# Control Charts for Variables

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Control Charts for Variables
EBB 341 Quality Control

Variation There is no two natural items in any category are the same.
Variation may be quite large or very small. If variation very small, it may appear that items are identical, but precision instruments will show differences.

3 Categories of variation
Within-piece variation One portion of surface is rougher than another portion. Apiece-to-piece variation Variation among pieces produced at the same time. Time-to-time variation Service given early would be different from that given later in the day.

Source of variation Equipment Material Environment Operator
Tool wear, machine vibration, … Material Raw material quality Environment Temperature, pressure, humadity Operator Operator performs- physical & emotional

Control Chart Viewpoint
Variation due to Common or chance causes Assignable causes Control chart may be used to discover “assignable causes”

Some Terms Run chart - without any upper/lower limits
Specification/tolerance limits - not statistical Control limits - statistical

Control chart functions
Control charts are powerful aids to understanding the performance of a process over time. Input PROCESS Output What’s causing variability?

Control charts identify variation
Chance causes - “common cause” inherent to the process or random and not controllable if only common cause present, the process is considered stable or “in control” Assignable causes - “special cause” variation due to outside influences if present, the process is “out of control”

Separate common and special causes of variation Determine whether a process is in a state of statistical control or out-of-control Estimate the process parameters (mean, variation) and assess the performance of a process or its capability

Control charts to monitor processes
To monitor output, we use a control chart we check things like the mean, range, standard deviation To monitor a process, we typically use two control charts mean (or some other central tendency measure) variation (typically using range or standard deviation)

Types of Data Variable data
Product characteristic that can be measured Length, size, weight, height, time, velocity Attribute data Product characteristic evaluated with a discrete choice Good/bad, yes/no

Control chart for variables
Variables are the measurable characteristics of a product or service. Measurement data is taken and arrayed on charts.

Control charts for variables
X-bar chart In this chart the sample means are plotted in order to control the mean value of a variable (e.g., size of piston rings, strength of materials, etc.). R chart In this chart, the sample ranges are plotted in order to control the variability of a variable. S chart In this chart, the sample standard deviations are plotted in order to control the variability of a variable. S2 chart In this chart, the sample variances are plotted in order to control the variability of a variable.

X-bar and R charts The X- bar chart is developed from the average of each subgroup data. used to detect changes in the mean between subgroups. The R- chart is developed from the ranges of each subgroup data used to detect changes in variation within subgroups

Control chart components
Centerline shows where the process average is centered or the central tendency of the data Upper control limit (UCL) and Lower control limit (LCL) describes the process spread

The Control Chart Method
X bar Control Chart: UCL = XDmean + A2 x Rmean LCL = XDmean - A2 x Rmean CL = XDmean  R Control Chart: UCL = D4 x Rmean LCL = D3 x Rmean CL = Rmean  Capability Study: PCR = (USL - LSL)/(6s); where s = Rmean /d2

Control Chart Examples
4/11/2017 Control Chart Examples UCL Nominal Variations LCL And finally, this chart shows a process with two points actually outside the control limits, an easy indicator to detect but not the only one. These rules, there are a few more, are commonly referred to as the Western Electric Rules and can be found in any advanced quality reference. Sample number Statistics 30

How to develop a control chart?

Define the problem Select a quality characteristic to be measured
Use other quality tools to help determine the general problem that’s occurring and the process that’s suspected of causing it. Select a quality characteristic to be measured Identify a characteristic to study - for example, part length or any other variable affecting performance.

Choose a subgroup size to be sampled
Choose homogeneous subgroups Homogeneous subgroups are produced under the same conditions, by the same machine, the same operator, the same mold, at approximately the same time. Try to maximize chance to detect differences between subgroups, while minimizing chance for difference with a group.

Collect the data Generally, collect subgroups (100 total samples) before calculating the control limits. Each time a subgroup of sample size n is taken, an average is calculated for the subgroup and plotted on the control chart.

Determine trial centerline
The centerline should be the population mean,  Since it is unknown, we use X Double bar, or the grand average of the subgroup averages.

Determine trial control limits - Xbar chart
The normal curve displays the distribution of the sample averages. A control chart is a time-dependent pictorial representation of a normal curve. Processes that are considered under control will have 99.73% of their graphed averages fall within 6.

