Download Basic Statistics --

Basic Statistics --- 1
Basic Statistics
How to get information about a production process?
Measurement  sample
Analysis  information for the engineers and managers
d
Basic Statistics --- 2
Some Engineering Fields where Statistical Tools are Applied
Statistical Process Control (SPC)
control charts
Basic Statistics --- 3
Six Sigma Process Improvement (6 )





DEFINE
MEASURE
ANALYZE
IMPROVE
CONTROL
Basic Statistics --- 4
Technical Diagnostics
condition monitoring with vibration analysis
Basic Statistics --- 5
Example: Disk of diameter
is produced
d
Nominal diameter: 25 mm
Tolerance range: 24.90 mm – 25.05 mm
Real (measured) diameter values (200-element sample):
24.96
25.04
24.66
25.00
25.22
24.98
24.67
24.87
24.83
24.96
24.96
24.96
24.85
24.81
24.81
25.03
24.62
24.87
24.94
24.80
24.97
24.89
24.76
24.92
24.86
24.84
24.79
24.91
24.62
24.84
24.82
24.76
25.00
24.79
24.98
25.05
24.88
24.94
25.00
24.89
24.98
24.71
25.07
24.75
24.63
25.11
24.89
24.97
24.85
24.75
24.44
24.99
24.86
24.87
24.96
24.69
24.79
25.07
24.99
24.71
24.57
24.81
24.93
25.04
24.91
24.73
24.95
24.72
24.76
24.80
24.72
24.86
24.77
24.86
24.83
24.84
24.82
25.08
24.99
24.91
24.68
24.66
24.61
25.29
25.05
24.94
24.75
24.73
24.98
25.13
24.65
24.83
24.89
24.60
25.03
24.84
25.19
24.84
24.96
24.73
24.81
24.72
24.96
25.10
24.81
24.97
24.82
24.92
25.08
24.92
24.74
24.77
24.87
24.87
24.77
24.93
25.09
24.86
24.87
24.57
24.78
24.69
24.84
24.98
25.13
25.17
24.88
24.85
25.02
24.97
25.01
25.02
24.68
24.98
24.89
24.79
24.80
25.09
24.78
24.71
24.93
25.20
24.90
24.95
24.76
25.15
24.92
24.98
24.76
25.03
24.96
24.73
24.91
24.83
24.92
24.86
24.87
24.75
24.85
25.01
25.16
24.73
25.26
25.13
24.93
24.85
24.80
25.02
24.80
24.83
25.03
24.97
25.03
24.98
24.84
24.82
25.02
24.80
25.08
25.02
24.94
24.90
24.95
25.43
24.88
24.67
25.19
24.99
24.80
25.29
24.98
24.94
24.78
24.74
24.63
24.84
24.74
25.00
24.85
25.17
Basic Statistics --- 6
Ordered sample:
24.44
24.71
24.76
24.81
24.85
24.88
24.93
24.97
25.01
25.08
24.57
24.72
24.77
24.81
24.85
24.88
24.93
24.97
25.01
25.09
24.57
24.72
24.77
24.81
24.85
24.89
24.94
24.97
25.02
25.09
24.60
24.72
24.77
24.82
24.85
24.89
24.94
24.98
25.02
25.10
24.61
24.73
24.78
24.82
24.85
24.89
24.94
24.98
25.02
25.11
24.62
24.73
24.78
24.82
24.85
24.89
24.94
24.98
25.02
25.13
24.62
24.73
24.78
24.82
24.86
24.89
24.94
24.98
25.02
25.13
24.63
24.73
24.79
24.83
24.86
24.90
24.95
24.98
25.03
25.13
24.63
24.73
24.79
24.83
24.86
24.90
24.95
24.98
25.03
25.15
24.65
24.74
24.79
24.83
24.86
24.91
24.95
24.98
25.03
25.16
24.66
24.74
24.79
24.83
24.86
24.91
24.96
24.98
25.03
25.17
24.66
24.74
24.80
24.83
24.86
24.91
24.96
24.98
25.03
25.17
24.67
24.75
24.80
24.84
24.87
24.91
24.96
24.99
25.04
25.19
24.67
24.75
24.80
24.84
24.87
24.92
24.96
24.99
25.04
25.19
24.68
24.75
24.80
24.84
24.87
24.92
24.96
24.99
25.05
25.20
Statistical parameters calculated from ordered sample
Minimum value, Maximum value, Range
Minimum value =
Maximum value =
Range = Maximum value – Minimum value =
Range is a measure of dispersion of the random variables.
24.68
24.75
24.80
24.84
24.87
24.92
24.96
24.99
25.05
25.22
24.69
24.76
24.80
24.84
24.87
24.92
24.96
25.00
25.07
25.26
24.69
24.76
24.80
24.84
24.87
24.92
24.96
25.00
25.07
25.29
24.71
24.76
24.81
24.84
24.87
24.93
24.97
25.00
25.08
25.29
24.71
24.76
24.81
24.84
24.88
24.93
24.97
25.00
25.08
25.43
Basic Statistics --- 7
Median (med)
Median is the “
-th” ordered sample element, practically
In our example:
Median is a measure of location
Basic Statistics --- 8
First (lower) quartile (
), Third (upper) quartile (
First (lower) quartile is the “
), Interquartile range
-th” ordered sample element, practically:
Otherwise
is the weighted average of elements neighboring to the
“
-th” element in the ordered sample.
In our example:
weights
0.25
50
50.25
0.75
51
Basic Statistics --- 9
Third (upper) quartile is the “
practically:
Otherwise
“
-th” ordered sample element,
is the weighted average of elements neighboring to the
-th” element in the ordered sample.
In our example:
weights
0.75
150
0.25
150.75
151
Basic Statistics --- 10
Interquartile range (range of the middle 50%) = Upper quartile – Lower quartile
In our example:
Further parameters
lower inner fence:
upper inner fence:
lower outer fence:
upper outer fence:
Outliers
A point beyond an inner fence on either side is considered a mild outlier.
A point beyond an outer fence is considered an extreme outlier.
Basic Statistics --- 11
Box-plot diagram
In our example
25.43
maximum value
25%
25%
upper quartile
median
lower quartile
25%
25%
24.98
24.88
24.79
25%
25%
25%
minimum value
25%
24.44
Basic Statistics --- 12
Box-plot diagram with outliers
outlier value
upper outer fence
25%
upper quartile
median
lower quartile
25%
25%
25%
lower outer fence
outlier value
whiskers: parts out of interquartile
Basic Statistics --- 13
Sample mean, Variance, Standard deviation
Sample
Sample mean
Sample mean is a measure of location.
Variance
Standard deviation
Variance and Standard deviation are measures of dispersion.
Basic Statistics --- 14
Calculations in MS Excel
number of
elements
mean
standard
deviation
maximum
element
minimum
element
median
=COUNT()
=AVERAGE()
=STDEV()
=MAX()
=MIN()
=MEDIAN()
Basic Statistics --- 15
Gauss curve (bell curve)
Parameters of products (random variables) are generally supposed to be
normally distributed. These types of variable are described with the “bell shaped”
Gauss curve.
The two parameters of the Gauss curve are


