1. (TCO 12) Six samples of subgroup size 2 ( n = 2) were collected. Determine th
ID: 429570 • Letter: 1
Question
1. (TCO 12) Six samples of subgroup size 2 (n = 2) were collected. Determine the upper control limit (UCL) for an X-bar chart if the mean of the sample averages is 4.7 and mean of the sample ranges is 0.35.
Factors for calculating control limits
n A2 D3 D4
2 1.880 0 3.267
3 1.023 0 2.574
4 0.729 0 2.282
5 0.577 0 2.114
6 0.483 0 2.004 (Points : 10)
Explanation / Answer
Before we start to explain the equation, we are going to understand few important terms necessary to equate the equation.
The Control Chart Defined
Control charts are generally used in a production or manufacturing environment and are used to control, monitor and IMPROVE a process. Common causes are always present and generally attributed to machines, material and time vs. temperature. This normally takes a minor adjustme ent to the process to make the correction and return the process to a normal output. HOWEVER, when making a change to the process, it should always be a MINOR change. If a plot is observed that shows a slight deviation trend upward or downward, the "tweaking" adjustment should be a slight change, and then another observation should be made. Too often people will over-correct by making too big of an adjustment which then causes the process to dramatically shift in the other direction. For that reason, all changes to the process should be SLIGHT and GRADUAL!
A control chart is a graph or chart with limit lines, called control lines. There are basically three kinds of control lines:
An Xbar & R Control Chart is one that shows both the mean value ( X ), and the range ( R ). The Xbar portion of the chart mainly shows any changes in the mean value of the process, while the R portion shows any changes in the dispersion of the process. This chart is particularly useful in that it shows changes in mean value and dispersion of the process at the same time, making it a very effective method for checking abnormalities within the process; and if charted while in progress, also points out a problem in the production flow in real time mode.
Steps In Making the Xbar and R Chart
STEP #1 - Collect the data. It is best to have at least 100 samples.
STEP #2 - Divide the data into sub groups, it is recommended the subgroups be of 4 or 5 data points each. The number of samples is represented by the letter " n " and the number of subgroups is represented by the letter " k ". The data should be divided into subgroups in keeping with the following conditions:
The data obtained should be from the same grouping of products produced.
A sub group should not include data from a different lot or different process.
STEP #3 - Record the data on a data sheet. Design the sheet so that it is easy to compute the values of X bar and R for each sub group (see the page in the class example).
STEP #4 - Find the mean value (Xbar). Use the following formula for each subgroup:
STEP #5 - Find the range, R. Use the following formula for each subgroup.
R = X (largest value) - X (smallest value) Example 14.0 - 12.1 = 1.9
X-bar
n is the number of observations
k is the number of subgroups
Upper control limit:
Lower control limit:
Range
k is the number of subgroups.
Upper control limit:
Lower control limit:
Tabular values for X-bar and range charts
Subgroup Size
A2
d2
D3
D4
2
1.880
1.128
-----
3.268
3
1.023
1.693
-----
2.574
4
0.729
2.059
-----
2.282
5
0.577
2.326
-----
2.114
6
0.483
2.534
-----
2.004
7
0.419
2.704
0.076
1.924
8
0.373
2.847
0.136
1.864
9
0.337
2.970
0.184
1.816
10
0.308
3.078
0.223
1.777
Xbar Control Chart:
Central Line (CL) = X double bar figure you calculated.
Upper Control Limit (UCL) = X double bar + A2 * R bar.
Lower Control Limit (LCL) = X double bar - A2 * R bar.
R Control Chart:
Central Line (CL) = R bar figure you calculated.
Upper Control Limit (UCL) = D4 * R bar.
Lower Control Limit (LCL) = D3 * R bar.
For our Class Exercise, the details are as follows:
X Control Chart CL = X double bar = 12.94
UCL = 12.94 + .577 * 1.35 = 13.719 Note that we are using 5 subgroups, so on the chart n = 5, and under the A2 column, 5 = 0.577. 1.35 is the figure you calculated for R bar.
LCL = 12.94 - .577 * 1.35 = 12.161
R Control Chart CL = R bar = 1.35
UCL = 2.115 * 1.35 = 2.86 Note that we are using 5 subgroups, so on the chart n = 5, and under the D4 column, 5 = 2.115.
LCL = Since our subgroups equal 5, if you look under the D3 column, there is no calculation coefficient to apply, thus there is no LCL.
n is the number of observations
k is the number of subgroups
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