# Chapter 3 - Control chart for variables

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### Control Charts for Variables:

Control Charts for Variables EBB 341 Quality Control

### Variation:

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:

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:

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

### Slide 5:

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

### Slide 6:

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

### Control chart functions:

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

### Control charts identify variation:

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”

Control charts help us learn more about processes 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:

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:

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:

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 :

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. S 2 chart In this chart, the sample variances are plotted in order to control the variability of a variable.

### X-bar and R charts:

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:

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:

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

### Control Chart Examples:

Control Chart Examples Nominal UCL LCL Sample number Variations

### How to develop a control chart?:

How to develop a control chart?

### Define the problem:

Define the problem 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 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:

Collect the data Generally, collect 20-25 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:

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:

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:

UCL & LCL calculation

### Determining an alternative value for the standard deviation:

Determining an alternative value for the standard deviation

### Determine trial control limits - R chart:

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 D 3 and D 4 (appendix 2)

### Example: Control Charts for Variable Data:

Example: Control Charts for Variable Data Slip Ring Diameter (cm) Sample 1 2 3 4 5 X R 1 5.02 5.01 4.94 4.99 4.96 4.98 0.08 2 5.01 5.03 5.07 4.95 4.96 5.00 0.12 3 4.99 5.00 4.93 4.92 4.99 4.97 0.08 4 5.03 4.91 5.01 4.98 4.89 4.96 0.14 5 4.95 4.92 5.03 5.05 5.01 4.99 0.13 6 4.97 5.06 5.06 4.96 5.03 5.01 0.10 7 5.05 5.01 5.10 4.96 4.99 5.02 0.14 8 5.09 5.10 5.00 4.99 5.08 5.05 0.11 9 5.14 5.10 4.99 5.08 5.09 5.08 0.15 10 5.01 4.98 5.08 5.07 4.99 5.03 0.10 50.09 1.15

### Calculation:

Calculation From Table above: Sigma X-bar = 50.09 Sigma R = 1.15 m = 10 Thus; X-Double bar = 50.09/10 = 5.009 cm R-bar = 1.15/10 = 0.115 cm Note: The control limits are only preliminary with 10 samples. It is desirable to have at least 25 samples.

### Trial control limit:

Trial control limit UCL x-bar = X-D bar + A 2 R-bar = 5.009 + (0.577)(0.115) = 5.075 cm LCL x-bar = X-D bar - A 2 R-bar = 5.009 - (0.577)(0.115) = 4.943 cm UCL R = D 4 R-bar = (2.114)(0.115) = 0.243 cm LCL R = D 3 R-bar = (0)(0.115) = 0 cm For A 2 , D 3 , D 4 : see Table B, Appendix n = 5

### Slide 30:

3-Sigma Control Chart Factors Sample size X-chart R-chart n A 2 D 3 D 4 2 1.88 0 3.27 3 1.02 0 2.57 4 0.73 0 2.28 5 0.58 0 2.11 6 0.48 0 2.00 7 0.42 0.08 1.92 8 0.37 0.14 1.86

