Economics 173 Business Statistics

1
Economics 173Business Statistics

• Lecture 2
• Fall, 2001
• Professor J. Petry
• http//www.cba.uiuc.edu/jpetry/Econ_173_fa01/

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Numerical Descriptive Measures

• Measures of central location
• arithmetic mean, median, mode, (geometric mean)
• Measures of variability
• range, variance, standard deviation, (coefficient
  of variation)
• Measures of association
• covariance, coefficient of correlation

3
Arithmetic mean
Measures of Central Location

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

Sample mean
Population mean
Sample size
Population size
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• Example

The mean of the sample of six measurements 7, 3,
9, -2, 4, 6 is given by
6
7
3
9
4
4.5

• Example

Calculate the mean of 212, -46, 52, -14, 66
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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
First, sort the salaries. Then, locate the values
in the middle
First, sort the salaries. Then, locate the value
in the middle
There are two middle values!
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
26,26,28,29, 30,32,60,31
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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.

The modal class
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• Example
• 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.
• What is the modal value?

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Relationship among Mean, Median, and Mode

• If a distribution is symmetrical, the mean,
  median and mode coincide

• If a distribution is non symmetrical, and skewed
  to the left or to the right, the three measures
  differ.

A positively skewed distribution (skewed to the
right)
Mode
Mean
Median
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• If a distribution is symmetrical, the mean,
  median and mode coincide

• If a distribution is non symmetrical, and skewed
  to the left or to the right, the three measures
  differ.

A negatively skewed distribution (skewed to the
left)
A positively skewed distribution (skewed to the
right)
Mean
Mode
Mean
Mode
Median
Median
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Measures of variability(Looking beyond the
average)

• Measures of central location fail to tell the
  whole story about the distribution.
• A question of interest still remains unanswered

How typical is the average value of all the
measurements in the data set?
or
How spread out are the measurements about the
average value?
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Observe two hypothetical data sets
Low variability data set
The average value provides a good representation
of the values in the data set.
High variability data set
This is the previous data set. It is now
changing to...
The same average value does not provide as good
presentation of the values in the data set as
before.
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The range

• The range of a set of measurements is the
  difference between the largest and smallest
  measurements.
• Its major advantage is the ease with which it can
  be computed.
• Its major shortcoming is its failure to provide
  information on the dispersion of the values
  between the two end points.

But, how do all the measurements spread out?
The range cannot assist in answering this question
Range
Largest measurement
Smallest measurement
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The variance

• This measure of dispersion reflects the values of
  all the measurements.
• The variance of a population of N measurements
  x1, x2,,xN having a mean m is defined as
• The variance of a sample of n measurementsx1,
  x2, ,xn having a mean is defined as

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Consider two small populations Population A 8,
9, 10, 11, 12 Population B 4, 7, 10, 13, 16
9-10 -1
11-10 1
8-10 -2
12-10 2
Thus, a measure of dispersion is needed that
agrees with this observation.
Let us start by calculating the sum of deviations
The sum of deviations is zero in both
cases, therefore, another measure is needed.
A
10
9
8
11
12
but measurements in B are much more
dispersed then those in A.
The mean of both populations is 10...
4-10 - 6
16-10 6
B
7-10 -3
13-10 3
7
4
10
13
16
15
9-10 -1
The sum of squared deviations is used in
calculating the variance.
11-10 1
8-10 -2
12-10 2
The sum of deviations is zero in both
cases, therefore, another measure is needed.
A
10
9
8
11
12
4-10 - 6
16-10 6
B
7-10 -3
13-10 3
7
4
10
13
16
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Let us calculate the variance of the two
populations
Why is the variance defined as the average
squared deviation? Why not use the sum of squared
deviations as a measure of dispersion instead?
After all, the sum of squared deviations
increases in magnitude when the dispersion of a
data set increases!!
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Which data set has a larger dispersion?
Let us calculate the sum of squared deviations
for both data sets
However, when calculated on per observation
basis (variance), the data set dispersions are
properly ranked
Data set B is more dispersed around the mean
A
B
1
3
1
3
2
5
SumA (1-2)2 (1-2)2 (3-2)2 (3-2)2 10
5 times
5 times
!
SumB (1-3)2 (5-3)2 8
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• Example
• Find the mean and the variance of the following
  sample of measurements (in years).
• 3.4, 2.5, 4.1, 1.2, 2.8, 3.7
• Solution

A shortcut formula
1/53.422.523.72-(17.7)2/6 1.075 (years)
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• The standard deviation of a set of measurements
  is the square root of the variance of the
  measurements.
• Example
• Rates of return over the past 10 years for two
  mutual funds are shown below. Which one have a
  higher level of risk?
• Fund A 8.3, -6.2, 20.9, -2.7, 33.6, 42.9, 24.4,
  5.2, 3.1, 30.05
• Fund B 12.1, -2.8, 6.4, 12.2, 27.8, 25.3, 18.2,
  10.7, -1.3, 11.4

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• Solution
• Lets use the Excel printout that is run from the
  Descriptive statistics sub-menu

Fund A should be considered riskier because its
standard deviation is larger
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Interpreting Standard Deviation

• The standard deviation can be used to
• compare the variability of several distributions
• make a statement about the general shape of a
  distribution.
• The empirical rule If a sample of measurements
  has a mound-shaped distribution, the interval

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• Example
• The duration of 30 long-distance telephone calls
  are shown next. Check the empirical rule for the
  this set of measurements.

• Solution
• First check if the histogram has an
  approximate
• mound-shape

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• Calculate the mean and the standard deviation
• Mean 10.26 Standard deviation 4.29.

• Calculate the intervals

Interval Empirical Rule Actual
percentage 5.97, 14.55 68 70 1.68,
18.84 95 96.7 -2.61, 23.13 100 100
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Measures of Association

• Two numerical measures are presented, for the
  description of linear relationship between two
  variables depicted in the scatter diagram.
• Covariance - is there any pattern to the way two
  variables move together?
• Correlation coefficient - how strong is the
  linear relationship between two variables

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The covariance
mx (my) is the population mean of the variable X
(Y) N is the population size. n is the sample
size.
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• If the two variables move the same direction,
  (both increase or both decrease), the covariance
  is a large positive number.

• If the two variables move in two opposite
  directions, (one increases when the other one
  decreases), the covariance is a large negative
  number.
• If the two variables are unrelated, the
  covariance will be close to zero.

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The coefficient of correlation

• This coefficient answers the question How strong
  is the association between X and Y.

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Strong positive linear relationship
1 0 -1
COV(X,Y)gt0
or
r or r
No linear relationship
COV(X,Y)0
Strong negative linear relationship
COV(X,Y)lt0
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• If the two variables are very strongly positively
  related, the coefficient value is close to 1
  (strong positive linear relationship).
• If the two variables are very strongly negatively
  related, the coefficient value is close to -1
  (strong negative linear relationship).
• No straight line relationship is indicated by a
  coefficient close to zero.

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• Example
• Compute the covariance and the coefficient of
  correlation to measure how advertising
  expenditure and sales level are related to one
  another.

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• Use the procedure below to obtain the required
  summations

Similarly, sy 8.839
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• Excel printout
• Interpretation
• The covariance (10.2679) indicates that
  advertisement expenditure and sales level are
  positively related
• The coefficient of correlation (.797) indicates
  that there is a strong positive linear
  relationship between advertisement expenditure
  and sales level.

Covariance matrix
Correlation matrix