UCL & LCL calculation

Determining an alternative value for the standard deviation

Determine trial control limits - R chart
The range chart shows the spread or dispersion of the individual samples within the subgroup. If the product shows a wide spread, then the individuals within the subgroup are not similar to each other. Equal averages can be deceiving. Calculated similar to x-bar charts; Use D3 and D4 (appendix 2)

Example: Control Charts for Variable Data
4/11/2017 Example: Control Charts for Variable Data Slip Ring Diameter (cm) Sample X R Example4.3 Statistics

Calculation From Table above: Sigma X-bar = 50.09 Sigma R = 1.15
Thus; X-Double bar = 50.09/10 = cm R-bar = 1.15/10 = cm Note: The control limits are only preliminary with 10 samples. It is desirable to have at least 25 samples.

Trial control limit UCLx-bar = X-D bar + A2 R-bar = (0.577)(0.115) = cm LCLx-bar = X-D bar - A2 R-bar = (0.577)(0.115) = cm UCLR = D4R-bar = (2.114)(0.115) = cm LCLR = D3R-bar = (0)(0.115) = 0 cm For A2, D3, D4: see Table B, Appendix n = 5

3-Sigma Control Chart Factors
Sample size X-chart R-chart n A2 D3 D4

X-bar Chart

R Chart

Run Chart

Another Example of X-bar & R chart

Given Data (Table 5.2) Subgroup X1 X2 X3 X4 X-bar UCL-X-bar X-Dbar
LCL-X-bar R UCL-R R-bar LCL-R 1 6.35 6.4 6.32 6.37 6.36 6.47 6.41 0.08 0.20 0.0876 2 6.46 0.1 3 6.34 0.06 4 6.69 6.64 6.68 6.59 6.65 5 6.38 6.44 6.39 6 6.42 6.43 0.09 7 0.05 8 6.33 9 6.48 6.45 0.04 10 0.11 11 0.03 12 13 0.12 14 0.07 15 6.5 16 6.29 17 18 6.28 6.58 0.3 19 20 6.56 6.55 6.51 21 22 23 24 25

Calculation From Table 5.2: Sigma X-bar = 160.25 Sigma R = 2.19 m = 25
Thus; X-double bar = /29 = 6.41 mm R-bar = 2.19/25 = mm

Trial control limit UCLx-bar = X-double bar + A2R-bar = (0.729)(0.0876) = 6.47 mm LCLx-bar = X-double bar - A2R-bar = 6.41 – (0.729)(0.0876) = 6.35 mm UCLR = D4R-bar = (2.282)(0.0876) = 0.20 mm LCLR = D3R-bar = (0)(0.0876) = 0 mm For A2, D3, D4: see Table B Appendix, n = 4.

X-bar Chart

R Chart

Revised CL & Control Limits
Calculation based on discarding subgroup 4 & 20 (X-bar chart) and subgroup 18 for R chart: = ( )/(25-2) = 6.40 mm = ( )/25 - 1 = = 0.08 mm

New Control Limits New value:
Using standard value, CL & 3 control limit obtained using formula:

From Table B: A = for a subgroup size of 4, d2 = 2.059, D1 = 0, and D2 = 4.698 Calculation results:

Trial Control Limits & Revised Control Limit
Revised control limits UCL = 6.46 CL = 6.40 LCL = 6.34 UCL = 0.18 CL = 0.08 LCL = 0

Revise the charts In certain cases, control limits are revised because: out-of-control points were included in the calculation of the control limits. the process is in-control but the within subgroup variation significantly improves.

Revising the charts Interpret the original charts Isolate the causes
Take corrective action Revise the chart Only remove points for which you can determine an assignable cause

Process in Control When a process is in control, there occurs a natural pattern of variation. Natural pattern has: About 34% of the plotted point in an imaginary band between 1s on both side CL. About 13.5% in an imaginary band between 1s and 2s on both side CL. About 2.5% of the plotted point in an imaginary band between 2s and 3s on both side CL.

The Normal Distribution -3 -2 -1 +1 +2 +3 Mean 68.26% 95.44%
4/11/2017 -3 -2 -1  +2 +3 Mean 68.26% 95.44% 99.74% The Normal Distribution  = Standard deviation LSL USL -3 +3 CL And at +/- 3 sigma, the most common choice of confidence/control limits in quality control application, the area is 99.97%. Statistics 20

34.13% of data lie between  and 1 above the mean ().
34.13% between  and 1 below the mean. Approximately two-thirds (68.28 %) within 1 of the mean. 13.59% of the data lie between one and two standard deviations Finally, almost all of the data (99.74%) are within 3 of the mean.

