İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
one. We call Figure 2 a probability distribution because if you choose an
M&M at random, the probability of getting, say, a brown M&M is equal
to the proportion of M&M's that are brown (0.30). 0 5 10 15 20 Brown
Red Yellow Green Blue Orange Frequency 41 F
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
It will serve to illustrate some features of all continuous distributions.
An example of a normal distribution is shown in Figure 4. Do you see
the bell? The normal distribution doesn't represent a real bell,
however, since the left and right tips extend
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
Figure 4 (their tails begin to hit the Xaxis further from the middle of the
curve for example, at 10 and 90 if drawn in place of Figure 4). Others
are less spread out (their tails might approach the Xaxis at 30 and 70).
More information on the normal d
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
two distributions that differ in this respect. The upper distribution has
relatively more scores in its tails; its shape is called leptokurtic. The
lower distribution has relatively fewer scores in its tails; its shape is
called platykurtic. Color Frequen
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
Let's say we have a variable X that represents the weights (in grams) of
4 grapes. The data are shown in Table 1. Table 1. Weights of 4 grapes.
We label Grape 1's weight X1, Grape 2's weight X2, etc. The following
formula means to sum up the weights of th
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
on a measure of shyness was 35 out of a possible 50, you would have
little idea how shy you are compared to other people. More relevant is
the percentage of people with lower shyness scores than yours. This
percentage is called a percentile. If 65% of the
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
color with a code of 3). Such nonsense arises because favorite color is a
nominal scale, and taking the average of its numerical labels is like
counting the number of letters in the name of a snake to see how long
the beast is. Does it make sense to compu
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
But numberofitems recalled is a more complicated case than it
appears at first. Consider the following example in which subjects are
asked to remember as many items as possible from a list of 10. Assume
that (a) there are 5 easy items and 5 difficult it
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
= 28 54 Linear Transformations by David M. Lane Prerequisites
None Learning Objectives 1. Give the formula for a linear
transformation 2. Determine whether a transformation is linear 3.
Describe what is linear about a linear transformation Often it is
ne
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
components of the CPI over time. Let us stress that it is misleading to
use a line graph when the Xaxis contains merely qualitative variables.
Figure 4 inappropriately shows a line graph of the card game data from
Yahoo, discussed in the section on quali
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
63 word or phrase like dear and read or funeral and real fun) or
scores on a vocabulary test? Questions from Case Studies Angry Moods
(AM) case study 13. (AM) Which variables are the participant variables?
(They act as independent variables in this study.
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
report should depend on how well different graphs reveal the aspects
of the data you consider most important. 15 20 25 30 Time F M 100 Bar
Charts by David M. Lane Prerequisites Chapter 2: Graphing Qualitative
Variables Learning Objectives 1. Create and in
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
such as gun or knife. The issue addressed by the experiment was
whether a preceding weapon word would speed up (or prime)
pronunciation of the aggressive word compared to a nonweapon
priming word. A positive difference implies greater priming of the
aggr
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
communicates the shape of the distribution. To provide experience in
constructing histograms, we have developed an interactive
demonstration (external link; Java required). The demonstration reveals
the consequences of different choices of bin width and o
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
simplify the table, we group scores together as shown in Table 1. Table
1. Grouped Frequency Distribution of Psychology Test Scores Interval's
Lower Limit Interval's Upper Limit Class Frequency 39.5 49.5 3 49.5 59.5
10 59.5 69.5 53 69.5 79.5 107 79.5 89.5
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
computer ownership. Comparing Distributions Often we need to
compare the results of different surveys, or of different conditions
within the same overall survey. In this case, we are comparing the
distributions of responses between the surveys or conditio
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
Computer 73 Figure 7. A line graph used inappropriately to depict the
number of people playing different card games on Sunday and
Wednesday. Summary Pie charts and bar charts can both be effective
methods of portraying qualitative data. Bar charts are bet
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
the number of buyers, yet Figure 5 is misleading because the viewer's
attention will be captured by areas. The areas can exaggerate the size
differences between the groups. In terms of percentages, the ratio of
previous Macintosh owners to previous Window
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
the 31 teams in the National Football League in the 2000 season. Table
1. Number of touchdown passes. A stem and leaf display of the data is
shown in Figure 1. The left portion of Figure 1 contains the stems. They
are the numbers 3, 2, 1, and 0, arranged
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
left). In the terminology of Chapter 3 (where we will study shapes of
distributions more systematically), the distribution is skewed. 79.5 89.5
147 320 89.5 99.5 130 450 99.5 109.5 78 528 109.5 119.5 59 587 119.5
129.5 36 623 129.5 139.5 11 634 139.5 149.
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
1000 2000 3000 4000 5000 6000 7000 111 Figure 4. An alternate way of
showing the number of people playing various card games on a Sunday
and on a Wednesday. The dot plot in Figure 4 makes it easy to compare
the days of the week for specific games while st
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
they are often useful for data sets with up to 200 observations. Figure 5
portrays the distribution of populations of 185 US cities in 1998. To be
included, a city had to have between 100,000 and 500,000 residents.
Figure 5. Stem and leaf display of popul
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
Box plots should be used instead since they provide more information
than bar charts without taking up more space. For example, a box plot
of the cursormovement data is shown in Figure 5. You can see that
Figure 5 reveals more about the distribution of m
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
correctly placed for the three chess positions. The maximum possible
score was 89. Student Dataset A Dataset B Dataset C You 3 3 3 John's 3
4 2 Maria's 3 4 2 Shareecia's 3 4 2 Luther's 3 5 1 125 Figure 1. Backtoback stem and leaf display. The left side s
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
information about a distribution. For example, a distribution with a
positive skew would have a longer whisker in the positive direction than
in the negative direction. A larger mean than median would also
indicate a positive skew. Box plots are good at p
İstanbul Kemerburgaz University  Mahmutbey Campus
Time Series Analysis
ECON 419

Spring 2017
made by the next three teams in the table. You can see for yourself
what the other rows represent. 32337 2001112223889 1
2244456888899 069 37 3233 2889 2001112223 156888899 1
22444 069 77 Figure 2. Stem and leaf display with the stems split in