8. control charts for variables
DESCRIPTION
8. Control Charts for VariablesTRANSCRIPT
Control Charts for Variables
Variables due to Variation
• Concept of variation– No 2 things are alike
• Variation exists– Even if variation small and appears same, precision
instruments show differences• Ability to measure variation is necessary before can control
Control Charts for Variables
Variation Area
Basically 3 categories of variation in piece part production (e.g. Light bulbs, washer, nuts, etc.) Within piece - surface roughness Piece to piece - dimensions Time to time - different outcomes e.g. morning and
afternoon, tool wear, workers’ tiredness
Control Charts for VariablesSources of Variation• Equipment
- tool wear, electrical fluctuations for welding• Material
- tensile strength, moisture content (e.g. raw material) • Environment
- temperature, light, humidity etc.• Operator
- method, SOP followed, motivation level, training• Inspection
- inspector, inspection equipment, environment
Control Charts for VariablesX-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.
Control Charts for Variables
Regardless of the distribution of population, the distribution of sample means drawn from the population will tend to follow a normal curve
1. The mean of the sampling distribution (x-double bar) will be the same as the population mean m
x = m
s nsx =
2. The standard deviation of the sampling distribution (sx) will equal the population standard deviation (s) divided by the square root of the sample size, n
Central Limit Theorem
Control Charts for VariablesStatistical Basis• Usually µ and are unknown, so they are estimated from
preliminary samples and subgroups.• 20-25 samples are taken usually each of which contains 4-6
observations• If m samples are taken and n observations are made in each
sample then the best estimator of process mean µ is the grand average.
• Where are the average of each sample
Control Charts for VariablesStatistical Basis (contd.)
• If x1, x2, . . . , xn is the observations for sample size n, then the range of the sample is the difference between the largest and smallest observations; that is,
• Let R1, R2, . . . , Rm be the ranges of the m samples. The average range is
Control Charts for Variables
In case of 3-sigma R chart, the control limits can be written as follows
Where is the standard deviation of range values of m samples
In case of 3-sigma x chart, the control limits can be written as follows
Where is the standard deviation of range values of m samples
Control Charts for Variables
Control Charts for VariablesFactor for x-Chart
A2 D3 D4
2 1.88 0.00 3.27
3 1.02 0.00 2.57
4 0.73 0.00 2.28
5 0.58 0.00 2.11
6 0.48 0.00 2.00
7 0.42 0.08 1.92
8 0.37 0.14 1.86
9 0.34 0.18 1.82
10 0.31 0.22 1.78
11 0.29 0.26 1.74
12 0.27 0.28 1.72
13 0.25 0.31 1.69
14 0.24 0.33 1.67
15 0.22 0.35 1.65
Factors for R-ChartSample Size
(n)
Control Charts for Variables
Example
Control Charts for Variables
Example
1 2 3 4 5 6 7 8 9 10 1112 13 14 15 16 1718 19 20 21 22-0.02
-0.01
-1.73472347597681E-17
0.00999999999999998
0.02
0.03
0.04
0.05
0.06
0.07
RUCLCLLCL
R- chart
Control Charts for Variables
Example
X- chart
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 2210.7
10.705
10.71
10.715
10.72
10.725
10.73
10.735
10.74
10.745
UCLCLLCL
Control Charts for Variables
R-chart(R-chart does not detect change in mean)
UCL
LCL
x-chart(x-chart detects shift in central tendency)
UCL
LCL
(Sampling mean is shifting upward but range is consistent)
Control Charts for Variables
R-chart(R-chart detects increase in dispersion)
UCL
LCL
(Sampling mean is constant but dispersion is increasing)
x-chart(x-chart does not detect the increase in dispersion)
UCL
LCL