Control charts for variablesVariables control chartsVariable data are measured on a continuous scale. For example: time, weight, distance or temperature can be measured in fractions or decimals.X-Bar, R-Charts, and S-ChartsThere are three types of variable control charts used determine if data is out of control, x-bar charts, r-charts and s-charts. An x-bar chart is often paired with either an r-chart or an s-chart to give a complete picture of the same set of data.

Pairing X-Bar with R-ChartsX-Bar (average) charts and R (range) -charts are often paired together. The X-Bar chart displays the centerline, which is calculated using the grand average, and the upper and lower control limits, which are calculated using the average range. The R-chart plots the average range and the limits of the range. The R-chart displays the dispersion of the subsets. X-Bar/R-Chart plot a subgroup average.

Pairing X-Bar with S-ChartsAlternatively, X-Bar charts can be paired with S-charts (standard deviation). This is typically done when the size of the subsets are large. For larger subsets, the range is a poor statistic to estimate the distributions of the subsets, and instead, standard deviation is used. In this case, the X-Bar chart will display control limits that are calculated using the average standard deviation. The S-Charts are similar to the R-charts; however, instead of the range, they track the standard deviation of multiple subsets.

Reading Control Charts

The quality of the individual points of a subset is determined unstable if any of the following occurs:Rule 1: Any point falls beyond 3σ from the centerline(this is represented by the upper and lower control limts).Rule 2: Two out of three consecutive points fall beyond 2σ on the same side of the centerline.

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Rule 3: Four out of five consecutive points fall beyond 1σ on the same side of the centerline. Rule 4: Nine or more consecutive points fall on the same side of the centerline

Table A: Table of Constants

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The following flow chart demonstrates the general method for constructing an X-bar chart, R-chart, or S-chart:

Calculating UCL and LCLFor the X-Bar chart the following equations can be used to establish limits, where is the grand average, is the average range, and is the average standard deviation.

Calculating Grand Average, Average Range and Average Standard Deviation

To calculate the grand average, first find the average of the n readings at each time point. The grand average is the average of the averages at each time point.

To calculate the grand range, first determine the range of the n readings at each time point. The grand range is the average of the ranges at each time point.

To calculate the average standard deviation, first determine the standard deviation of the n readings at each time point. The average standard deviation is the average of the standard deviations at each time point.

Note: You will need to calculate either the grand range or the average standard deviation, not both.

For X-bar charts, the UCL and LCL may be determined as follows:

Alternatively, can be used as well to calculate UCL and LCL:

The centerline is simply .

For R-charts, the UCL and LCL may be determined as follows:

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The centerline is the value RA

For S-charts, the UCL and LCL may be determined as follows:UCL = B4 S A

LCL = B3 S A

The center line is simply, CL = SA

Calculating Region Boundaries

To determine if your system is out of control, you will need to section your data into regions A, B, and C, below and above the grand average. These regions are shown in Figure III. To calculate the boundaries between these regions, you must first calculate the UCL and LCL. The boundaries are evenly spaced between the UCL and LCL. One way to calculate the boundaries is shown below.

Boundary between A and B above XGA = XGA + (UCL − XGA) * 2 / 3

Boundary Between B and C above XGA = XGA + (UCL − XGA) * 1 / 3

Boundary Between A and B below XGA = LCL + (XGA − LCL) * 1 / 3

Boundary Between B and C below XGA = LCL + (XGA − LCL) * 2 / 3

Example 1Assume that in the manufacture of 1 kg Mischmetal ingots, the product weight varies with the batch. Below are a number of subsets taken at normal operating conditions (subsets 1-7), with the weight values given in kg. Construct the X-Bar, R-charts, and S-charts for the experimental data (subsets 8-11). Measurements are taken sequentially in increasing subset number.

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Solution:First, the average, range, and standard deviation are calculated for each subset.

Next, the grand average XGA, average range RA, and average standard deviation SA are computed for the subsets taken under normal operating conditions, and thus the centerlines are known. Here n=4.