X-bar Chart

R Chart

Run Chart

### Slide 34:

Another Example of X-bar & R chart

### Slide 35:

Subgroup X 1 X 2 X 3 X 4 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 6.35 0.08 0.20 0.0876 0 2 6.46 6.37 6.36 6.41 6.4 6.47 6.41 6.35 0.1 0.20 0.0876 0 3 6.34 6.4 6.34 6.36 6.36 6.47 6.41 6.35 0.06 0.20 0.0876 0 4 6.69 6.64 6.68 6.59 6.65 6.47 6.41 6.35 0.1 0.20 0.0876 0 5 6.38 6.34 6.44 6.4 6.39 6.47 6.41 6.35 0.1 0.20 0.0876 0 6 6.42 6.41 6.43 6.34 6.4 6.47 6.41 6.35 0.09 0.20 0.0876 0 7 6.44 6.41 6.41 6.46 6.43 6.47 6.41 6.35 0.05 0.20 0.0876 0 8 6.33 6.41 6.38 6.36 6.37 6.47 6.41 6.35 0.08 0.20 0.0876 0 9 6.48 6.44 6.47 6.45 6.46 6.47 6.41 6.35 0.04 0.20 0.0876 0 10 6.47 6.43 6.36 6.42 6.42 6.47 6.41 6.35 0.11 0.20 0.0876 0 11 6.38 6.41 6.39 6.38 6.39 6.47 6.41 6.35 0.03 0.20 0.0876 0 12 6.37 6.37 6.41 6.37 6.38 6.47 6.41 6.35 0.04 0.20 0.0876 0 13 6.4 6.38 6.47 6.35 6.4 6.47 6.41 6.35 0.12 0.20 0.0876 0 14 6.38 6.39 6.45 6.42 6.41 6.47 6.41 6.35 0.07 0.20 0.0876 0 15 6.5 6.42 6.43 6.45 6.45 6.47 6.41 6.35 0.08 0.20 0.0876 0 16 6.33 6.35 6.29 6.39 6.34 6.47 6.41 6.35 0.1 0.20 0.0876 0 17 6.41 6.4 6.29 6.34 6.36 6.47 6.41 6.35 0.12 0.20 0.0876 0 18 6.38 6.44 6.28 6.58 6.42 6.47 6.41 6.35 0.3 0.20 0.0876 0 19 6.35 6.41 6.37 6.38 6.38 6.47 6.41 6.35 0.06 0.20 0.0876 0 20 6.56 6.55 6.45 6.48 6.51 6.47 6.41 6.35 0.11 0.20 0.0876 0 21 6.38 6.4 6.45 6.37 6.4 6.47 6.41 6.35 0.08 0.20 0.0876 0 22 6.39 6.42 6.35 6.4 6.39 6.47 6.41 6.35 0.07 0.20 0.0876 0 23 6.42 6.39 6.39 6.36 6.39 6.47 6.41 6.35 0.06 0.20 0.0876 0 24 6.43 6.36 6.35 6.38 6.38 6.47 6.41 6.35 0.08 0.20 0.0876 0 25 6.39 6.38 6.43 6.44 6.41 6.47 6.41 6.35 0.06 0.20 0.0876 0 Given Data (Table 5.2 )

### Calculation:

Calculation From Table 5.2: Sigma X-bar = 160.25 Sigma R = 2.19 m = 25 Thus; X-double bar = 160.25/29 = 6.41 mm R-bar = 2.19/25 = 0.0876 mm

### Trial control limit:

Trial control limit UCL x-bar = X-double bar + A 2 R-bar = 6.41 + (0.729)(0.0876) = 6.47 mm LCL x-bar = X-double bar - A 2 R-bar = 6.41 – (0.729)(0.0876) = 6.35 mm UCL R = D 4 R-bar = (2.282)(0.0876) = 0.20 mm LCL R = D 3 R-bar = (0)(0.0876) = 0 mm For A 2 , D 3 , D 4 : see Table B Appendix, n = 4.

X-bar Chart

R Chart

### Revised CL & Control Limits:

Revised CL & Control Limits Calculation based on discarding subgroup 4 & 20 (X-bar chart) and subgroup 18 for R chart: = (160.25 - 6.65 - 6.51)/(25-2) = 6.40 mm = (2.19 - 0.30)/25 - 1 = 0.079 = 0.08 mm

### New Control Limits:

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

### Slide 42:

From Table B: A = 1.500 for a subgroup size of 4, d 2 = 2.059, D 1 = 0, and D 2 = 4.698 Calculation results:

### Slide 43:

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

### Revise the charts:

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:

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:

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 1 s on both side CL. About 13.5% in an imaginary band between 1 s and 2 s on both side CL. About 2.5% of the plotted point in an imaginary band between 2 s and 3 s on both side CL.

### Slide 47:

The Normal Distribution -3  -2  -1  +1  +2  +3  Mean 68.26% 95.44% 99.74%  = Standard deviation LSL USL -3  +3  CL

### Slide 48:

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.

### Slide 49:

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 Normal Distribution Review

Common Causes

### Process Out of Control:

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.

### Slide 52:

Assignable Causes (a) Mean Grams Average

### Slide 53:

Assignable Causes (b) Spread Grams Average

### Slide 54:

Assignable Causes (c) Shape Grams Average

### Slide 55:

Control Charts UCL Nominal LCL Assignable causes likely 1 2 3 Samples

### Control Chart Examples:

Control Chart Examples Nominal UCL LCL Sample number Variations

### Control Limits and Errors:

Control Limits and Errors LCL Process average UCL (a) Three-sigma limits Type I error: Probability of searching for a cause when none exists

### Control Limits and Errors:

Control Limits and Errors Type I error: Probability of searching for a cause when none exists UCL LCL Process average (b) Two-sigma limits

### Control Limits and Errors:

Type II error: Probability of concluding that nothing has changed Control Limits and Errors UCL Shift in process average LCL Process average (a) Three-sigma limits

### Control Limits and Errors:

Type II error: Probability of concluding that nothing has changed Control Limits and Errors UCL Shift in process average LCL Process average (b) Two-sigma limits

### Achieve the purpose:

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:

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:

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:

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  +1  +2  +3  Mean 68.26% 95.44% 99.74%

### Chart zones:

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.