Question

s Chart: In this section we described control charts for R and x based on ranges. Control charts for monitoring variation and center (mean) can also be based on standard deviations. An s chart for monitoring variation is constructed by plotting sample standard deviations with a centerline at s (the mean of the sample standard deviations) and control limits at B4s and B3s,where B4and B3 are found in Table 14-2 on page 660 in this section. Construct an s chart for the data of Table 14-1 on page 655. Compare the result to the R chart given in Example 3 “R Chart of Altimeter Errors.”

Short answer

The s-chart is:

The referred R-chart is following the same pattern as the s-chart with different scale of observations.

Step-by-step solution

Given information

Refer to the table 14-1 is taken to construct the s-chart.

Day	S
1	4.02
2	5.18
3	2.07
4	3.91
5	5.03
6	10.18
7	5.55
8	12.72
9	12.71
10	15.76
11	12.86
12	6.69
13	13.13
14	11.37
15	17.42
16	22.99
17	26.73
18	16.47
19	12.19
20	28.5

Step 2:Define a s-chart

S-chart contains a sequential mapping of sample standard deviation measure over time.It includes three values:

• Centerline\(\left( {\bar s} \right)\): mean of all sample standard deviation.
• Lower control limit (LCL)
• Upper control limit (UCL)

From the given values of sample standard deviation, the mean for sample standard deviation is computed as,

\(\begin{array}{c}\bar s = \frac{{\sum s }}{{{\rm{Number}}\;{\rm{of}}\;{\rm{days}}}}\\ = \frac{{4.02 + 5.18 + ... + 28.5}}{{20}}\\ = 12.274\end{array}\)

Thus, the centerline of the s-chart is 12.274.

The upper and lower control limits are computed as,

\(\begin{array}{l}L.C.L = {B_3}\bar s\\U.C.L = {B_4}\bar s\end{array}\)

The values of control limits constant is taken from table 14-2 (control tables constant) as\({B_4} = 2.089,\;{B_3} = 0\).

Thus, the values are:

\(\begin{array}{c}L.C.L = 0\left( {12.274} \right)\\ = 0\\U.C.L = 2.089\left( {12.274} \right)\\ = 25.640\end{array}\)

Sketch the s-chart

Steps to construct the s-chart are:

1. Draw two axis with horizontal axis scaled for days and vertical axis scaled for sample standard deviation.

1. Mark the sample standard deviation measure corresponding to days using dots and connect each consecutive using a line segment.

1. Draw three horizontal lines as centerline, U.C.L and L.C.L parallel to the horizontal axis.

Thus, the s-chart is described as follows:

Compare to R-chart

Refer to R-chart from example 3 for altimeter errors, which has the following measure of centerline, lower control limit and upper control limit:

\(\begin{array}{c}\bar R = 30.65\\U.C.L = 64.79\\L.C.L = 0\end{array}\)

It is observed that the pattern of R-chart is almost same as the s-chart expressed above, though the reference scale for the observations are different.