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### Chapter 6

Control Charts for Variables

6-1. Introduction

- Variable - a single quality characteristic that can be measured on a numerical scale.
- We monitor both the mean value of the characteristic and the variability associated with the characteristic.

Variables data

- Examples
- Length, width, height
- Weight
- Temperature
- Volume

6-2. Control Charts for and R

Notation for variables control charts

- n - size of the sample (sometimes called a subgroup) chosen at a point in time
- m - number of samples selected
- = average of the observations in the ith sample (where i = 1, 2, ..., m)
- = grand average or “average of the averages (this value is used as the center line of the control chart)

6-2. Control Charts for and R

Notation and values

- Ri = range of the values in the ith sample
Ri = xmax - xmin

- = average range for all m samples
- is the true process mean
- is the true process standard deviation

6-2. Control Charts for and R

Statistical Basis of the Charts

- Assume the quality characteristic of interest is normally distributed with mean , and standard deviation, .
- If x1, x2, …, xn is a sample of size n, then the average of this sample is
= (X1 + X2 + … + Xn)/n

is normally distributed with mean, , and standard deviation,

6-2. Control Charts for and R

Statistical Basis of the Charts

- If n samples are taken and their average is computed, the probability is 1- α that any sample mean will fall between:
- The above can be used as upper and lower control limits on a control chart for sample means, if the process parameters are known.

Control Charts for

- When the parameters are not known
- Compute
- Compute R(bar)

- Now
- UCL =
- LCL =

Control Charts for

- But, sest =
- So, can be written as:
- Let A2 =
- Then,

6-2. Control Charts for and R

Control Limits for the chart

- A2 is found in Appendix VI for various values of n.

Control chart for R

We need an estimate of sR

We will use relative range W = R/s, R = Ws

Let d3 be the standard deviation of W

StdDev [R] = StdDev [Ws]

sR = d3s

Use R to estimate sR

Control chart for R

sest,R = d3(R(bar)/d2)

UCL = R(bar) + 3 sest,R = R(bar) + 3d3(R(bar)/d2)

CL = R(bar)

LCL = R(bar) - 3 sest,R = R(bar) - 3d3(R(bar)/d2)

Let D3 = 1 – 3(d3/d2)

And, D4 = 1 + 3(d3/d2)

6-2. Control Charts for and R

Control Limits for the R chart

- D3 and D4 are found in Appendix VI for various values of n.

6-2. Control Charts for and R

Trial Control Limits

- The control limits obtained from equations (6-4) and (6-5) should be treated as trial control limits.
- If this process is in control for the m samples collected, then the system was in control in the past.
- If all points plot inside the control limits and no systematic behavior is identified, then the process was in control in the past, and the trial control limits are suitable for controlling current or future production.

6-2. Control Charts for and R

Trial control limits and the out-of-control process

- Eqns 6.4 and 6.5 are trial control limits
- Determined from m initial samples
- Typically 20-25 subgroups of size n between 3 and 5

- Any out-of-control points should be examined for assignable causes
- If assignable causes are found, discard points from calculations and revise the trial control limits
- Continue examination until all points plot in control
- If no assignable cause is found, there are two options
- Eliminate point as if an assignable cause were found and revise limits
- Retain point and consider limits appropriate for control

- If there are many out-of-control points they should be examined for patterns that may identify underlying process problems

- Determined from m initial samples

6-2. Control Charts for and R

Estimating Process Capability

- The x-bar and R charts give information about the performance or Process Capability of the process.
- Assumes a stable process.
- We can estimate the fraction of nonconforming items for any process where specification limits are involved.
- Assume the process is normally distributed, and x is normally distributed, the fraction nonconforming can be found by solving:
P(x < LSL) + P(x > USL)

Example 6-1: Estimating process capability

- Determine sest = R(bar)/d2 = .32521/2.326=0.1398
- Where d2 values are given in App. Table VI

- Specification limits are 1.5 + .5 microns.
- (1.00, 2.00)

- Assuming N(1.5056, 0.13982)
- Compute p = P(x < 1.00) + P(x > 2.00)

Example 6-1: Estimating process capability

- P = F([1.00–1.5056]/0.1398) +1 – F([2.00-1.5056]/.1398) = F(-3.6166) + 1 – F(3.53648) = .00035
- These values from the APPENDIX 2 / 693
- So, about 350 ppm will be out of specification

6-2. Control Charts for and R

Process-Capability Ratios (Cp)

- Used to express process capability.
- For processes with both upper and lower control limits,
Use an estimate of if it is unknown.