mean:
standard deviation:
2,5

2

1,5
1
0,5
0
24, 2
24, 4
24, 6
24, 8
24.74
25, 0
24.89
m
25.04
25, 2
25, 4
25, 6
Basic Statistics --- 16
Mean ( or ) determines the position of the Gauss curve on the real line.
Standard deviation ( ) and Variance ( ) determine the shape of the Gauss curve.
Basic Statistics --- 17
How to use the Gauss curve
probability = area
E.g. the probability of that the diameter is between 24.6 mm and 25.0 mm is
2,5
2
1,5
P
1
0,5
0
24, 2
24, 4
24, 6
24, 8
25, 0
25, 2
25, 4
25, 6
Basic Statistics --- 18
Significance of the mutual position of the Gauss curve (related to the actual status of the
production) and the tolerance range.
When the tolerance range contains the
not less than
.
interval the ratio of good parts is
2,5
2
waste production
68.26%
1,5
31.74%
1
0,5


0
24, 2
24, 4
24, 6
m
24, 8
25, 0
m
m
tolerance range
25, 2
25, 4
25, 6
Basic Statistics --- 19
When the tolerance range contains the
not less than
.
interval the ratio of good parts is
2,5
2
waste production
95.44%
1,5
4.56%
1
0,5
2
2
0
24, 2
24, 4
24, 6
m  2
24, 8
25, 0
m
tolerance range
25, 2
m  2
25, 4
25, 6
Basic Statistics --- 20
When the tolerance range contains the
not less than
.
interval the ratio of good parts is
2,5
2
waste production
99.72%
1,5
0.28%
1
0,5
3
3
0
24, 2
24, 4
m  3
24, 6
24, 8
25, 0
m
25, 2
25, 4
25, 6
m  3
tolerance range
Remark: In “six sigma” production the maximum number of defects is 3.4 per one
million opportunities (DPMO).
Basic Statistics --- 21
Example
40-element sample from the length of a product
38.6700
38.8219
38.1284
37.5954 Mean
37.3548
38.7021
39.3412
38.5422
38.9243
38.7088
37.9876
38.4057
37.6489
37.3971
37.4518
37.6334
36.1390
39.2069
38.4276
37.6716
39.1592
37.0019
38.0310
38.4234
38.4190
37.5946
38.1790
38.0846
38.6394
38.9653
38.9765
38.2542
37.3501
37.4205
38.1952
39.2728
37.6612
38.4600
38.0421
38.9821
Standard deviation
40.0000
39.5000
39.0000
38.5000
38.0000
37.5000
37.0000
36.5000
36.0000
35.5000
0
5
10
15
20
25
30
35
40
45
Basic Statistics --- 22
Histogram (frequency density)
class
frequency
36-36.5
1
36.5-37
0
37-37.5
6
37.5-38
7
38-38.5
12
38.5-39
10
39-39.5
4
total:
40
relative frequency
0.025
0
0.15
0.175
0.3
0.25
0.1
1
height of bin in the histogram
0.05
0
0.3
0.35
0.6
0.5
0.2
0.6
0.35
0.3
0.3
0.25
0.15
0.175
0.1
0.05
0.025
36
36.5
37
37.5
38
38.5
39 39.5
Basic Statistics --- 23
Frequency distribution and Cumulative frequency distribution
Time
1
2
3
4
5
6
7
8
9
10
Frequency (number of failures) 5
6
7
9
3
7
5
2
4
2
Relative frequency
0.1 0.12 0.14 0.18 0.06 0.14 0.1 0.04 0.08 0.04
Cumulative frequency
5
Relative cumulative frequency
11
18
27
30
37
42
44
48
0.1 0.22 0.36 0.54 0.6 0.74 0.84 0.88 0.96
10
0.2
9
0.18
8
0.16
7
0.14
6
0.12
5
0.1
4
0.08
3
0.06
2
0.04
1
0.02
0
50
1
0
1
2
3
4
5
6
7
8
9
10
1
60
1.2
50
1
40
0.8
30
0.6
20
0.4
10
0.2
2
3
4
5
6
7
8
9
10
4
5
6
7
8
9
10
0
0
1
2
3
4
5
6
7
8
9
10
1
2
3
Basic Statistics --- 24
Example (in-process sampling)
11 five-element samples are taken from a production line
1
37.4377
38.2228
38.5868
38.9704
39.2809
mean 38.4997
std 0.7149
2
38.2602
37.1559
37.6804
37.4569
39.8453
38.0797
1.0666
3
37.7691
38.7237
38.0653
38.8220
37.6646
38.2089
0.5365
4
37.2184
37.2043
38.5417
38.0758
38.0628
37.8206
0.5887
5
39.1935
38.4041
38.3385
39.3114
37.6369
