINITAB, the ICHART command is used, where the default value for the moving range is length 2. Enter the individual observations into one column and type

MTB > ICHART C

Historical values for µ and s can be specified using subcommands, where required.

You need at least 20 observations to have an adequate sample size with which to estimate the process standard deviation using the moving range.

Individuals charts are not going to be very powerful in detecting small to moderate shifts in the process mean.

2s limits

This lack of power could influence one to sometimes use " 2s limits" for individuals charts. This will necessarily lead to more false alarms, but the increased chance of detecting true special causes may be justified. The cost of investigating special causes would obviously be a factor in this decision.

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