- Cest,p = (2.0 – 1.0)/[6(.1398)] = 1.192

6-2. Control Charts for and R

- If Cp > 1, then a low # of nonconforming items will be produced.
- If Cp = 1, (assume norm. dist) then we are producing about 0.27% nonconforming.
- If Cp < 1, then a large number of nonconforming items are being produced.

6-2. Control Charts for and R

Process-Capability Ratios (Cp)

- The percentage of the specification band that the process uses up is denoted by
**The Cp statistic assumes that the process mean is centered at the midpoint of the specification band – it measures potential capability.

Example 6-1: Estimating bandwidth used

- Pest = (1/1.192)100% = 83.89%
- The process uses about 84% of the specification band.

Phase II Operation of Charts

- Use of control chart for monitoring future production, once a set of reliable limits are established, is called phase II of control chart usage (Figure 6.4)
- A run chart showing individuals observations in each sample, called a tolerance chart or tier diagram (Figure 6.5), may reveal patterns or unusual observations in the data

6-2. Control Charts for and R

Control Limits, Specification Limits, and Natural Tolerance Limits (See Fig 6.6)

- Control limits are functions of the natural variability of the process (LCL, UCL)
- Natural tolerance limits represent the natural variability of the process (usually set at 3-sigma from the mean) (LNTL, UNTL)
- Specification limits are determined by developers/designers. (LSL, USL)

6-2. Control Charts for and R

Control Limits, Specification Limits, and Natural Tolerance Limits

- There is no mathematical relationship between control limits and specification limits.

Rational Subgroups

- charts monitor between-sample variability (variability in the process over time)
- R charts measure within-sample variability
(the instantaneous process variability at a given time)

- Standard deviation estimate of used to construct control limits is calculated from within-sample variability
- It is not correct to estimate using

6-2. Control Charts for and R

Guidelines for the Design of the Control Chart

- Specify sample size, control limit width, and frequency of sampling
- If the main purpose of the x-bar chart is to detect moderate to large process shifts, then small sample sizes are sufficient (n = 4, 5, or 6)
- If the main purpose of the x-bar chart is to detect small process shifts, larger sample sizes are needed (as much as 15 to 25).

6-2. Control Charts for and R

Guidelines for the Design of the Control Chart

- R chart is insensitive to shifts in process standard deviation when n is small
- The range method becomes less effective as the sample size increases
- May want to use S or S2 chart for larger values of n > 10

6-2. Control Charts for and R

Guidelines for the Design of the Control Chart

Allocating Sampling Effort

- Choose a larger sample size and sample less frequently? Or, choose a smaller sample size and sample more frequently?
- The method to use will depend on the situation. In general, small frequent samples are more desirable.

Changing sample size of X-bar and R charts

- 1- variable sample size: the center line changes continuously and the chart is difficult to interpret. (X-bar and s is more appropriate)
- 2- Permanent or semi-permanent change:

Introduction to Statistical Quality Control, 4th Edition

6-2.3 Charts Based on Standard Values

- If the process mean and variance are known or can be specified, then control limits can be developed using these values:
- Constants are tabulated in Appendix VI

6-2.4 Interpretation of and R Charts

- Patterns of the plotted points will provide useful diagnostic information on the process, and this information can be used to make process modifications that reduce variability.
- Cyclic Patterns ( systematic environmental changes )
- Mixture ( tend to fall near or slightly outside the control limits – over control and adjustment too often)
- Shift in process level ( new workers, raw material,…)
- Trend ( continuous movement in one direction- wear out )
- Stratification ( a tendency for the point to cluster artificially around the center line- e.g. Error in control limits computations )

6-2.6 The Operating Characteristic Function

- How well the and R charts can detect process shifts is described by operating characteristic (OC) curves.
- Consider a process whose mean has shifted from an in-control value by k standard deviations. If the next sample after the shift plots in-control, then you will not detect the shift in the mean. The probability of this occurring is called the b-risk.