Normal Distribution Review
Define the 3-sigma limits for sample means as follows: What is the probability that the sample means will lie outside 3-sigma limits? Note that the 3-sigma limits for sample means are different from natural tolerances which are at

4/11/2017 Common Causes The first set of slides presents the various aspects of common and assignable causes. This slide and the two following show a normal process distribution and how it allows for expected variability which is termed common causes. Statistics 2

Process Out of Control The term out of control is a change in the process due to an assignable cause. When a point (subgroup value) falls outside its control limits, the process is out of control.

Assignable Causes (a) Mean Average Grams 4/11/2017
The new distribution will look as shown in this slide. Grams Statistics 7

Assignable Causes (b) Spread Average Grams 4/11/2017
The new distribution has a much greater spread (higher standard deviation). Grams Statistics 9

Assignable Causes (c) Shape Average Grams 4/11/2017
A skewed (non-normal) distribution will result in a different pattern of variability. Grams Statistics 11

Control Charts Assignable causes likely UCL Nominal LCL 1 2 3 Samples
4/11/2017 Control Charts Assignable causes likely UCL Nominal However, the third sample plots outside the original distribution, indicating the likely presence of an assignable cause. LCL Samples Statistics 24

Control Chart Examples
4/11/2017 Control Chart Examples UCL Nominal Variations LCL And finally, this chart shows a process with two points actually outside the control limits, an easy indicator to detect but not the only one. These rules, there are a few more, are commonly referred to as the Western Electric Rules and can be found in any advanced quality reference. Sample number Statistics 30

Control Limits and Errors
4/11/2017 Control Limits and Errors Type I error: Probability of searching for a cause when none exists (a) Three-sigma limits UCL Process average As long as the area encompassed by the control limits is less than 100% of the area under the distribution, there will be a probability of a Type I error. A Type I error occurs when it is concluded that a process is out of control when in fact pure randomness is present. Given +/- 3 sigma, the probability of a Type I error is = , a very small probability. LCL Statistics 32

Control Limits and Errors
4/11/2017 Control Limits and Errors Type I error: Probability of searching for a cause when none exists (b) Two-sigma limits UCL Process average When the control limits are changed to +/- 2 sigma, the probability of a Type I error goes up considerably, from to = While this is still a small probability, it is a change that should be carefully considered. In practice, this would mean more samples would be identified inappropriately as out-of-control. Even if they were subsequently ‘Okd’, there would be increased costs due to many more cycles through the four step improvement process shown previously. LCL Statistics 33

Control Limits and Errors
4/11/2017 Control Limits and Errors Type II error: Probability of concluding that nothing has changed (a) Three-sigma limits UCL Shift in process average Process average Returning to the 3 sigma limits, we can see the probability of making a Type II error, in this case failing to detect a shift in the process mean. LCL Statistics 34

Control Limits and Errors
4/11/2017 Control Limits and Errors Type II error: Probability of concluding that nothing has changed (b) Two-sigma limits UCL Shift in process average Process average By reducing the control limits to +/- 2 sigma, we see the probability of failing to detect the shift has been reduced. LCL Statistics 35

Achieve the purpose Our goal is to decrease the variation inherent in a process over time. As we improve the process, the spread of the data will continue to decrease. Quality improves!!

Improvement

Examine the process A process is considered to be stable and in a state of control, or under control, when the performance of the process falls within the statistically calculated control limits and exhibits only chance, or common causes.

Consequences of misinterpreting the process
Blaming people for problems that they cannot control Spending time and money looking for problems that do not exist Spending time and money on unnecessary process adjustments Taking action where no action is warranted Asking for worker-related improvements when process improvements are needed first

Process variation When a system is subject to only chance causes of variation, 99.74% of the measurements will fall within 6 standard deviations If 1000 subgroups are measured, 997 will fall within the six sigma limits. -3 -2 -1  +2 +3 Mean 68.26% 95.44% 99.74%

Chart zones Based on our knowledge of the normal curve, a control chart exhibits a state of control when: Two thirds of all points are near the center value. The points appear to float back and forth across the centerline. The points are balanced on both sides of the centerline. No points beyond the control limits. No patterns or trends.