presentation on statistical process control (SPC) by SSQCE

1. STATISTICAL PROCESS CONTROL
(SPC)
A presentation by
THE SOCIETY
OF
STATISTICAL QUALITY CONTROL ENGINEERS
BHOPAL

2. SPC is prevention oriented and data
based quality control which focuses on
detection and elimination of any abnormality
(assignable cause) in the process and thus
resulting in consistent quality.

3. Statistical Process Control requires
knowledge of :
1) Basic statistics
2) Fundamental law of variation
3) Histogram
4) Process capability
5) Control charts and run chart

4. BASIC STATISTICS

5. WHAT IS STATISTICS?
STATISTICS is science of DATA
IT INVOLVES - Collection }
- Analysis } of DATA
- Interpretation }
DATA - Collection of numerical values

6. DATA TYPES
Attribute Data : which can be counted.
Example: Number of defectives
in a lot, Number of
defects per piece etc.
Variable data : which can be measured and can
take on any value within a given
range.
Example: Length, weight, Age,
time, temperature etc.

7. DATA
VARIATION
CENTRAL
VALUE

8. MEASURES OF CENTRAL TENDENCY
(CENTRAL VALUE)
• MEAN (AVERAGE)
• MEDIAN
• MODE

9. MEASURES OF DISPERSION (VARIATION)
• RANGE
• STANDARD DEVIATION
• VARIANCE

10. EXAMPLE -1
The table shows weights of 10 samples taken
from a bulk lot of a chemical (volume of each
sample is constant).
FIND OUT MEASURES OF CENTRAL TENDENCY
AND MEASURES OF DISPERSION

11. DATA TABLE -1
SAMPLE
NUMBER
WEIGHT IN GRAMMES
(1) 6
(2) 2
(3) 6
(4) 3
(5) 6
(6) 9
(7) 3
(8) 4
(9) 5
(10) 6

12. DETERMINING AVERAGE
AVERAGE = ∑x/n
= 6+2+6+3+6+9+3+4+5+6
10
= 50/10
= 5 gms

13. DETERMINING MEDIAN
After arranging the values in ascending order, we have:
2, 3, 3, 4, 5, 6, 6, 6, 6, 9 – (EVEN “n”)
Median = Middle value if “n” is ODD
= Average of the two middle values if “n” is EVEN
(n = No. of observations)
MEDIAN = (5+6)/2= 5.5 gms

14. DETERMINING MODE
MODE = Observation occurring most
frequently
= 6

15. DETERMINING RANGE
Range = Maximum weight – Minimum weight
= 9 – 2
= 7 gms

16. DETERMINING STANDARD DEVIATION AND VARIANCE
x (x – x) (x – x)²
6 + 1 1
2 - 3 9
6 + 1 1
3 - 2 4
6 + 1 1
9 + 4 16
3 - 2 4
4 - 1 1
5 0 0
6 + 1 1
∑ 50 38

17. CALCULATION OF STANDARD DEVIATION
AND VARIANCE
STANDARD DEVIATION FOR SAMPLE
“ ” = ∑ (X-X)²/(n-1)
STANDARD DEVIATION FOR POPULATION
“σ” = ∑ (X-X)²/n
STANDARD DEVIATION ( SAMPLE), “ ” = 38/9= 4.2
= 2.05 gms
VARIANCE = (Std. Deviation)² = 4.2

18. EXERCISE- 2
The performance of employees in a section has
been assessed on a scale 0 to 10 (“0” being worst
and “10” being best). The performance score is
presented in the Data Table – 2.
Find out mean, median, mode, range, standard
deviation and variance.

19. DATA TABLE – 2
PERFORMANCE SCORE OF EMPLOYEES
EMPLOYEE PERFORMANCE SCORE
(1)
(2)
(3)
(4)
(5)
(6)
(7)
5
9
6
7
5
7
3

20. ANSWERS – Example 2
Mean = 6
Median = 6
Mode = 5, 7
Range = 6
Standard Deviation = 1.9
Variance = 3.6

21. FUNDAMENTAL
LAW OF VARIATION

22. UNDERSTANDING CHANCE CAUSES AND
ASSIGNABLE CAUSES OF VARIATION
 Variation is inevitable in any process /operation/component
 Variations occur due to the following two causes:
(i) Chance causes
(ii) Assignable causes

23. VARIATION DUE TO CHANCE CAUSES
(Also called natural variation or inherent variation)
• These variations are due to a large number of factors which are either
uncontrollable or uneconomical to control.
Examples of such factors are: temperature, humidity variations, variation in
other environmental conditions, slight variations in raw material etc.
• These variations are unavoidable. It is not possible to enumerate each and
every chance cause and eliminate it.
• The factors causing chance variations are many, but the effect of a single
cause is very less. Variation due to chance cause is the sum total of
variations due to all chance causes – known or unknown.
• Variations due to chance causes follow certain standard probability
distributions like Normal distribution in case of variable type of data and
binomial/poisson distribution in case of attribute type of data.