X-Bar limits are computed (using ).

X-Bar limits are computed (using ).

Note: Since n=4 (a relatively small subset size), both and can be used to accurately calculate the UCL and LCL.

R-chart limits are computed.

S-chart limits are computed.

The individual points in subsets 8-11 are plotted below to demonstrate how they vary with in comparison with the control limits.

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Figure E-1: Chart of individual points in subsets 8-11.

The subgroup averages are shown in the following X-Bar chart:

Figure E-2: X-Bar chart for subsets 8-11.

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The R-chart is shown below:

Figure E-3: R-chart for subsets 8-11.

The S-chart is shown below:

Figure E-4: S-chart for subsets 8-11.

The experimental data is shown to be in control, as it obeys all of the rules given above.

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Example 2It’s your first day on the job as a chemical engineer in a plant, and one of your responsibilities is to monitor the pH of a particular process. You are asked by your boss to monitor the stability of the system. She gives you some baseline data for the process, and you collect data for the process during your first day. Construct X-bar and R-Charts to report your results.

Table 1: Baseline data

To be consistent with the baseline data, each hour you take four pH readings. The data you collect is displayed below.

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Table 2: Experimental data

Solution

For this situation, there are k=24 subsets because there are 24 data sets. For each subset, n=4 because there are four pH measurements taken each hour. The first thing you do is calculate the mean and range of each subset. The means are calculated using the AVERAGE() Excel function and the ranges are calculated using MAX() – MIN(). Once these values are calculated, the Grand Average XGA and average range RA are calculated. These values are simply the means of each subset’s mean and range. This data is displayed below.

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Table 3: Data used to calculate and grand average and Range.

Now that you know XGA = 7.01 and RA = 0.12, you can calculate the upper control limit, UCL, and lower control limit, LCL, for the X-bar control chart.

From Table A, A2 = 0.729 when n=4. Using equations UCL and LCL for X-bar charts listed above:

Then the UCL = 7.0982, LCL = 6.9251 and XGA = 7.01 are plotted in Excel along with the average values of each subset from the experimental data to produce the X-bar control chart.

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Table 4: Average subset values and ranges plotted on the X-bar and R-chart

Figure E-5: X-bar control chart

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Then, to construct the Range charts, the upper and lower control limits were found. For n=4, D3 = 0 and D4 = 2.282 so then:

Then, UCL = 0.2710, LCL = 0, RA = 0.12, and the ranges for each subset were plotted vs. time in Excel (Figure E-6).

Figure E-6: Range control chart

From both of these charts, the process is in control because all rules for stabilty are met.

It's important that both of these charts be used for a given set of data because it is possible that a point could be beyond the control band in the Range chart while nothing is out of control on the X-bar chart.

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Control charts for AttributesAttributes control chartsAttribute data are counted and cannot have fractions or decimals. Attribute data arise when you are determining only the presence or absence of something: success or failure, accept or reject, correct or not correct. For example, a report can have four errors or five errors, but it cannot have four and a half errors. Types of attribute control charts

• p chart (proportion chart) • np chart • c chart (count chart) • u chart

Defect vs. Defective‘Defect’ – a single nonconforming quality characteristic.

‘Defective’ – items having one or more defects.

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1. P-ChartA P-Chart is an attribute control chart used when plotting:1) DEFECTIVES2) BINOMIAL ASSUMPTIONS SATISFIED3) VARIABLE SAMPLE SIZE (will also plot constant sample size)Below is a sample set of data which will help illustrate the creation of the P-chart, its centerline and control limits.

Example on P chart:Alpha electronic company manufactures cathode ray tubes on mass production basis. At some intermediate point of production line, 15 samples of size 50 each have been taken. Tubes within each sample were classified into good or bad. The related data are given in the following table. Construct a P-chart with 3 sigma limit and comment on the process.SampleNumber

Number of Defective Tubes Proportion of Defective Tubes

1 10 0.20

2 10 0.20

3 9 0.18

4 10 0.20

5 4 0.08

6 6 0.12

7 2 0.04

8 3 0.06

9 9 0.18

10 4 0.08

11 8 0.16

12 11 0.22

13 8 0.16

14 10 0.20

15 9 0.18

The limits are incorporated in the following Fig. 18.7 along with the sample observations.

