Transcript Chap 2 Introduction to Statistics

Chap 2 Introduction to Statistics
This chapter gives overview of statistics
including histogram construction,
measures of central tendency, and
dispersion
INTRODUCTION TO STATISTICS
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Statistics – deriving relevant information
from data
Deals with
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Collection of data – census, GDP, football,
accident, no. of employees (male, female ,
department, etc)
Collection , tabulation, analysis,
interpretation, an presentation of quantitative
data – can make some conclusions on sample
or population studied, make decisions on
quality
INTRODUCTION TO STATISTICS
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Use of statistics in quality deals with
second meaning. – inductive statistics
Examples :
What can we learn from the data?
What conclusions can be drawn?
What does the data tell about our process
and product performance? etc.
INTRODUCTION TO STATISTICS
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Understand the use of statistics vital
in business
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to make decisions based on facts
in conducting business improvements
in controlling and monitoring process,
products or service performance
Application of statistics to real life
problems such as for quality
problems will result in improved
organizational performance
Collection of data
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Collect Data – direct observation or indirect
through written or verbal questions (market
research, opinion polls)
Direct observation measured, visual checking,
classified as variables and attributes
Variables data – measurable quality
characteristics
Attributes – characteristics not measured but
classified as conforming or non-conforming
Collection of data
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Data collected with purpose
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Find out process conditions
For improvement
Variables – quality characteristics that are
measurable and countable
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CONTINUOUS - Dimensions, weight, height,
etc. (meter, gallon, p.s.i., etc.)
DISCRETE - numbers that exhibit gaps,
countable, (no. of defective parts, no. of
defects/car, Whole numbers, 1, 2, 3….100)
Collection of data
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Attributes - quality characteristics that are nonmeasurable and ‘those we do not want to
measure’
Example : surface appearance, color,
Acceptable, non-acceptable conforming, nonconf.
Data collected in form of discrete values
Variables (weight of sugar) CAN be classified as
attributes
 weight within limits – number of conforming
 outside limits – no. of non conforming
Summarizing Data
 Consider this data set on number of Daily Billing errors
0
1
3
0
1
0
1
0
1
5
4
1
2
1
2
0
1
0
2
0
0
2
0
1
2
1
1
1
2
1
1
0
4
1
3
1
1
1
1
3
4
0
0
0
0
1
3
0
1
2
2
3
 Data in this from
Meaningless
Not effective
Difficult to use
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Need to summarize data in the form of:
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Graphical – Freq. Dist., Histogram, Graphs,
Charts, Diagrams
Analytical – Measures of central tendency,
Measure of dispersion
Frequency Distribution (FD)
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Summary of how data (observations) occur
within each subdivision or groups of observed
values
Help visualize distribution of data
Can see how total frequency is distributed
Two types :
Ungrouped data – listing of observed values
Grouped data – lump together observed values
20.5  21.5


