Control Charts: Sample Number
Control Charts: Sample Number
Control Charts: Sample Number
UCL
UWL
Statistics
Target
LWL
LCL
1 2 3 4 5 6 7 8
Sample Number
1. Introduction
Introduction
Stop
Investigate/Adjust
Go
Stop
Take action ( Investigate/Adjust )
To ensure
that the output of the process is-
Normal
Whether Output is Normal?
Both histogram and control chart can tell us whether the output
is normal? However,
Histogram views the process as history ,
as the entire output together.
Control chart views the process in real time,
at different time intervals as the process progresses.
Histogram a History of Process Output
16
14
Frequency
12
10
8
6
4
2
0
47 48 49 50 51 52 53 54
kg
Control Chart Views Process in Real Time
Output of the process in real time
Target
Mean
UCLx
Target
LCLx
UCLr
Range
Time Intervals
Why Control Chart?
It helps in finding
is there any change in location of process mean
in real time
Change in Location of Process Mean
43 44 45 46 47 48 49 50 51 52 53
Why Control Chart?
It helps in finding
Spread due
Larger spread due
to common causes
to special causes
43 44 45 46 47 48 49 50 51 52 53
Why Control Chart?
By maintaining control chart we avoid 100% inspection, and thus save cost
of verification.
Why Control Chart?
Step No. 1
Identify quality characteristics of product or process that affects
“fitness for use”.
Step No . 2
Step No. 3
Take samples at different intervals and plot statistics of the
sample measurements on control chart.
Step No. 4
Target
1 2 3 4 5
Sample Number
5. Types of control chart
Types of Control Chart
For tracking
Accuracy
Mean
control chart
Variable
Control
Chart
For tracking
Precision
Range
control chart
6. Concepts behind control charts
Understanding effect of shift of process
mean
Case When Process Mean is at Target
Target Process
L Mean
U
-3s +3 s U-L=6s
42 43 44 45 46 47 48 49 50 51 52 53
42 43 44 45 46 47 48 49 50 51 52 53
42 43 44 45 46 47 48 49 50 51 52 53
When there is no shift in the process nearly all the observations fall within -3
s and + 3 s.
When there is small shift in the mean of process some observations fall
outside original -3 s and +3 s zone.
Chances of an observation falling outside original -3 s and + 3 s zone
increases with the increase in the shift of process mean.
Our Conclusion from Normal Distribution
Since on the control charts for accuracy we plot and watch the
trend of the means and ranges of the samples, it is necessary
that we should understand the behaviour of
Suppose we have a lot of 1000 tablets, and let us say, weight of the tablets
follows a normal distribution having a standard deviation, s.
Let us take a sample of n tablets. Calculate mean of the sample and record
it. Continue this exercise of taking samples, calculating the mean of
samples and recording, 1000 times.
The mean of samples shall have normal distribution with standard deviation,
Sm = (s÷ n). Distribution of population and ‘means of sample’ shall have
same means.
Distribution - Population Vs Sample Means
Distribution of
means of samples
[standard deviation = (s÷ n)]
Distribution of population
(standard deviation = s
43 44 45 46 47 48 49 50 51 52 53
Quality Characteristics
Control and Warning Limits for Mean Control Chart
1 2 3 4 5 6 7
Sample Number
8. Establishing Control Charts
Establishing Control Chart
Step No.1
- Weight
- Length
- Viscosity
- Tensile Strength
- Capacitance
Establishing Control Chart
Step No.2
Step No. 3
If the shift in the process average causes more loss, then take
smaller samples more frequently.
If our lot size in a shift is say 3000, then in a shift we require 50 units. If the
sample size n, is say 4 then
Number of visits to the process is = 50÷4 = 12
Step No. 4
Collect data on a special control chart data
collection sheet. ( Minimum 100 observations)
The data collection sheet has following main portions:
1. General details for part, department etc.
2. Columns for date and time sample taken
3. Columns for measurements of sample
4. Column for mean of sample
5. Column for range of sample
Typical Data Collection Sheet
Measurement
SN Date Time
Mean Range
X1 X2 X3 X4
…..
25
Establishing Control Chart
Step No. 5
Fill up the control chart data sheet
1 47 45 48 52 51
2 48 52 47 50 50
3 49 48 52 50 49
4 49 50 52 50 49
5 51 50 53 50 48
6 50 50 49 51 47
7 51 48 50 50 54
8 50 48 50 50 52
9 48 48 49 50 51
10 49 50 50 52 51
Example - Calculation of Subgroup No.1
1 47 45 48 52 51 48.6 7
2 48 52 47 50 50 49.4 5
3 49 48 52 50 49 49.6 4
4 49 50 52 50 49 50.0 3
5 51 50 53 50 48 50.4 5
6 50 50 49 51 47 49.4 4
7 51 48 50 50 54 50.6 6
8 50 48 50 50 52 50.0 4
9 48 48 49 50 51 49.2 3
10 49 50 50 52 51 50.2 3
Establishing Control Chart
Step No. 6
In our case
(7 + 5 +4 3 + 5 + 4 + 6 + 4 + 3 + 3 )
R=
Total number of subgroups
Establishing Control Chart
Step No. 7
Using following table of constants find trial control limit for mean and
range control chart’
Sub Group
A2 D4 D3
Size
2 1.880 3.267 0
3 1.023 2.527 0
4 0.729 2.282 0
5 0.577 2.115 0
6 0.483 2.004 0
7 0.419 1.924 0.076
Establishing Control Chart
Step No. 8
Calculate Trial control Limits with target value, T
Size of Subgroup, n = 5
Target value, T = 50
Step No. 8
Step No. 9
Discard the outliers
Outliers are those observations which do not belong to normal
population. If Outliers are included in the calculation, then the
information is distorted.
