ISE 390
Chapter 1
Introduction
Spring 2005
Probability and Statistics for Modern Engineering , Second Edition, Lawrence L. Lapin
Class Objectives
• Develop an understanding of the terms used in the study of Statistics.
• Understand the difference in populations and samples.
• Demonstrate capability to collect, organize, present, analyze, and
interpret numerical data.
• Understand Statistical Measures used to describe populations and
samples including:
Location
Variability
•
Central Tendency
•
Range
•
Mean
•
Variance
•
Median
•
Standard Deviation
•
Mode
•
Coefficient of Variation
• Position
• Percentiles, Fractiles, and
Quartiles
Definitions
• Statistic (Singular) – A data point (Boeing closed at $45.00.)
• Statistics (Plural) – The Dow was down 25 points today. (Aggregate of
30 stock
prices)
• Statistics (Discipline) - The collection, organization, presenting,
analyzing, and interpreting of numerical data for the purpose of
providing information with which to reach a decision or communicate
information in the face of uncertainty.
• Statistical Population – The collection of ALL possible members of a
specific group of interest.
• Sample – A subset of the statistical population used to represent the
population from which information is inferred regarding the
population.
• Quantitative Population – Observations are numerical values.
• Qualitative Population – Observations are attributes described in
terms of color, shape, sex, etc.
Types of Quantitative Data
• Ratio Data – Operations of addition, subtraction,
multiplication, and division are applicable, Examples
include time or physical data such as minutes, height,
weight, etc.
• Interval Data – Only addition or subtraction is
applicable. Temperature scales are examples.
• Ordinal Data – No arithmetic operations apply.
Some relative value is associated with the scale. This
includes scales such as hurricane or tornado ratings or
the Beaufort wind scale.
• Nominal Data – Numbers that represent arbitrary
code. No relative meaning. Codes for offices or colors
are examples.
Analysis of Data
Sample Times to Inspect Test Devices for Calibration
Frequency Distribution
Number of
Inspections
30
20
10
0
10
15
Histogram
20
Frequency Polygon
Frequency Curve
Stem and Leaf Plot
Frequency
2
16
29
27
11
6
4
2
2
1
100
Stem
11
12
13
14
15
16
17
18
19
20
Leaf (Decimal)
7 3
8 9 6 7 6
7 0 1 0 5
5 2 7 3 8
0 6 2 5 7
9 1 8 8 7
9 0 0 3
9 5
0 1
1
3
6
4
3
0
9
3
2
7
5
0
3
2
2
3
5
6
9
9
6
5
7 8 0 9 8 9
0 0 1 2 1 6 7 0 2 3 0 1 9 8 2 3 2 2 7
8 6 2 0 3 4 7 7 7 2 2 2 2 0 2 7 3
4
Advantage Over Other Displays of Data: No data is lost.
Relative and Cumulative Frequency Distributions
Music preferences in 200 young adults 14 to 19
Rap
Alternative
Rock and Roll
Country
Classical
100
50
26
20
4
The pie chart quickly tells you that
• the majority of students like rap best (50%), and
• the remaining students prefer alternative (25%), rock and roll (13%),
country (10%) and classical (2%).
Tip! When drawing a pie chart, ensure that the segments are ordered by size
(largest to smallest) and in a clockwise direction.
Source (or "Adapted from", if appropriate): Statistics Canada's Internet Site, full URL of source pages, and date of extraction.
Statistics Canada information is used with the permission of Statistics Canada. Users are forbidden to copy the data and redisseminate them, in an original or modified