24. VARIATION DUE TO ASSIGNABLE CAUSES
(Also called variation due to special causes/
unnatural causes)
• These variations are due to just one or few individual causes.
• Even a single assignable cause results in a large variation.
• Assignable causes being few, they can be detected and
eliminated.
• Examples of assignable causes: variations due to defective raw
material, faulty setup or untrained operator.
• Variations due to assignable causes do not follow any probability
distribution and hence they can easily be “singled out” from
chance variations.

27. HISTOGRAM
Popularly known as horoscope of the process, histogram is
a simple tool which gives the following vital information
about the process:
 Whether the process is “capable”
 Whether the process is well centered
 Whether any abnormality (assignable cause) is present
in the process
4) Whether the process is under statistical control
Though an off-line technique, its simplicity renders is very useful
for making a preliminary assessment of the process.

28. HISTOGRAM : EXAMPLE -1
TEST SCORE OF 50 STUDENTS (RAW DATA)
MAXIMUM MARKS = 100
46 68 22 78 65 89 59 69 70
78
64 75 92 80 73 75 64 82 25
68
71 17 65 59 74 38 71 25 90
76
63 58 86 73 96 67 72 64 32
90
69 23 58 79 60 87 70 77 63

29. FREQUENCY DISTRIBUTION TABLE OF TEST SCORES
------------------------------------------------------------------------------------------------------
CLASS FREQUENCY
INTERVAL
-------------------------------------------------------------------------------------------------------
0 – 10 NIL
10 – 20 1
20 – 30 4
30 – 40 2
40 – 50 1
50 - 60 5
60 - 70 14
70 - 80 15
80 - 90 6
90 - 100 2
------------------------------------------------------------------------------------------------------
Total: 50
------------------------------------------------------------------------------------------------------

33. HISTOGRAM: EXAMPLE -2
RAW DATA
Length of Cast Bars in mm
26, 28, 35, 42, 49, 63, 70, 77, 86, 30, 37,
43, 50, 57, 64, 71, 80, 34, 40, 44, 51, 58,
65, 72, 40, 45, 52, 59, 66, 72, 41, 46, 53,
60, 67, 41, 47, 54, 61, 69, 35, 48, 55, 54,
62, 46, 53, 60, 53, 58
TOTAL OBSERVATIONS = 50
SPECIFIED LENGTH = 65 +/- 10MM
OBJECTIVE:
To construct Histogram and draw conclusions regarding
the casting process.

34. FREQUENCY DISTRIBUTION TABLE
--------------------------------------------------------------------------------------------
CLASS FREQUENCY
INTERVAL
---------------------------------------------------------------------------------------------
20 – 27 1
27 – 34 3
34 – 41 7
41 – 48 8
48 – 55 10
55 – 62 8
62 – 69 6
69 – 76 4
76 – 83 2
83 – 90 1
-----------------------------------------------------------------------------------------
Total: 50
-------------------------------------------------------------------------------------------

36. PROCESS
CAPABILITY

37. REVISITING
NORMAL DISTRIBUTION

39. PROCESS CAPABILITY:
RELATED TERMS
x = Process average
μ = Specified nominal value (Target)
σ = Process standard deviation
USL= Upper specification limit
LSL= Lower specification limit
Tolerance (T) = USL – LSL
Cp = Process capability index
Cpk= Process centering index

40. PROCESS CAPABILITY
AND
PROCESS CAPABILITY INDEX (Cp)
If the process is under statistical control i.e. no assignable
causes are present, then
PROCESS CAPABILITY = 6 X Standard Deviation
= 6 X σ
USL – LSL
PROCESS CAPABILITY INDEX (Cp) =
6 σ
For process to be capable, Cp > 1

