Question

\(\bar x\)- Chart Based on Standard Deviations An x chart based on standard deviations (instead of ranges) is constructed by plotting sample means with a centerline at x and control limits at x + A3s and x - A3s, where A3 is found in Table 14-2 on page 660 and s is the mean of the sample standard deviations. Use the data in Table 14-1 on page 655 to construct an xchart based on standard deviations. Compare the result to the x chart based on sample ranges in Example 5 “x Chart of Altimeter Errors.”

Short answer

The \(\bar x\)-chart is:

The pattern in the referred chart is different from the chart formed here. Here, both charts show statistically out of control processes.

Step-by-step solution

Given information

The data for sample mean is taken to construct the\(\bar x\)chart based on standard deviation.

Day	S	\(\bar x\)
1	4.02	2.8
2	5.18	1.6
3	2.07	2.6
4	3.91	36.6
5	5.03	8.4
6	10.18	-0.8
7	5.55	8.4
8	12.72	11.4
9	12.71	-1
10	15.76	-1
11	12.86	3.6
12	6.69	4.2
13	13.13	-7.6
14	11.37	-12.2
15	17.42	-15
16	22.99	9
17	26.73	16.4
18	16.47	7.4
19	12.19	-11
20	28.5	-7

Step 2:Definea \(\bar x\)-chart based on standard deviations

\(\bar x\)-chart is a graph that maps sample mean values corresponding to the time period.When the chart is based on standard deviation measures, \(\bar s\) (mean of standard deviation is used to compute the control limits.

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

The mean for samplemean and sample standard deviations is computed as,

\(\begin{array}{c}\bar \bar x = \frac{{\sum {\bar x} }}{{{\rm{Number}}\;{\rm{of}}\;{\rm{days}}}}\\ = \frac{{2.8 + 1.6 + ... + \left( { - 7} \right)}}{{20}}\\ = 2.84\end{array}\)

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

The upper and lower control limits are computed as,

\(\begin{array}{l}L.C.L = \bar \bar x - {A_3}\bar s\\U.C.L = \bar \bar x + {A_3}\bar s\end{array}\)

The control chart constants table is referred for obtaining the constant value measures as\({A_3} = 1.427\).

Thus, the values are:

\(\begin{array}{c}L.C.L = 2.84 - 1.427\left( {12.274} \right)\\ = - 14.675\\U.C.L = 2.84 + 1.427\left( {12.274} \right)\\ = 20.355\end{array}\)

Sketch the control chart

Steps to construct the\(\bar x\)-chart based on the standard deviation are:

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

1. Mark the sample mean measures 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 \(\bar x\)-chart is described as follows:

Compare to R-chart

Refer to\(\bar x - \)chart from example5, which has the following measure of centerline, lower control limit and upper control limit:

\(\begin{array}{c}\bar \bar x = 1.19\\U.C.L = 18.88\\L.C.L = - 16.50\end{array}\)

It is observed that the pattern in both charts isdifferent, and both reflect out of control processes.