form, for commercial purposes, without the expressed permission of Statistics Canada. Information on the availability of the wide range of data from Statistics Canada
can be obtained from Statistics Canada's Regional Offices, its World Wide Web site at http://www.statcan.ca, and its toll-free access number 1-800-263-1136.
Ogive for Cumulative Frequency Distribution
Normal Probability Distributions
x
Probability Density Function
a
b
Cumulative Probability Density
Function
SPSS Scatter Diagram for Concrete Strength vs. Pulse
Velocity
Scatter Diagram
Scatter Diagrams
Statistical Measures
Population descriptors are called Parameters
Sample descriptors are called Sample Statistics
Location
Variability
• Central Tendency
• Range
•
Mean
• Variance
•
Median
• Standard Deviation
•
Mode
• Coefficient of Variation
• Position
• Percentiles, Fractiles,
and Quartiles
Central Tendency – Mean
Population Mean
N = number of members in population
N
m
x
i
i 1
Sample Mean Approximated
from Grouped Data
N
Sample Mean - X
n = number of
members in sample
n
X
m (lower case Greek letter Mu)
x
i 1
n
i
Central Tendency – Median and Mode
Median – That value above and below which an
equal number of observations lie. Denoted by m
584
633
693
663
755
In the case where there is an even number of samples or observations,
the Median is the average of the middle two samples.
584
633
640
680
693
755
The Median is (680 + 640)/2 = 660.
Mode – The most frequently occurring value
0 0 3 7 1
0 2 10 27 15
0 is the Mode
0 0 1 2
Where the data is taken from a continuum, the Mode is taken to be the
midpoint of the class interval with the highest frequency.
Frequency Distribution Forms and Summary Measures
Positional Comparisons of the Three Measures of Central Tendency for
Symmetrical and Skewed distributions
Bimodal Frequency Distributions
Using Stem and Leaf Plot to Find Median and Mode
Frequency
2
16
29
27
11
6
4
2
2
1
100
Stem
11
12
13
14
15
16
17
18
19
20
Leaf (Decimal)
3 7
0 2 3 5 6
0 0 0 0 0
0 0 2 2 2
0 2 2 3 4
0 1 7 8 8
0 0 3 9
5 9
0 1
1
6
0
2
5
9
7
0
2
5
7
1
2
6
8
1
2
6
8
1
2
7
8 9 9 9 9 9
1 2 2 2 2 2 3 3 3 3 5 6 6 7 7 7 8 9 9
3 3 3 3 4 4 5 5 6 6 7 7 7 7 7 8 8
7
Median – The median will be between the 50th and 51st observation.
Adding 2 + 16 + 29 = 47, we know that the Median will be on the stem for
14. Counting in on the 14 stem, we find that the 50th and 51st observations
are both 14.2. The Median is (14.2 + 14.2)/2 = 14.2.
Mode – Look for the fraction that occurs most frequently on one stem.
In this data, that is the 2 in the 14 stem. The Mode is 14.2.
Finding Percentiles from Ungrouped Data
1.
Sort the data in ascending order.
2.
Establish the decimal allowable range.
1
n 1
d 
n
n
1
10  9
d 
10
10
.1 < d < .9
(d = decimal equivalent of desired percentile; n = sample size)
3.
Find the relative position of the desired percentile. ( n + 1) d (10 + 1) .1
Let k be the largest integer such that:
k < ( n + 1) d
1 < (1.11) k = 1
The desired percentile will lie between Xk and Xk + 1 or X1 and X2
4.
Compute the percentile value.
Qd = Xk + [( n + 1) d - k](Xk + 1 – Xk) = 5.3 + [(10 + 1) .1 – 1] (5.4 – 5.3)
Q.1 = 5.3 + [(11).1 – 1](.1) = 5.3 + (.1)(.1) = 5.31
k < (11) .25 = 2.75 k = 2
Q.25 = 5.4 + [(11).25 – 2](5.7 - 5.4) = 5.4 + (0.75)(.3) = 5.4 + .225 = 5.63
k < (11) .5 = 5.5 k = 5