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Sample numberFig. 18.7 P-chartProblem 2- Variable sample size

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From this figure, it is clear that all the points are within the control limits. But, there is an upswing towards the right hand side of the figure. This means that there is predominant uptrend which may take the process out of control in future, if no corrective action is taken.

From the data set above the following graph is produced. Many statistical programs offer a variety of options for establishing the limits and centerline descriptions.

Problem 3Twenty samples of n = 200 were taken by an operator at a workstation in a production process. The number of defective items in each sample were recorded as follows.

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Management wants to develop a p-chart using 3-sigma limits. Set up the p-chart and plot the observations to determine if the process was out of control at any point.

Solution:

Step 1. Compute p:

Step 2. Determine the control limits:

Step 3. Construct the p-chart with p = 0.08, UCL = 0.137, and LCL = 0.023. The process does not appear to be out of control.

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2. Pattern Tests

In the preceding problem, even though the control chart indicates that the process is in control, management wants to use pattern tests to further determine if the process is in control.

Solution:

Step 1. Determine the "up-and-down" and "above-and-below" runs and zone observations. Construct the zone boundaries on the control chart as follows.

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Because 4 out of 5 consecutive points are in Zone B (i.e., points 16 to 20) it suggests that nonrandom patterns may exist and that the process may not be in control.

2. NP-Chart

NP-Chart is an Attribute control chart used when plotting:

1) DEFECTIVES

2) BINOMIAL ASSUMPTIONS SATISFIED

3) CONSTANT (fixed) SAMPLE SIZE (subgroup size)

Each observation is independent.

Use this chart to develop upper and lower control limits (UCL and LCL) and determine performance of process over time. Plots the number of non-conforming units or pieces. Each piece or unit may have one or more defects that make it defective.

Below is sample set of data and the NP-Chart follows.

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The chart indicates that the process is in control. The control tests that were used all passed in this case.Another Example to calculate NP-Chart Control LimitsWith 50 samples (n=50), a process has a p-bar of 0.25. What are the upper and lower three sigma limits (UCL/LCL)?

A) 0.25 and -0.25B) 12.5 and -12.5C) 21.69 and 3.31D) 15.56 and 9.44Answer: C

3. C-Chart

Attribute (Discrete) Control Charts

The C-Chart is an attribute control chart used when plotting:

1) DEFECTS

2) POISSON ASSUMPTIONS SATISFIED

3) CONSTANT (fixed) SAMPLE SIZE (subgroup size)

Shown below is a sample of data taken from 25 runs of a constant sample size.

In all the runs, we sampled 12 widgets. In the first run, there were 10 DEFECTS found among the 12 widgets.

It does not say if all 10 DEFECTS were found in one widget or which amount, it simply says 10 in total were found between all 12 samples in Run #1.

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The chart below is the C-Chart representing the data shown above.

The chart indicates that the process is in control. The control tests that were used all passed in this case.

Recall there are a variety of control tests and most statistical software programs allow you to select and modify these criteria.

4. U-Chart

Attribute (Discrete) Control Charts

U-Chart is an attribute control chart used when plotting:

1) DEFECTS

2) POISSON ASSUMPTIONS SATISFIED

3) VARIABLE SAMPLE SIZE (subgroup size)

Each observation is independent.

Used to develop upper control limit and lower control limit (UCL and LCL) and determine performance of process over time. Plots the number of defects per unit sampled in a variable sized sample.

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Shown below is the data set plotted using a U-Chart.

The chart indicates that the process is in control. The control tests that were used all passed in this case.

Recall there are a variety of control tests and most statistical software programs allow you to select and modify these criteria.

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http://www.six-sigma-material.com/P-Chart.html

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One standard deviation