42. PROCESS CENTERING INDEX (Cpk)
Cpk is a measure of process centering and defined as:
USL - x x - LSL
Let Z (u) = , Z(L) =
3σ 3σ
Then, Cpk = smaller of Z (U) and Z (L)

44. EXAMPLE
Length of a moulded component has been specified as
100 ± 3 mm. Length was measured on 60 components.
Average length and standard deviation has been worked
out as 101 mm and 0.9 mm respectively.
(i) Find out process capability and hence comment
whether the process is capable.
(ii) Find out process capability Index (Cp) and process
centering index (Cpk)

45. SOLUTION
CALCULATION OF PROCESS CAPABILITY
Process Average ( x ) = 101 mm
Standard deviation (σ) = 0.9 mm
Upper specification Limit (USL) = 100 + 3 = 103 mm
Lower specification Limit (LSL) = 100 - 3 = 97 mm
Tolerance = (USL – LSL) = 103 – 97
= 6 mm
PROCESS CAPABILITY = 6 x σ = 6 x 0.9 = 5.4 mm
As process capability is less than tolerance, the process of
manufacturing moulded components is a capable process.

46. Calculation of process capability index
USL – LSL 103 – 97 6
Cp = = = = 1.11
6 σ 6 x 0.9 5.4

47. Calculation of process centering index
USL –X 103 – 101 2
Z(U) = = = = 0.74
3σ 3 x 0.9 2.7
X - LSL 101 – 97 4
Z(L) = = = = 1.48
3σ 3 x 0.9 2.7
The smaller of Z(U) and Z(L) is 0.74
Hence Cpk = 0.74

48. KEY TO STATISTICAL PROCESS CONTROL
(1) Ensure that no assignable causes are present
in the process.
(Check by control chart/ histogram/ normal probability
graph/ “p” value)
(2) Ensure that the process is “capable”.
(check by Cp value. The greater the Cp, the better)
(3) Ensure that the process is well centered.
(check by Cpk value. The Cpk value to be as close
to Cp as possible)

49. PUZZLE
(1) What is the process capability index
(Cp) for a six sigma process ?
(2) How much “shift” is normally assumed
for a six sigma process and what is the
corresponding value of Cpk ?

50. CONTROL CHARTS
(SHEWHART CHARTS)

51. WHAT IS CONTROL CHART
A powerful SQC tool for on-line
process control

52. HOW CONTROL CHART WORKS
Control Chart detects assignable causes ( also called special
causes )
in the process. Once the assignable causes are eliminated, the
process runs under the influence of chance causes alone and the
process is said to be under statistical control. Variations are, then,
controlled within natural limits called “control limits” which are
generally narrower than the specification limits.
CONTROL CHARTS ADVANTAGES
(1) Control chart prevents defects in the process
(2) Control chart keeps the variations to a minimum, generally
narrower than the specification limits.
The result : process delivers better and consistent quality
products.

53. CONTROL CHART TYPES
(1) Control charts for VARIABLES
(2) Control charts for ATTRIBUTES

56. CONTROL CHART FOR ATTRIBUTES
CONTROL CHART TYPE QUALITY CHARACTERISTIC
p – chart Proportion defective or percent defective
np – chart Number of defectives
c – chart Number of defects in a defined “unit”
u – chart Number of defects per piece

62. WHEN TO ACT ON A CONTROL CHART
The requirements in a control chart are:
(i) All the points should lie within control limits
(ii) The points should be randomly distributed.
(i.e. the points should not depict any rising/falling trend, cyclic
pattern, clustering of points above or below central line, shift of
process average etc.)
Violation of either of the above two conditions indicates presence
of assignable cause in the process which must be investigated and
eliminated.
In case of p-chart/np-chart/c-chart/u-chart, if a point goes below the
lower control limit, it should not be ignored thinking that it is an
improvement. The cause of improvement must be investigated and if it
is true, the same should be implemented in the process.

64. RUN CHART

65. WHAT IS A RUN CHART
Run chart is a process control chart much simpler but less powerful than control chart.
Run charts generally precede control charts.
In a run chart individual observations (not averages) on plotted on time scale in order
of production. The chart shows central line (nominal value) and specification limits.
WHEN TO ACT ON A RUN CHART
The requirements in a run chart are:
(i) All the points should lie within specification limits
(ii) The points should be randomly distributed.
Violation of either of the above two conditions calls for investigation and corrective action.