Transcript Describing Variation & Distributions

IENG 486 Lecture 04
Describing Variation
& Distributions
7/21/2015
IENG 486: Statistical Quality & Process Control
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Assignment:
 Print off Review Data from link on Materials pg.

Bring the data and your exam calculator to next class
 Reading:

Chapter 1: (1.1, 1.3 – 1.4.5)


Chapter 2: (2.2 – 2.7)


Cursory – get Fig. 1.12., p.34; Deming Management,1.4.4 Liability
Cursory – Define, Measure, Analyze, Improve, Control
Chapter 3: (3.1, 3.3.1, 3.4.1)
 HW 1: Chapter 3 Exercises:



7/21/2015
1, 3, 4 – using exam calculator
10 (use Normal Plots spreadsheet from Materials page)
43, 46, 47 (use Exam Tables from Materials page – Normal Dist.)
IENG 486: Statistical Quality & Process Control
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What is Quality
Many definitions:
Better performance
 Better service
 Better value
 Whatever the customer says it is…

For SPC, quality means better:
Understanding of process variation,
 Control of the variation in the process, and
 Improvement in the process variation.

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Understanding Process Variation
Three Aspects:
Location
 Spread
 Shape

Basic Statistics:
Quantify
 Communicate

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Location: Mode
 The mode is the value (or values) that occurs
most frequently in a distribution.
 To find the mode:
1. Sort the values into order (with no repeats),
2. Tally up how many times each value appears in
the original distribution.
3. The mode (or modes) has the largest tally


Dist. 1 has two modes: 20 and 15 (four times, ea.)
Dist. 2 has one mode: 15 (appearing seven times)
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Location: Median
Half of the values will fall above and half of the
values will fall below the median value.
To estimate the median:

Sort the values (keeping the duplicates in the list), and
then count from one end until you get to one half
(rounding down) of the total number of values.
 For
an odd number of values, the median is the next value.
 For an even number of values, the median value is half of the
sum of the current value and the next sorted value.
 Dist. 1 median is 19.5
 Dist. 2 median is 15
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Location: Mean
 The mean has a special notation: x for a
sample ( for the entire population)
 To calculate the mean:
1. add up all of the values
2. divide the sum by the number of values
n
x


x
i
i 1
n
Dist. 1 mean is 18.6,
Dist. 2 mean is 15.0 Mean is influenced by outliers
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Spread: Range
Range is the difference between the maximum
and the minimum values, denoted R.
R  max( xi )  min( xi )
This value gives us the extreme limits of the
distribution spread.
Much easier to calculate than other measures
 Very sensitive to outliers
 Range of Dist. 1 is 11
 Range of Dist. 2 is 4

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Spread: Variance
Variance has the symbol 2 when referring to the
entire population (S2 for a sample variance)

The formula for the variance is:
 x
n
S2 
i
x
i 1

2
n 1
Measures the dispersion with less emphasis on outliers
 Units for variance aren’t very intuitive
If population is
 Calculation is unpleasant
known, use n
in denominator!
(calculating equation could be used)
 The variance for Dist. 1 is 10.58, for Dist. 2 it is 1.63

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Spread: Standard Deviation
The standard deviation ( for the population, or S
for a sample) is the square root of the variance.

Defn. Special calculating formula:
 x
n
S  S2 
i 1
i
x

2
n 1


  xi 
n
2
 i 1 
x


i
n
S  i 1
n 1

Not as easily influenced by outliers

Has the same units as measure of location.
 Std deviation for Dist. 1 is 3.25
 Std deviation for Dist. 2 is 1.28
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IENG 486: Statistical Quality & Process Control
n
2
If population is
known, use n
in denominator!
10
Shape: Prob. Density Functions
The shape of a distribution is a function that
maps each potential x-value to the likelihood
that it would appear if we sampled at random
from the distribution. This is the probability
density function
(PDF).
  1 :68.26% of the total area
  2 :95.46% of the total area
  3 :99.73% of the total area
-3
-2
-

+
+2
+3
Area Under the Normal Curve
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Shape: Stem-and-Leaf Plot
48
59
53
54
49
47
52
49
51
45
52
64
63
79
60
65
53
62
64
60
 Divide each number into:
 StatGraphics Output:
Stem – one or more of the
leading digits
Leaf – remaining digits
(may be ordered)
Choose between 4 and 20
stems
Stem-and-Leaf Display
5
4|57899
6
5|122334
1
5|9
6
6|002344
1
6|5
0
7|
1
7|9



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Shape: Box (and Whisker) Plot
Box-and-Whisker Plot
Max value
85
80
Third quartile
Value
75
70
65
Mean
Median
60
55
50
45
First quartile
 Visual display of

Min value
central tendency, variability, symmetry, outliers
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Shape: Histogram
A histogram is a vertical bar chart that takes the
shape of the distribution of the data. The
process for creating a histogram depends on
the purpose for making the histogram.
One purpose of a histogram is to see the shape of a
distribution. To do this, we would like to have as
much data as possible, and use a fine resolution.
 A second purpose of a histogram is to observe the
frequency with which a class of problems occurs.
The resolution is controlled by the number of
problem classes.

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Goals of Statistical Quality Improvement
 Find
special
causes
 Head off
shifts in
process
 Obtain
predictable
output
 Continually
improve the
process
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Statistical Quality Control and Improvement
Improving Process Capability and Performance
Continually Improve the System
Characterize Stable Process Capability
Head Off Shifts in Location, Spread
Time
Identify Special Causes - Bad (Remove)
Identify Special Causes - Good (Incorporate)
Reduce Variability
Center the Process
LSL
0
USL
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