Review
Distribution
The distribution of a variable describes WHAT values the variable takes
and HOW often it takes these values.
Stat 226 – Introduction to Business Statistics I
Spring 2009
Professor: Dr. Petrutza Caragea
Section A
Tuesdays and Thursdays 9:30-10:50 a.m.
Depending on the type of the data (categorical or quantitative) we need to
use different graphical and numerical tools to analyze and summarize the
data at hand.
We will start by describing data graphically:
Chapter 1, Section 1.1
bar graphs, pie charts and pareto charts can be used to graphically
summarize categorical data.
a common graphical display for quantitative data is a histogram.
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Introduction to Business Statistics I
Chapter 1, Section 1.1
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Introduction to Business Statistics I
Chapter 1, Section 1.1
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Chapter 1.1 – Displaying Distributions with Graphs
Strategies for data analysis:
Chapter 1.1 – Displaying Distributions with
Graphs
examine each variable in data set individually and look for possible
relationships among variables
start with graphs and add numerical summaries as needed
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Introduction to Business Statistics I
Chapter 1, Section 1.1
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Chapter 1, Section 1.1
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Chapter 1.1 – Displaying Distributions with Graphs
Chapter 1.1 – Displaying Distributions with Graphs
One of the first graphical displays is the so-called “coxcomb” by Florence
Nightingale:
The text in the lower left corner reads:
“The Areas of the blue, red, & black wedges are each measured from the center as the
common vertex.
The blue wedges measured from the center of the circle represent area for the deaths
from Preventable or Mitigable Zymotic diseases, the red wedges measured from the
center the deaths from wounds, & the black wedges measured from the center the
deaths from all other causes.
The black line across the red triangle in Nov. 1854 marks the boundary of the deaths
from all other causes during the month.
In October 1854, & April 1855, the black area coincides with the red, in January &
February 1855, the blue coincides with the black.
The entire areas may be compared by following the blue, the red, & the black lines
enclosing them.”
Source: Nightingale, Florence. Notes on Matters Affecting the Health, Efficiency and
Hospital Administration of the British Army, 1858.
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Chapter 1, Section 1.1
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Chapter 1.1 – Displaying Distributions with Graphs
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∗
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Chart
24.6
Example: data on loan amounts owed by Asia to several foreign loaners:
percentage
35.8
11.7
9.1
8.8
5
29.8
≈ 100∗∗
31.7
80.9
loan amount
loan amount∗
97.2
31.7
24.6
23.8
16.3
80.9
274.5
Chapter 1, Section 1.1
Chapter 1.1 – Displaying Distributions with Graphs
Categorical variables: Bar graphs and Pie charts
country
Japan
Germany
France
US
Great Britain
others
Total
Introduction to Business Statistics I
13.6
23.8
97.2
country
country
loan amount in Billions of Dollars,
due to rounding percentages will add up to 100.2
France
Japan
Germany
US
Great Britain
others
∗∗
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Chapter 1, Section 1.1
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Chapter 1.1 – Displaying Distributions with Graphs
Chapter 1.1 – Displaying Distributions with Graphs
Pie Chart
A pie chart shows the amount of data that belong to each category as a
proportional part of the circle
Chart
9.1%
useful when only one variable is of interest
11.7%
loan amount
29.8%
easy to compare: relative size of the parts to each other as well as the
size of each part compared to the total
5.0%
percentage have to add up to 100, i.e. we need to account for all
possible categories∗
8.8%
35.8%
∗
Assume, we did not have complete information on loan amounts for all
other countries: a pie chart can no longer be constructed because loan
amounts for Japan, Germany, France, US and Great Britain do not
account for 100% of loan amount given to Asia. (see also AYK problem
1.3, page 9)
country
France
Japan
country
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Germany
US
Great Britain
others
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Chapter 1, Section 1.1
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Chapter 1.1 – Displaying Distributions with Graphs
Chart
125
125
35.8%
50
9.1%
11.7%
8.8%
75
Japan
16.6%
9.1%
0
country
US
Japan
France
Germany
US
others
Japan
France
Germany
Great Britain
Introduction to Business Statistics I
Chapter 1, Section 1.1
country
Great Britain country
others
11.7%
France
12.5%
25
50
75
100
125
loan amount
country
Germany
US
5.0%
Germany
12.9%
0
France
Japan
35.8%
Great Britain
7.1%
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8.8%
50
25
0
29.8%
US
5.0%
country
others
50.9%
Great Britain
25
10 / 37
Bar graphs are valuable presentation tools as they are effective at
reinforcing differences in magnitude (note that bars have to be of
equal widths and are equally spaced)
Bar graphs are (obviously) useful when the observed outcomes of the
variable, our data, can be placed into different categories
Bar graphs can be either horizontal or vertical
country
75
Chapter 1, Section 1.1
Chart
100
29.8%
loan amount
loan amount
100
Introduction to Business Statistics I
Chapter 1.1 – Displaying Distributions with Graphs
Bar Graph
A bar graph shows the amount of data that belong to each category as
proportionally sized rectangular areas. They are more flexible than pie
charts because we don’t need to account for all possible categories of the
variable.
