statistical process control
DESCRIPTION
Statistical Process Control. Overview. Variation Control charts R charts X-bar charts P charts. Statistical Quality Control (SPC). Measures performance of a process Primary tool - statistics Involves collecting, organizing, & interpreting data Used to: - PowerPoint PPT PresentationTRANSCRIPT
Statistical Process Control
Overview
• Variation
• Control charts
–R charts
–X-bar charts
–P charts
• Measures performance of a process• Primary tool - statistics• Involves collecting, organizing, &
interpreting data • Used to:
–Control the process as products are produced
–Inspect samples of finished products
Statistical Quality Control (SPC)
Bottling Company
• Machine automatically fills a 20 oz bottle.• Problem with filling too much? Problems with
filling to little?• So Monday the average is 20.2 ounces.• Tuesday the average is 19.6 ounces.• Is this normal? Do we need to be concerned?• Wed is 19.4 ounces.
Natural Variation• Machine can not fill every
bottle exactly the same amount – close but not exactly.
Natural variation
19.820.020.220.420.620.821.021.2
1 2 3 4 5
Bottle
Oun
ces
Assignable variation
• A cause for part of the variation
Assignable variation
19.820.020.220.420.620.821.021.2
1 2 3 4 5
Bottle
Oun
ces
SPC
• Objective: provide statistical signal when
assignable causes of variation are present
ControlCharts
RChart
VariablesCharts
AttributesCharts
XChart
PChart
CChart
Continuous Numerical Data
Categorical or Discrete Numerical Data
Control Chart Types
• Characteristics for which you focus on defects
• Classify products as either ‘good’ or ‘bad’, or count # defects– e.g., radio works or not
• Categorical or discrete random variables
Attributes
Measuring quality
Characteristics that you measure, e.g., weight, length
May be in whole or in fractional numbers
Continuous random variables
Variables
• Show changes in data pattern– e.g., trends
• Make corrections before process is out of control
• Show causes of changes in data– Assignable causes
• Data outside control limits or trend in data– Natural causes
• Random variations around average
Control Chart Purposes
Figure S6.7
Steps to Follow When Using Control Charts
TO SET CONTROL CHART LIMITS
1. Collect 20-25 samples of n=4 or n=5 a stable
process
compute the mean of each sample.
2. Calculate control limits
Compute the overall means
Calculate the upper and lower control limits.
Steps to Follow When Using Control Charts - continued
TO MONITOR PROCESS USING THE CONTROL CHARTS:1. Collect and graph data
Graph the sample means and ranges on their respective control charts
Determine whether they fall outside the acceptable limits.
2. Investigate points or patterns that indicate the process is out of control. Assign causes for the variations.
3. Collect additional samples and revalidate the control limits.
Control Charts for Variables
Glacier Bottling• Manage at Glacier Bottling is concerned about their
filling process. In particular, they want to know whether or not the machines are really filling the bottles with 16 ounces.
• Create an Xbar chart that will be used to monitor the process.
• Collected data for 25 days. Each day, pulled 4 bottles from the filling line and measured the amount in the bottle.
Bottle Volume in OuncesSample Num Obs 1 Obs 2 Obs 3 Obs 4
1 15.85 16.02 15.83 15.932 16.12 16.00 15.85 16.013 16.00 15.91 15.94 15.834 16.20 15.85 15.74 15.935 15.74 15.86 16.21 16.106 15.94 16.01 16.14 16.037 15.75 16.21 16.01 15.868 15.82 15.94 16.02 15.949 16.04 15.98 15.83 15.9810 15.64 15.86 15.94 15.8911 16.11 16.00 16.01 15.8212 15.72 15.85 16.12 16.1513 15.85 15.76 15.74 15.9814 15.73 15.84 15.96 16.1015 16.20 16.01 16.10 15.8916 16.12 16.08 15.83 15.9417 16.01 15.93 15.81 15.6818 15.78 16.04 16.11 16.1219 15.84 15.92 16.05 16.1220 15.92 16.09 16.12 15.9321 16.11 16.02 16.00 15.8822 15.98 15.82 15.89 15.8923 16.05 15.73 15.73 15.9324 16.01 16.01 15.89 15.8625 16.08 15.78 15.92 15.98