Checking for Outliers
In our case
- None of the subgroup mean is more than 52.5
- None of the subgroup mean is less than 47.5
- None of the range is more than 9.3
- None of the range is less than 0
Step No. 10
Compute warning limits for mean control chart
2 x A2 x R
Upper warning limit, UWLx = T +
3
2 x A2 x R
Lower warning limit, LWLx = T -
3
Calculation of Control Limits for Mean Control Chart
2 x 0.577 x 4.4
Uwlx = 50 +
3
= 51.7
2 x 0.577 x 4.4
Lwlx = 50 -
3
= 48.3
Action and Warning Limits for Mean Control chart
UCLx
UWLx
Target
Mean
LWLx
LCLx
1 2 3 4 5 6 7
Sample Number
Action and Warning Limits for Mean Control Chart for Example
UCLx=52.5
UWLx=51.7
Target=50
Mean
LWLx=48.3
LCLx= 47.5
1 2 3 4 5 6 7
Sample Number
Constants for Range Control chart
Sample
D4 D3 DWLR DWUR
size, n
2 3.27 0 0.04 2.81
Step No. 11
Compute warning limits for range control chart
Upper Warning Limit, UWLr = DWUR x R
Target
Mean
LWLx
LCLx
UCLr
UWLr
Range
R
LWLr
1 2 3 4 5 6 7
Sample Number
Calculation of Warning Limits for Range Control Chart
In our case
Target = 50
LWLx = 48.3
LCLx = 47.5
UCLr = 9.3
UWLr = 8
Range
R = 4.4
LWLr = 1.6
1 2 3 4 5 6 7
Sample Number
Flow Chart for Establishing Control Chart
Start
Record observations
UCLx = T + A2 x R
LCLx = T - A2 x R
UCLr = D4 x R
LCLr = D3 x R
Is any Yes
sub-group mean or range Drop that
out side the control Group
limit ?
No
Flow Chart for Control Chart
Stop
10. Interpreting control charts
Interpreting Control Chart
Zone - 1 If the plotted point falls in this zone, do not make any
adjustment, continue with the process.
Zone - 2 If the plotted point falls in this zone then special cause may be
present. Be careful watch for plotting of another sample(s).
Zone - 3 If the plotted point falls in this zone then special cause has crept
into the system, and corrective action is required.
Zones for Mean Control Chart
Zone - 3 Action
UCL
Zone - 2 Warning
UWL
Zone - 1 Continue
Sample Mean
Target
Zone - 1 Continue
Zone - 2 Warning
LWL
LCL
Zone - 3 Action
1 2 3 4 5 6 7
Sample Number
Interpreting Control Chart
Because the basis for control chart theory follows the normal distribution,
the same rules that governs the normal distribution are used to interpret
the control charts. These rules include:
- Randomness.
- Symmetry about the centre of the distribution.
- 99.73% of the population lies between - 3 s of and + 3 s the centre line.
- 95.4% population lies between -2 s and + 2 s of the centre line.
Interpreting Control Chart
UCL
UWL
Statistics
Target
LWL
LCL
1 2 3 4 5 6 7 8
Sample Number
Interpreting Control Chart
Seven consecutive points are falling on one side of the centre line.
This pattern indicates a shift in the process output from changes in the
equipment, methods, or material or shift in the measurement system.
Interpreting Control Chart
Seven consecutive points on one
side of the centre line
UCL
UWL
Statistics
Target
LWL
LCL
1 2 3 4 5 6 7 8
Sample Number
Interpreting Control Chart
UCL
UWL
Statistics
Target
LWL
LCL
1 2 3 4 5 6 7 8
Sample Number
Interpreting Control Chart
UCL
UWL
Statistics
Target
LWL
LCL
1 2 3 4 5 6 7 8
Sample Number
Interpreting Control Chart
A trend of seven points in a row upward or downward
demonstrates nonrandomness.
Target
LWL
LCL
1 2 3 4 5 6 7 8
Sample Number
Interpreting Control Chart
Seven consecutive points having
downward trend
UCL
UWL
Statistics
Target
LWL
LCL
1 2 3 4 5 6 7 8
Sample Number