Chart
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France
Japan
Germany
US
Great Britain
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France
Japan
Germany
US
Great Britain
others
Introduction to Business Statistics I
Chapter 1, Section 1.1
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Chapter 1.1 – Displaying Distributions with Graphs
Chapter 1.1 – Displaying Distributions with Graphs
Chart
Comments:
125
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Chapter 1, Section 1.1
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75
50
25
Chapter 1.1 – Displaying Distributions with Graphs
Great Britain
US
France
Germany
Introduction to country
Business Statistics I
Stat 226 (Spring 2009, Section A)
country
others
0
Japan
Sometimes, however, it is more useful to arrange the bars with
respect to their magnitude, i.e. order them from tallest to shortest
in order to highlight the categories with the highest frequencies (most
important classes).
This type of bar graph is called a Pareto chart.
100
loan amount
Often classes have a “natural order”, so it makes sense to put the
bars in that order.
For example consider how many Fr, So, J, S are enrolled in this 226
section.
Japan
France
others
US
Chapter 1, Section 1.1
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Germany
Great Britain
Chapter 1.1 – Histograms
Example: Ages and annual salaries for the CEOs of the 60 top ranked small companies
in America in 1993. Source: Forbes, Nov. 8, 1993, ”America’s Best Small Companies,”
Quantitative Variables: Frequency Tables and Histograms
Distribution of Variable Age
Numerical data (observations are numbers rather than categories) are very
common in business. One of the most common ways to summarize
observations from a quantitative variable is a histogram.
Idea:
group values into “classes” (intervals) of equal width
count how many observations fall into each class
draw a bar graph for the counts (keeping the intervals in numerical
order, adjacent but non-overlapping)
30 35 40 45 50 55 60 65 70 75
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Chapter 1.1 – Histograms
Chapter 1.1 – Histograms
General Guidelines for Drawing a Histogram
Distribution of Salary in Thousands
all classes have same width
classes must not overlap, i.e. each observation can only belong to one
class
√
reasonable number of classes ( n for samples of size n ≤ 150
observations)
there is no “best choice” of class width and number of classes ⇒ use
good judgement to decide (see also next example for effect of class
width)
0
200
400
600
800
frequency f for each class corresponds to number of observations that
belong to that class
1000 1200
sum of all class frequencies must add to n, the size of the sample
Note: Classifying data into classes leads to a loss of information, always
be aware of that!!!
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Chapter 1, Section 1.1
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Chapter 1.1 – Histograms
Distributions
Age
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Chapter 1, Section 1.1
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Chapter 1.1 – Histograms
Distributions
Distributions
Effect
of class width Age
Age
Different shapes of distributions/histograms
We distinguish the following main shapes of distributions
symmetric
30 34 38 42 46 50 54 58 62 66 70 74 78
30 36 42 48 54 60 66 72 78
30 35 40 45 50 55 60 65 70 75
skewed to the right
skewed to the left
Distributions
Age
Distributions
Distributions
Age
Age
uniform/rectangular
J-shaped
35 40 45 50 55 60 65 70 75
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40
50
60
70
80
Introduction to Business Statistics I
30
45
60
in addition we distinguish between bimodal and multi-modal
distributions as opposed to uni-modal distributions.
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Chapter 1.1 – Histograms
Chapter 1.1 – Histograms
Example: Boston Housing Data - housing values for Boston suburbs, 506
observations on 14 different variables
Example: Boston Housing Data cont’d — different shapes of distributions
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
variable
Crime rate
Residential zone
nonretail
river
NO concentration
#of rooms
age
distance
Access to higway
taxrate
pt ration
Black people
Lower status
M value of home
explanation
per capita crime rate
proportion of residential land zoned for large lots
proportion of nonretail business acres
Charles river
nitric oxides concentration
average number of rooms per dwelling
proportion of owner-occupied units built prior to 1940
weighted distances to five Boston employment centers
index of accessibility to radial highways
full-value property tax-rate per $ 10,000
pupil--teacher ratio
1000(B - 0.63)^2I(B less than 0.63)where B is the proportion of blacks
% lower status of the population
median value of owner-occupied homes in $1000
type
quantitative
quantitative
quantitative
categorical
quantitative
quantitative
quantitative
quantitative
quantitative
quantitative
quantitative
quantitative
quantitative
quantitative
NO concentration
# of rooms
value of home
comments
1 if tract bounds river, 0 otherwise
.4
.5
.6
.7
.8
5
Tax rate
Bar chart for the only categorical variable (variable 4)
6
7
8
10
lower status
20
30
40
50
black people
400
N
300
200
100
200
0
0
300
400
500
600
700
0
5
10
15
20
25
30
35
50
100 150 200 250 300 350 400
1
chas
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0
1
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Chapter 1.1 – Histograms
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Introduction to Business Statistics I
140
150
Histogram of Amount Spent on Most Recent Haircut
100
100
80
60
possible outliers
Number of Students
4
40
spread
0
0
20
3
50
center
Number of Students
2
120
Describing distributions/histograms
overall shape of histogram
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Chapter 1.1 – Histograms
Histogram of Time Spent Sleeping the Night
before Classes Began
1
Chapter 1, Section 1.1
0
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4
6
8
10
12
0
20
Sleep Time in Hours
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40
60
80
100
Amount Spent in Dollars
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Chapter 1, Section 1.1
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Chapter 1.1 – Stemplots
Chapter 1.1 – Stem-and-Leaf Plot (Stemplot)
Example: random sample of package design ratings ranging from 0 - 45:
shows the actual digits
Each numerical value is divided into two components
22 21 31 20 25 21 32 26 43 30 27 30 27 36 28 33 38 35 19 30 34 41
leading digits – stem
trailing digits – leaf
How to make a stem-and-leaf plot:
1
Separate each observation into a stem consisting of all but the last (most right)
digit and a leaf (the last digit). Stems may have as many digits as needed, but
each leaf contains only a single digit.