38.5769
0.6874
6
39.9876
40.0846
39.3294
39.6510
38.6839
39.5473
0.5671
7
38.6964
39.5734
40.0056
41.3098
39.0332
39.7237
1.0186
8
40.2873
40.0175
38.1670
39.6610
38.3053
39.2876
0.9865
9
40.9404
39.2120
40.2001
39.5555
40.4035
40.0623
0.6867
10
39.4997
40.5900
40.8394
41.8119
39.9059
40.5294
0.8935
11
37.9616
39.2604
41.4546
39.0278
41.0610
39.7531
1.4649
Basic Statistics --- 25
Mean
41.0000
40.5000
40.0000
39.5000
39.0000
38.5000
38.0000
37.5000
0
2
4
6
8
10
12
10
12
Standard deviation
1.6000
1.4000
1.2000
1.0000
0.8000
0.6000
0.4000
0.2000
0.0000
0
2
4
6
8
Basic Statistics --- 26
Probability of an event
Probability is the measure of the likelihood that an event will occur.
Probability is quantified as a number between 0 and 1
(0 indicates impossibility and 1 indicates certainty).
The higher the probability of an event, the more certain that the event will occur.
Basic Statistics --- 27
Example: Tossing of a fair (unbiased) coin
Two outcomes ("head" and "tail") are both equally probable.
The probability is 1/2 (or 50%) of either "head" or "tail":
This type of probability is also called a priori probability.
Basic Statistics --- 28
Example: Rolling of a fair (unbiased) dice
Six outcomes (1,2,3,4,5,6) are equally probable:
This type of probability is also called a priori probability.
Basic Statistics --- 29
Classical Probability
If all the outcomes of a trial are equally likely, then the probability of an event can be
calculated by the formula:
Example
Random experiment: rolling a dice
Basic Statistics --- 30
Calculations with probability
Multiplication rule for independent events:
Example: Reliability of a series system
Reliability block diagram
Basic Statistics --- 31
Addition rule:
If A and B are independent
Example: Reliability of a parallel system
Reliability block diagram
Basic Statistics --- 32
“Not” rule:
Basic Statistics --- 33
Random variables
Random experiment: outcome is uncertain (e.g. measurements)
Random variable: outcome is a random number
Sample space (range): set of possible outcomes
Example: throw with two dice, random variable is the sum of the two numbers
Sample space = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
Probability distribution (discrete variable):
k
2
3
4
5
6
7
8
9
10
11
12
p 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36
Basic Statistics --- 34
Calculations with probability distribution
Example: throw with two dice, random variable is the sum of the two numbers
k
2
3
4
5
6
7
8
9
10
11
12
p 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36
k
2
3
4
5
6
7
8
9
10
11
12
p 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36
Basic Statistics --- 35
Density Function and Probability function (continuous random variables)
Random variable:
Probability Density Function
(Cumulative) Probability Function
p
p
a
a
b
= area under the curve (area between the
curve and the horizontal axis)
b
x
Basic Statistics --- 36
Normal Distribution
Parameters of products (random variables) are generally supposed to be
normally distributed.
The density function of normal distribution is the “bell shaped” Gauss curve.
The two parameters of the Gauss curve are the mean (
deviation ( )
or ) and the standard
2,5

2

1,5
1
0,5
0
24, 2
24, 4
24, 6
24, 8
24.74
25, 0
24.89
m
25.04
25, 2
25, 4
25, 6