21.5

22.5


22.5  23.5
FD - Ungrouped Data
1.
2.
3.
Establish array,
arrange in ascending
or descend (as in
column 1)
Tabulate the
frequency – place
tally marking in
column 2
Present in graphical
form – Histogram,
Relative freq. distr.
No of
errors
Tally mark
Frequency
0
///////////
13
1
////
2
/////
3
////
4
5
4 graphical representations
1.
Frequency histogram
2.
Relative freq histogram
3.
Cumulative frequency histogram
4.
Relative cum frequency histogram
Frequency
FD – Ungrouped data
14
12
10
8
6
4
2
0
1
2
3
4
No error
Freq
Relative
freq
Cumulative
freq
Rel cum
freq
0
15
0.29
15
0.29
1
20
0.38
35
0.67
2
8
0.15
43
0.83
3
5
0.10
48
0.92
4
3
0.06
51
0.98
5
1
0.02
52
1.00
Total
52
5
Frequency Distribution For
Grouped Data
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Data which are continuous variable need grouping
Steps
1. Collect data and construct tally sheet
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Make tally - coded if necessary
Too many data – group into cells
Simplify presentation of distribution
Too many cells – distort true picture
Too few cells – too concentrated
No of cells – judgment by analyst – trial and error
Generally 5-20 cells
Less than 100 data – use 5 –9 cells
100 – 500 data – use 8 to 17 cells
More than 500 – use 15 to 20 cells
Cell interval (i)
CELL
Midpoint
UPPER BOUNDARY
CELL NOMENCLATURE
2. Determine the range
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R = XH - XL
R = range
XH = highest value of data
XL = lowest value of data
Example :
If highest number is 2.575 and lowest number
is 2.531, then
R = XH - XL
= 2.575 – 2.531
= 0.044
3. Determine the cell interval
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Cell interval = distance between adjacent cell midpoints.
If possible, use odd interval values e.g. 0.001, 0.07, 0.5 ,
3; so that midpoint values will have same no. decimal
places as data values.
Use Sturgis rule.
i = R/(1+ 3.322 log n)
Trial and error
h = R/i ;h= number of cells or cllases
Assume i = 0.003; h = 0.044/0.003 = 15 cells
Assume i = 0.005; h = 0.044/0.005 = 9 cells
Assume ii = 0.007; h = 0.044/0/.007 = 6 cells
Cell interval 0.005 with 9 cells will give best presentation
of data. Use guidelines in step 1.
4. Determine cell midpoints
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MPL = XL + i/2 (do not round)
= 2.531 + 0.005/2 = 2.533
1st cell have 5 different values (also the other
cells)
2.531
2.535
2.532
2.533
2.534
2.5
33
2.53
8
5. Determine cell boundaries
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Limit values of cell
lower
upper
To avoid ambiguity in putting data
Boundary values have an extra decimal
place or sig. figure in accuracy that
observed values
+ 0.0005 to highest value in cell
- 0.0005 to lowest value in cell
6. Tabulate cell frequency
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Post amount of numbers in each cell
Frequency distribution table
Cell boundary
Cell MP
Freq.
2.531 – 2.535
2.533
6
2.536 – 2.540
2.538
8
2.541 – 2.545
2.543
12
2.546 – 2.550
2.553
13
2.551 – 2.555
2.553
20
2.556 – 2.560
2.563
19
2.561 – 2.565
2.563
13
2.566 – 2.570
2.568
11
2.571 – 2.575
2.573
8
110
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Freq dist gives better view of central value and
how data dispersed than the unorganized data
sheet
Histogram – describes variation in process
Used to
solve problems
determine process capability
compare with specifications
suggest shape of distribution
indicate data discrepancies, e.g. gaps
Characteristics Of Frequency
Distribution
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Symmetry, Number of modes (one, two
or multiple), Peakedness of data
Bi-modal
Sym.
Skew
Right
Skew
Left
‘very peak’
leptokurtic
flatter
platykurtic
Characteristics of Frequency
Distribution
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F.D. can give sufficient info to provide basis for
decision making.
Distributions are compared regarding:-
Location
Spread
Shape
Descriptive Statistics
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Analytical method allow comparison between
data
2 main analytical methods for describing data
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Measures of central tendency
Measures of dispersion
Measures of central tendency of a distribution a numerical value that describes the central
position of data
3 common measures
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mean
median
mode
Measure of Central Tendency
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Mean - most common measure used
What is middle value? What is average
number of rejects, errors, dimension of
product?
Mean for Ungrouped Data - unarranged
x (x bar)
n
X

x
i1
n
i

x1  x 2   x n
n
Mean
Example
A QA engineer inspects 5 pieces of a tyre’s
thread depth (mm). What is the mean thread
depth?
x1 = 12.3 x2 = 12.5 X3 = 12.0.
x4 = 13.0 x5 = 12.8
x

Σx i
5

62.5
5
 12.5 mm
Mean - Grouped Data
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When data already grouped in frequency
distribution
h
f
i
xi
x

fi
fi
n
xi
(n)= sum. of freq.
= freq in the ith cell
= no. of cells/class
= mid point in ith cell
i1
Σfi 
Mean - Grouped Data
Cell (i)
Class
boundary
Mid
Point
(xi)
Freq
(fi)
Fixi
fi
fixi
1
1 – 20
10
2
20
2
2
21 – 40
30
10
300
12
3
41 - 60
50
20
1000
32
4
61 – 80
70
12
840
44
5
81 -100
90
6
540
50
Totals
x

 fi xi
fi
2700
= 2700/50 = 54
Weighted average
Tensile tests aluminium alloy conducted with different
number of samples each time. Results are as follows:
n
1st test : x1 = 207 MPa
n=5
 w i xi
nd
2 test : x2 = 203 MPa
n=6
xw
 i 1n
 wi
3rd test : x3 = 206 MPa
n=3
i 1
xw