6-2.6 The Operating Characteristic Function

- The probability of not detecting a shift in the process mean on the first sample is b
= P{LCL <Xbar< UCL m = m1 = m0 + ks}

- L= multiple of standard error in the control limits
k = shift in process mean (# of standard deviations).

Example

- We are using a 3s limit Xbar chart with sample size equal to 5
- L = 3
- n = 5

- Determine the probability of detecting a shift to m1 = m0 + 2s on the first sample following the shift
- k = 2

Example, continued

So, b=F[3–2 SQRT(5)]-F[-3–2 SQRT(5)]

= F(-1.47) – F(-7.37) = .0708

P(not detecting on first sample) = .0708

P(detecting on first sample) =1- .0708=.9292

6-2.6 The Operating Characteristic Function

- The operating characteristic curves are plots of the value against k for various sample sizes

If the shift is 1.0σ and the sample size is n = 5, then

β = 0.75.

Use of Figure 6-13

- Let k = 1.5
- When n = 5, b = .35
- P(detection on 2nd sample) = .35(.65)
- P(detection on 3rd sample) = .352(.65)

- When n = 5, b = .35
- P(detection on rth sample) = br-1(1-b)
- ARL = 1/(1-b)
- 1/(1-.35) = 1/.65 = 1.54

OC curve for the R chart

- Use l = s1/s0 ( the ratio of new to old process standard deviation )in Fig. 6.14
- R chart is insensitive when n is small
- But, when n is large, W = R/s
loses efficiency, and S chart is better

6-3.1 Construction and Operation of and S Charts

- S is an estimator of c4
where c4 is a constant depends on sample size n.

- The standard deviation of S is

6-3.1 Construction and Operation of and S Charts

- If a standard is given the control limits for the S chart are:
- B5, B6, and c4 are found in the Appendix VI for various values of n.

6-3.1 Construction and Operation of and S Charts

No Standard Given

- If is unknown, we can use an average sample standard deviation

6-3.1 Construction and Operation of and S Charts

Chart when Using S

The upper and lower control limits for the chart are given as

where A3 is found in the Appendix VI

6-3.2 The and S Control Charts with Variable Sample Size

- The and S charts can be adjusted to account for samples of various sizes.
- A “weighted” average is used in the calculations of the statistics.
m = the number of samples selected.

ni = size of the ith sample

6-3.2 The and S Control Charts with Variable Sample Size

- The grand average can be estimated as:
- The average sample standard deviation is:

6-3.2 The and S Control Charts with Variable Sample Size

- Control Limits

6-4. The Shewhart Control Chart for Individual Measurements

- What if you could not get a sample size greater than 1 (n =1)? Examples include
- Automated inspection and measurement technology is used, and every unit manufactured is analyzed.
- The production rate is very slow, and it is inconvenient to allow samples sizes of n > 1 to accumulate before analysis

- The X and MR charts are useful for samples of sizes n = 1.

6-4. The Shewhart Control Chart for Individual Measurements

Moving Range Control Chart

- The moving range (MR) is defined as the absolute difference between two successive observations:
MRi = |xi - xi-1|

which will indicate possible shifts or changes in the process from one observation to the next.

6-4. The Shewhart Control Chart for Individual Measurements

X and Moving Range Charts

- The X chart is the plot of the individual observations. The control limits are
where

6-4. The Shewhart Control Chart for Individual Measurements

X and Moving Range Charts

- The control limits on the moving range chart are:

6-4. The Shewhart Control Chart for Individual Measurements

Example

Ten successive heats of a steel alloy are tested for hardness. The resulting data are

Heat Hardness Heat Hardness

1 52 6 52

2 51 7 50

3 54 8 51

4 55 9 58

5 50 10 51

6-4. The Shewhart Control Chart for Individual Measurements

Example

6-4. The Shewhart Control Chart for Individual Measurements

Interpretation of the Charts

- MR charts cannot be interpreted the same as R or charts.
- Since the MR chart plots data that are “correlated” with one another, then looking for patterns on the chart does not make sense.
- MR chart cannot really supply useful information about process variability.

6-4. The Shewhart Control Chart for Individual Measurements

- The normality assumption is often taken for granted.
- When using the individuals chart, the normality assumption is very important to chart performance.

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