Glacier Bottling
Bottle Volume in Ounces
Sample Num Obs 1 Obs 2 Obs 3 Obs 4
1 15.85 16.02 15.83 15.93
2 16.12 16.00 15.85 16.01
3 16.00 15.91 15.94 15.83
4 16.20 15.85 15.74 15.93
5 15.74 15.86 16.21 16.10
Remember: There are 25 samples of size 4 to calculate the control limits. We are doing the first 5 right now…
Bottle Volume in Ounces
Sample Num Obs 1 Obs 2 Obs 3 Obs 4 R
1 15.85 16.02 15.83 15.93
2 16.12 16.00 15.85 16.01
3 16.00 15.91 15.94 15.83
4 16.20 15.85 15.74 15.93
5 15.74 15.86 16.21 16.10
Glacier Bottling
16.02 – 15.83 = 0.19
Bottle Volume in OuncesSample Num Obs 1 Obs 2 Obs 3 Obs 4 R
1 15.85 16.02 15.83 15.93 0.19
2 16.12 16.00 15.85 16.01 0.27
3 16.00 15.91 15.94 15.83 0.17
4 16.20 15.85 15.74 15.93 0.46
5 15.74 15.86 16.21 16.10 0.47
6 15.94 16.01 16.14 16.03 0.20
7 15.75 16.21 16.01 15.86 0.46
8 15.82 15.94 16.02 15.94 0.20
9 16.04 15.98 15.83 15.98 0.21
10 15.64 15.86 15.94 15.89 0.30
11 16.11 16.00 16.01 15.82 0.29
12 15.72 15.85 16.12 16.15 0.43
13 15.85 15.76 15.74 15.98 0.24
14 15.73 15.84 15.96 16.10 0.37
15 16.20 16.01 16.10 15.89 0.31
16 16.12 16.08 15.83 15.94 0.29
17 16.01 15.93 15.81 15.68 0.33
18 15.78 16.04 16.11 16.12 0.34
19 15.84 15.92 16.05 16.12 0.28
20 15.92 16.09 16.12 15.93 0.20
21 16.11 16.02 16.00 15.88 0.23
22 15.98 15.82 15.89 15.89 0.16
23 16.05 15.73 15.73 15.93 0.32
24 16.01 16.01 15.89 15.86 0.15
25 16.08 15.78 15.92 15.98 0.30
Rbar = 0.29 ounce
Glacier Bottling
R-Charts
UCLR = D4RLCLR = D3R
R = 0.29
Control Chart FactorsFactor for UCL Factor for Factor
Size of and LCL for LCL for UCL forSample x-Charts R-Charts R-Charts
(n) (A2) (D3) (D4)
2 1.880 0 3.2673 1.023 0 2.5754 0.729 0 2.2825 0.577 0 2.1156 0.483 0 2.0047 0.419 0.076 1.924
This chart is in your text and will be provided for exams if needed.
Glacier Bottling
R-Charts
UCLR = D4RLCLR = D3R
R = 0.29 D4 = 2.282
D3 = 0
UCLR = 2.282 (0.29) = 0.654 ounce
LCLR = 0(0.29) = 0 ounce
Glacier Bottling
R-Charts
UCLR = D4RLCLR = D3R
R = 0.29 D4 = 2.282
D3 = 0
UCLR = 2.282 (0.29) = 0.654 ounceLCLR = 0(0.29) = 0 ounce
Glacier Bottling
1 3 5 7 9 11 13 15 17 19 21 23 250.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
RUCLRLCLRRbar
Figure S6.7
Bottle Volume in OuncesSample Num Obs 1 Obs 2 Obs 3 Obs 4
1 15.85 16.02 15.83 15.932 16.12 16.00 15.85 16.013 16.00 15.91 15.94 15.834 16.20 15.85 15.74 15.935 15.74 15.86 16.21 16.106 15.94 16.01 16.14 16.037 15.75 16.21 16.01 15.868 15.82 15.94 16.02 15.949 16.04 15.98 15.83 15.9810 15.64 15.86 15.94 15.8911 16.11 16.00 16.01 15.8212 15.72 15.85 16.12 16.1513 15.85 15.76 15.74 15.9814 15.73 15.84 15.96 16.1015 16.20 16.01 16.10 15.8916 16.12 16.08 15.83 15.9417 16.01 15.93 15.81 15.6818 15.78 16.04 16.11 16.1219 15.84 15.92 16.05 16.1220 15.92 16.09 16.12 15.9321 16.11 16.02 16.00 15.8822 15.98 15.82 15.89 15.8923 16.05 15.73 15.73 15.9324 16.01 16.01 15.89 15.8625 16.08 15.78 15.92 15.98
Bottle Volume in Ounces
Sample Num Obs 1 Obs 2 Obs 3 Obs 4 R Xbar
1 15.85 16.02 15.83 15.93 0.19
2 16.12 16.00 15.85 16.01 0.27
3 16.00 15.91 15.94 15.83 0.17
4 16.20 15.85 15.74 15.93 0.46
5 15.74 15.86 16.21 16.10 0.47
Glacier Bottling
(15.85+16.02+15.83+15.93)/4 = 15.908
Bottle Volume in Ounces
Sample Num Obs 1 Obs 2 Obs 3 Obs 4 R Xbar
1 15.85 16.02 15.83 15.93 0.19 15.908
2 16.12 16.00 15.85 16.01 0.27
3 16.00 15.91 15.94 15.83 0.17
4 16.20 15.85 15.74 15.93 0.46
5 15.74 15.86 16.21 16.10 0.47