2
Write stems in a vertical column with smallest at the top, then draw a vertical line
at the right if this column
3
Write each leaf in the row to the right of its stem, in increasing order out from
the stem
4
add legend on how to read stem plot
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Chapter 1, Section 1.1
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Introduction to Business Statistics I
Chapter 1.1 – Stemplots
Chapter 1.1 – Stemplots
Split Stem-and Leaf Plot:
Back-to-Back Stem-and Leaf Plot:
Sometimes it might be more informative to have a so-called Split
Stem-and Leaf Plot where we split the stem further, namely into two
groups for the leading digits: 0 to 4 and 5 to 9.
Stat 226 (Spring 2009, Section A)
Introduction to Business Statistics I
Chapter 1, Section 1.1
411
555
4210
99885
3322110
98776655
100
876
4
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2
3
3
4
4
5
5
6
6
7
7
Chapter 1, Section 1.1
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234
5589
023
566788
0023345
55679
0134
68
24
5
Introduction to Business Statistics I
Chapter 1.1 – Time Plots
Chapter 1.1 – Time Plots
A trader at the stock exchange in Frankfurt.
Photo from NY Times, January 21, 2008
Stocks Plunge in Europe and Asia on U.S. Recession Fear
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Chapter 1.1 – Time Plots
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Chapter 1, Section 1.1
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Chapter 1.1 – Time Plots
Time Series average number of occupied rooms By month
average number
of occupied rooms
Time Plots
A time-plot displays data that are observed over a given period of
time.
Helps us understand the behavior of the data over time.
1100
1000
900
800
700
600
500
0
Always put time on the horizontal axis of your plot and the variable of
interest in the vertical axis
12
24
36
48
60
72
84
96 108 120 132 144 156 168
month
important features:
trend
seasonal variation
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Chapter 1.1 – Time Plots
Chapter 1.1 – Time Plots
A trend in a time series refers to a long-term, persistent rise or fall,
i.e. a positive (upward) or negative (downward trend)
A pattern in a time series that repeats itself at known regular
intervals is called seasonal variation.
Time Series average number of occupied rooms By month
1100
average number
of occupied rooms
average number
of occupied rooms
Time Series average number of occupied rooms By month
1000
900
800
700
600
1100
1000
900
800
700
600
500
500
0
12
24
36
48
60
72
84
0
96 108 120 132 144 156 168
12
24
36
48
60
72
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Chapter 1, Section 1.1
84
96 108 120 132 144 156 168
month
month
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Chapter 1.1 – Time Plots
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Chapter 1.1 – Time Plots
The southern oscillation is defined as the barametric pressure difference between Tahiti and the Darwin Islands at
sea level.
The southern oscillation is a predictor of El Nino which in turn is thought to be a driver of world-wide weather.
Specifically, repeated southern oscillation values less than -1 typically defines an El Nino.
Time Series Sunspots By Year
Mean
Std
N
150
46.93
37.17775
100
Time Series Oscillation By Year
Mean -0.103509
Std
1.0898398
N
456
2
Oscillation
1
100
Sunspots
3
50
0
-1
0
-2
-3
1770
1780
1790
1800
1810
1820
1830
1840
1850
1860
1870
Year
-4
1954 1956 1958 1960 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994
Year
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Chapter 1, Section 1.1
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Chapter 1.1 – Time Plots
Time Series number of employed persons in thousands By date
Mean 138862.22
Std
3761.5691
N
103
number of employed
persons in thousands
145000
140000
135000
01/2007
01/2006
01/2005
01/2004
01/2003
01/2002
01/2001
01/2000
01/1999
130000
date
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Chapter 1, Section 1.1
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