(5)(207)  6 (203)  3 (206)
xw = weighted avg.
 205 MPa
563
wi = weight of ith
or use sum of weights equals 1.00
W1 = 5/(5+6+3) = 0.36
W2 = 6/(5+6+3) = 0.43
W3 = 3/(5+6+3) = 0.21 Total = 1.00
average
Median – Ungrouped Data
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Median – value of data which divides total
observation into 2 equal parts
Ungrouped data – 2 possibilities
When total number of data (N) is a) odd or b)
even
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If N is odd ; (N+1/2)th value is median
eg. 3
4
5
6
8
N+1/2=6/2=3 ,
3rd no.
If N is even
eg. 3
5
7
9
½ of (5+7)=6
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NOTE:
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ORDER THE NUMBERS FIRST!
Median – Grouped Data
Need to find cell / class having middle value &
interpolating in the cell using
n

  cfm 
i
x 0.5  Lm   2
 fm 


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Lm = lower boundary of cell with the median
Cfm
= Cum. freq. of all cells below Lm
fm =class/cell freq. where median occurs
i =
cell interval
Example
MD
=
40.5 + 10
=
53.5
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Measures of dispersion
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describes how the data are spread out or
scattered on each side of central value
both measures of central tendency & dispersion
needed to describe data
Exams Results
Class 1 – avg. :
60.0 marks
highest
:
95
lowest
:
25
Class 2 – avg. :
60.0 marks
highest
:
100
lowest
:
15 marks
Measures of dispersion
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Main types – range, standard deviation,
and variance
Range – difference bet. highest & lowest
value
R = XH - XL
Standard deviation
Variance – standard deviation squared
Large value shows greater variability or
spread
Standard deviation
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For Ungrouped Data
s =
sample std. dev.
n
s
 x  x 
2
i
i1
n 1
xi =
x
n
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=
=
observed value
average
no. of observed value
or use
n
s
n
i1
 n

x i    x i 
 i1

n n 1
2
2
Standard deviation – grouped
data
fixi
Class
boundary
Mid
Point
(xi)
Freq
(fi)
1
1 – 20
10
2
20
2
2
21 – 40
30
10
300
12
3
41 - 60
50
20
1000
32
4
61 – 80
70
12
840
44
5
81 -100
90
6
540
50
h
s
n
1
Fixi
fi
Cell (i)
Totals
2700
 h
2
fi x i    fi x i  
 1

n (n 1)
50 (166,600)  (2700)2

 424.49  20.6
50 49

NOTE:
2

DO NOT ROUND OFF fixi & fixi2
ACCURACY AFFECTED
Concept Of Population and Sample
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Total daily prod. of steel shaft.
Population
Year’s Prod. Volume of calculators
Compute x and s sample
statistics
True Population Parameters
Sample
 and 
Why sample?
not possible measure population
costs involved
100% manual inspection –
accuracy/error
Concept Of Population and Sample
SAMPLE
Statistics,
x , s
POPN.
Parameter
 - mean
 - std. dev.
Normal Distribution
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Also called Gaussian distribution
Symmetrical, unimodal, bell-shaped dist
with mean, median, mode same value
Popn. curve – as sample size  cell
interval  - get smooth polygon
ND
Normal Distribution
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Much of variation in nature & industry
follow N.D.
Variation in height of humans, weight of
elephants, casting weights, size piston
ring
Electrical properties, material – tensile
strength, etc.
Example - ND
Characteristics of ND
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Can have different mean but same
standard deviation
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Different standard deviation but same
mean
Relationship between std
deviation and area under curve
Normal Distribution Example
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Need estimates of mean and standard
deviation and the Normal Table
Example :
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From past experience a manufacturer
concludes that the burnout time of a
particular light bulb follows a normal
distribution. Sample has been tested and
the average (x ) found to be 60 days with
a standard deviation () of 20 days. How
many bulbs can be expected to be still
working after 100 days.
Solution
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Problem is actually to find area under the curve beyond 100 days
Sketch Normal distribution and shade the area needed
Calculate z value corresponding to x value using formula
Z=(xi - )/ = (100-60)/20 = +2.00
Look in the Normal Table for z = +2.00 – gives area under curve as
0.9773
But, we want x >100 or z > 2.00. Therefore Area = 1.000 – 0.9773
= 0.0227, i.e. 2.27% probability that life of light bulb is > 100
hours
σ =20
0
μ = 60 100
x
Test For Normality
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To determine whether data is normal
Probability Plot - plot data on normal
probability paper
Steps
Order the data
Rank the observations
Calculate the plotting position
i= rank , n=sample size,
100(i  0.5)
PP 
PP= plotting position in %
n
4.
5.
6.
7.
Label data scale
Plot the points on normal probability paper
Attempt to fit by eye ‘best line’
Determine normality
Example