Lecture 5: Graphical Techniques and Numerical Measures

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Lecture 5 Graphical Techniques and Numerical
Measures

• Professor Aurobindo Ghosh
• E-mail ghosh_at_galton.econ.uiuc.edu

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Graphical Excellence

• Graphical excellence deals with the effective
  use of graphical techniques.
• Effective graphical techniques are
• informative,
• concise,
• clear presentation of the data to the viewer.

How can we achieve graphical excellence?
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• Graphical excellence is achieved when
• The graph presents large data sets concisely and
  coherently.
• The ideas and concepts to be delivered are
  clearly understood to the viewer.
• The graph encourages the viewer to compare
  variables.
• The display induces the viewer to address the
  substance of the data and not the form of the
  graph.
• There is no distortion of what the data reveal.

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Graphical Deception

• It is important to be able to evaluate critically
  the information presented by graphical
  techniques.
• Things to be cautious about when observing a
  graph
• Is there a missing scale on one axis.
• Do not be influenced by a graphs caption.
• Are changes presented in absolute values only, or
  in percent form too.

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Is there a missing scale on one axis.
Are changes presented in absolute values only,
or in percent form too.
?
(3)
120.0
(2)
110.0
(1)
100.0
Time
Time
Has any axis been stretched?
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Dollars
10
Aug. 98
Sept. 98
1980
1985
1990
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Measures of Central Location

• Usually, we focus our attention on two aspects of
  measures of central location
• Measure of the central data point (the average).
• Measure of dispersion of the data about the
  average.

The central data point reflects the locations
of all the actual data points.
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Measures of Central Location (Central Tendency)

• Usually, we focus our attention on two aspects of
  measures of central location
• Measure of the central data point (the average).
• Measure of dispersion of the data about the
  average.

If the third data point appears exactly in the
middle of the current range, the
central location should not change (because it
is currently residing in the middle).
With two data points, the central location
should fall in the middle between them (in order
to reflect the location of both of them).
But if the third data point appears on the left
hand-side of the midrange, it should pull the
central location to the left.
With one data point clearly the central
location is at the point itself.
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Arithmetic mean

• This is the most popular and useful measure of
  central location

Sample mean
Population mean
Sample size
Population size
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• Example 4.1

The mean of the sample of six measurements 7, 3,
9, -2, 4, 6 is given by
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7
3
9
4
4.5
42.19
15.30
53.21
43.59
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• Example 4.3

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The median

• The median of a set of measurements is the value
  that falls in the middle when the measurements
  are arranged in order of magnitude.

Even number of observations
There are two middle values!
26,26,28,29, 30,32,60,31
29.5,
26,26,28,29,30,32,60,31
26,26,28,29, 30,32,60,31
26,26,28,29, 30,32,60,31
First, sort the salaries. Then, locate the value
in the middle
First, sort the salaries. Then, locate the values
in the middle
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The mode

• The mode of a set of measurements is the value
  that occurs most frequently.
• Set of data may have one mode (or modal class),
  or two or more modes.

For large data sets the modal class is much more
relevant than the a single- value mode.
The modal class
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• Example 4.5
• The manager of a mens store observes the waist
  size (in inches) of trousers sold yesterday 31,
  34, 36, 33, 28, 34, 30, 34, 32, 40.
• The mode of this data set is 34 in.

This information seems valuable (for example,
for the design of a new display in the store),
much more than the median is 33.2 in..
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• Example 4.6

A professor of statistics wants to report the
results of a midterm exam, taken by 100
students. The data appear in file XM04-06. Find
the mean, median, and mode, and describe the
information they provide.
The mean provides information about the over-all
performance level of the class. It can serve as
a tool for making comparisons with other
classes and/or other exams.
The Median indicates that half of the class
received a grade below 81, and half of the
class received a grade above 81.
The mode must be used when data is qualitative.
If marks are classified by letter grade, the
frequency of each grade can be calculated.Then,
the mode becomes a logical measure to compute.