Glacier Bottling
(16.12+16.00+15.85+16.01)/4 = 15.995
Bottle Volume in Ounces
Sample Num Obs 1 Obs 2 Obs 3 Obs 4 R Xbar
1 15.85 16.02 15.83 15.93 0.19 15.908
2 16.12 16.00 15.85 16.01 0.27 15.995
3 16.00 15.91 15.94 15.83 0.17 15.920
4 16.20 15.85 15.74 15.93 0.46 15.930
5 15.74 15.86 16.21 16.10 0.47 15.978
6 15.94 16.01 16.14 16.03 0.20 16.030
7 15.75 16.21 16.01 15.86 0.46 15.958
8 15.82 15.94 16.02 15.94 0.20 15.930
9 16.04 15.98 15.83 15.98 0.21 15.958
10 15.64 15.86 15.94 15.89 0.30 15.833
11 16.11 16.00 16.01 15.82 0.29 15.985
12 15.72 15.85 16.12 16.15 0.43 15.960
13 15.85 15.76 15.74 15.98 0.24 15.833
14 15.73 15.84 15.96 16.10 0.37 15.908
15 16.20 16.01 16.10 15.89 0.31 16.050
16 16.12 16.08 15.83 15.94 0.29 15.993
17 16.01 15.93 15.81 15.68 0.33 15.858
18 15.78 16.04 16.11 16.12 0.34 16.013
19 15.84 15.92 16.05 16.12 0.28 15.983
20 15.92 16.09 16.12 15.93 0.20 16.015
21 16.11 16.02 16.00 15.88 0.23 16.003
22 15.98 15.82 15.89 15.89 0.16 15.895
23 16.05 15.73 15.73 15.93 0.32 15.860
24 16.01 16.01 15.89 15.86 0.15 15.943
25 16.08 15.78 15.92 15.98 0.30 15.940
RBar = 0.29 ounce
XBarBar = 15.9469 ounces
• Shows % of nonconforming items
• Attributes control chart
– Nominally scaled categorical data
• e.g., good-bad
p Chart
p Chart Control Limits
# Defective Items in Sample i
Size of sample i
z = 2 for 95.5% limits;
z = 3 for 99.7% limits
s
ii
p
p
n
nppzpLCL
nppzpUCL
1
i
s
1ix
p
)1(
)1(
HOMETOWN BANK
Hometown Bank
The operations manager of the booking services department of Hometown Bank is concerned about the number of wrong customer account numbers recorded by Hometown personnel. Each week a random sample of 2,500 deposits is taken, and the number of incorrect account numbers is recorded. The records for the past 12 weeks are shown in the following table. Is the process out of control? Use 3-sigma control limits.
Hometown Bank
UCLp = p + zp
LCLp = p - zp
p = p(1 - p)/n
Sample WrongNumber Account
Number 1 15 2 12 3 19 4 2 5 19 6 4 7 24 8 7 9 1010 1711 1512 3
Total 147
Total defectivesTotal observationsp =
n =
Control Charts for Attributes
Control Chartsfor Attributes
Hometown Bank
UCLp = p + zp
LCLp = p – zp
p = p(1 – p)/n
n = 2500 p = 0.0049
Control Chartsfor Attributes
Hometown Bank
UCLp = p + zp
LCLp = p – zp
p = 0.0049(1 – 0.0049)/2500
n = 2500 p = 0.0049
Control Chartsfor Attributes
Hometown Bank
UCLp = p + zp
LCLp = p – zp
p = 0.0014
n = 2500 p = 0.0049
Control Chartsfor Attributes
Hometown Bank
p = 0.0014
n = 2500 p = 0.0049
UCLp = 0.0049 + 3(0.0014)LCLp = 0.0049 – 3(0.0014)
Control Chartsfor Attributes
Hometown Bank
p = 0.0014
n = 2500 p = 0.0049
UCLp = 0.0049 + 3(0.0014)LCLp = 0.0049 – 3(0.0014)
Why 3?3-sigma limits
Also to within 99.7%
UCLp = 0.0091LCLp = 0.0007
Control Chartsfor Attributes
Hometown Bank
p = 0.0014
n = 2500 p = 0.0049
p-ChartWrong Account Numbers
Figure S6.7
Which control chart is appropriate?
• Webster Chemical Company produces mastics and caulking for the construction industry. The product is blended in large mixers and then pumped into tubes and capped.
• Webster is concerned whether the filling process for tubes of caulking is in statistical control. The process should be centered on 8 ounces per tube. Several samples of eight tubes are taken and each tube is weighed in ounces.
Which control chart is appropriate?
• A sticky scale brings Webster’s attention to whether caulking tubes are being properly capped. If a significant proportion of the tubes aren’t being sealed, Webster is placing their customers in a messy situation. Tubes are packaged in large boxes of 144. Several boxes are inspected. The number of leaking tubes in each box is recorded.