Q..50 = 6.10 + [(11).5 – 5](6.1 – 6.1) = 6.10 + (0.5)(0) = 6.10 + 0 = 6.10
k < (11) .75 =8. 25 k = 8
Q.75 = 6.4 + [(11).75 – 8](6.5 - 6.4) = 6.4 + (0. 25)(.1) = 6.4 + .025 = 6.43
1
2
3
4
5
6
7
8
9
10
5.3
5.4
5.7
6.0
6.1
6.1
6.2
6.4
6.5
6.6
Percentiles, Fractiles, and Quartiles
Percentile – The point below which the stated percentage lies.
Example: 17 of 20 observations fall below 23. What percentile is 23? 17/20
= .85. 23 is the 85th percentile.
Fractile - The point below which the stated fraction lies.
Example: 17 of 20 observations fall below 23. What fractile is 23? 17/20 =
.85. 23 is the .85-fractile.
Quartile – Divides data into four groups of equal frequency
First Quartile – Same as the 25th percentile (.25-fractile)
Second Quartile - Same as the 50th percentile (.50-fractile) and also the
Median.
Third Quartile - Same as the 75th percentile (.75-fractile)
Finding Percentiles from Ungrouped Data
Frequency
2
16
29
27
11
6
4
2
2
1
100
Stem
11
12
13
14
15
16
17
18
19
20
Leaf (Decimal)
3 7
0 2 3 5 6
0 0 0 0 0
0 0 2 2 2
0 2 2 3 4
0 1 7 8 8
0 0 3 9
5 9
0 1
1
6
0
2
5
9
7
0
2
5
7
1
2
6
8
1
2
6
8
1
2
7
8 9 9 9 9 9
1 2 2 2 2 2 3 3 3 3 5 6 6 7 7 7 8 9 9
3 3 3 3 4 4 5 5 6 6 7 7 7 7 7 8 8
7
First Quartile
(100 + 1) .25 = 25.25 Q.25 = 13.0 + (25.25 – 25.0)(13.1 – 13.0) = 13.025
Second Quartile
(100 + 1) .5 = 50.5 Q.5 = 14.2 + (50.5 – 50.0)(14.2 – 14.2) = 14.2
Third Quartile
(100 + 1) .75 = 75.75 Q.75 = 15.0 + (75.75 – 75.0)(15.2 – 15.0) = 15.15
Fourth Quartile
Finding Percentiles Graphically
14.2
Q.50
Finding Percentiles from Grouped Data
(Raw data unavailable, but grouped into frequency distribution)
Qd = Xk + [( n + 1) d - k]
(
Xk + 1 – Xk
h-k
)
(73 + 1)(.25) = 18.5
The greatest cumulative frequency not exceeding 18.5 is 7 for the first interval , so
k = 7. Q.25 will fall in the second interval, and its cumulative frequency is 30.
105.0 – 100.0
Q.25 = 100.0 + [( 18.5 - 7]
= 102.5
30 - 7
(
)
Variability*
*The term “Dispersion” is also used.
Variability – Interquartile Range
(Difference in the third quartile, Q.75 and the first quartile, Q.25)
Boxplot
Range
Interquartile Range
Interquartile Range = 6.43 – 5.63 = 0.80
From Slide 16; Q.25 = 5.63, Q.50 = 6.10, Q.75 = 6.43
1
2
3
4
5
6
7
8
9
10
5.3
5.4
5.7
6.0
6.1
6.1
6.2
6.4
6.5
6.6
Variance and Standard Deviation
(Measures deviation of individual observations around the mean.)
Variance
Standard Deviation
N
Population
 – Lower case Greek
letter Sigma
2 
 (x  m 
2
i
i 1
N
 (x  X 
2
n
Sample
s2 
i
i 1
Approximated
from Grouped
Data
s2 
s
2
s  s2
n 1
n
Calculation
by Hand
  2
x
2
 nX
2
s  s2
i 1
f


n 1
k
x  nX
2
k
n 1
2
s  s2
The sum of the deviations of the samples about the sample mean = 0.
Composite Summary Measures
Kurtosis – Measures the
“peakedness”of a distribution
Coefficient of Variation

v

Population –
m
s
Sample - v 
X
Coefficient of Skewness –
Provides direction and
degree of skewness of
frequency distribution
(
3 Xm
SK 
s

A.
Which has the greatest Coefficient of
Variation?
B.
Mesokurtic – Meso means intermediate.
C.
Leptokurtic – Lepto means slender.
D.
Platykurtic – Platy means flat.
m = Median
SK > 0, Positively Skewed
SK < 0, Negatively Skewed
SK = 0, No Skew
Proportion
Population Proportion

Number of Observatio ns in Category
Population Size
Sample Proportion
P
Number of Observatio ns in Category
Sample Size