Page 75 - DMGT206_PRODUCTION_AND_OPERATIONS_MANAGEMENT

Production and Operations Management

Notes The range was calculated by visually examining the observation data in each row and subtracting
the largest number from the smallest number. The range is a measure of dispersion, i.e. it is the
difference between the maximum and minimum values in a sample.
The mean of the means was found by selecting the S symbol and averaging the values of ‘µ’. This
time we are averaging a column and must ensure that the correct column is selected. The mean
of the ranges are also calculated in a similar manner by selecting the S symbol and averaging the
values of ‘R’. The mean values are shown in row 15 in the table.
Standard tables are available that give the number of defective pieces acceptable in each lot
based on the size of the sample. Using such a table, with n = 3, we get; d2 = 1.693. The standard
deviation of the sample can be calculated by dividing the value of the mean of the range (µ) by
the value d2.
The standard error of the sample is obtained using the formula given below and is shown in row
18 of the figure.
sµ = s / v n
Though the Upper Control Limit (UCL) and the Lower Control Limit (LCL), by convention and
due to standard statistical practice, it is normally fixed at 3 times the standard deviation. Based
on this rationale, the UCL and LCL have been shown in rows 20 and 21 in the figure.

The data in the table and the additional calculations can now be converted into a graphical
presentation using the Chart Wizard in Excel. Use the ‘line type’ chart type and provide values
for the 4 series, i.e., means of samples, UCL, CL and LCL. The y-axis should represent the
variable measurement and the x-axis represents the sequence of samples. A control chart for the
example we have been following is shown as Figure 4.3.

Figure 4.3: R Chart using MS Excel

2.010

2.005 UCL

2.000
CL

1.995
LCL
1.990
1.985
1 2 3 4 5 6 7 8 9 10

The R chart indicates that the process is in control. All the points fall between the UCL and LCL.
As a matter of fact, the process seems to be in excellent control, all the points are very close to the
CL. However, this needs to be verified. When dealing with a quality characteristic that is a
variable, it is a standard practice to control both the mean value of the quality characteristic and
its variability. This is done by using the R chart.
The R Chart: The basic data for the R chart was shown in Table 4.2 and its calculation explained.
From standard tables, with n = 3, we get the value of d3; which relates to the variability of the
sample. In this case, d3 = 1.693. This value is used to obtain the standard deviation of the range.

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