Measure of Central Tendency TOT 2
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Transcript of Measure of Central Tendency TOT 2
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ECO 72 - INTRODUCTION TO ECONOMIC STATISTICS
Topic 2
Measures of
Central TendencyThese slides are copyright 2003 by Tavis Barr. This material may be distributed only subject
to the terms and conditions set forth in the Open Publication License, v1.0 or later (the latestversion is presently available at http://www.opencontent.org/openpub/).
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Measures of Central TendencyThis chapter looks at three different concepts of how wedescribe a typical element of a data set.
Mean Median Mode
There is no one best concept for all cases; we willdiscuss the advantages and disadvantages of each.
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Mean
The meanis what is most
commonly called theaverage.
If a population is finite, ofsize N, we can write thepopulation mean as
EXAMPLE:
There are three countries in
North America (N=3)
Their land areas are:
Canada 9,093,507 km2
Mexico 1,923,040 km2
U.S. 9,161,923 km2
Total 2,017,840 km2
Average land area:
2,017,840/3 = 6,726,157 km2
Source: 2005 CIA World Factbooki=1
i=NX
i
N=X
1X
2X
N
N
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Mean sample mean For a sample of size
n, we can write sample
mean as
Example:
Ten people are asked how
many hours of TV theywatched last night.
Their responses are 1, 2, .5,0, 4, 0, 2, 1.5, 0, 3.
Mean:i=1
i=nX
i
n=
X1X
2X
n
n1+2+0.5+4+2+1.5+3
10=1.4
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Advantages of the sample mean1.It takes all values in the sample into account.
2.It is unique: Each sample and population has only
one mean.
3.The sum of X minus the mean is zero, so the
mean acts as a balancing point.
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Disadvantages of the Mean
1. It only exists for quantitative data
What is the mean between good, fair, poor? Between red, yellow, and blue?
2. It can be affected strongly by outliers.
Example: In Whoville, there are 10 people who earn$10,000 a year and one person who earns $1,000,000
What is the mean? Is it a typical income?
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Weighted MeanWeighted meansoccur when we have someobservations that we wish to place more importance on
than others.
They require a weighting variablethat indicates the
importance to place on a given observation.
We denote the original variable by Xiand the weighting
variable by Wi.
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Weighted Mean Formula
Formula for the weighted mean:
i=1
i=n
WiX
i
i=1
i=n
Wi
=
W1X
1W
2X
2W
nX
n
W1W
2W
n
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Weighted Mean ExampleLife Expectancy in a group of northern African countries:
Country Life ExpectancyAlgeria 68
Egypt 66Libya 73
Morocco 67
Nigeria 41
Sudan 55Tunisia 72
Sum 442
Mean 442/7 = 63.14
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Weighted Mean Example (cont'd)Life Expectancy in a group of northern African countries:
Country Life Expectancy Population (mil)Algeria 68 31
Egypt 66 70
Libya 73 5Morocco 67 29
Nigeria 41 126
Sudan 55 31Tunisia 72 10
Sum 442 302
Mean 442/7 = 63.14
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Weighted Mean Example (cont'd)Life Expectancy in a group of northern African countries:
Country Life Expectancy Population (mil) LE x Popn
Algeria 68 31 2108Egypt 66 70 4620Libya 73 5 365
Morocco 67 29 1943
Nigeria 41 126 5166
Sudan 55 31 1705
Tunisia 72 10 720
Sum 442 302 16627
Mean 442/7 = 63.14
Weighted Mean: 16627/302 = 55.05
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Median Looks at midpoint of data when they are sorted
from highest to lowest.
If even number of observations, take average oftwo midpoints.
Example: Hours of television watched, sorted:
0, 0, 0, .5, 1, 1.5, 2, 2, 3, 4
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Median Looks at midpoint of data when they are sorted
from highest to lowest.
If even number of observations, take average oftwo midpoints.
Example: Hours of television watched, sorted:
0, 0, 0, .5, 1, 1.5, 2, 2, 3, 4
Median: (1 + 1.5)/2 = 1.25
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Advantages of Median1. Works on ordered data as well as quantitativedata.
Example: 20 Opinions of Hillwood CafeExcellent: 3
Good: 6
Fair: 7
Poor: 4
Pick midpoint from: Poor, Poor, Poor, Poor, Fair, Fair, Fair, Fair, Fair,
Fair, Fair, Good, Good, Good, Good, Good, Good, Excellent,
Excellent, Excellent.
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Advantages of Median (cont'd)2. Median is unaffected by outliers:
There are 11 people in Whoville; ten make $10,000 per
year and one makes $1 million per year. What is themedian income?
3. Median is unique: A sample has only one.
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Disadvantage of MedianDisadvantage: Not affected by changes in data away
from center.
Example: In Whoville, what would happen to themedian income if the millionaire suddenly started
making only $25,000?
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Mode Asks which value is observed most.
Example: 20 Opinions of Hillwood Cafe
Excellent: 3
Good: 6
Fair: 7
Poor: 4
Here, Fair is the most common response.
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Mode Advantage Advantage: Works on category data.
Example: Ethnic groups in Ethiopia (millions)
Amharic/Tigray 22.6
Oromo 28.2
Shankella 4.2
Sidamo 6.3
Somali 4.2
Other 4.9
Source: 2005 CIA World Factbook
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Disadvantages of Mode May not be unique. Consider the following sample
of 10 people
Favorite Flavor of Ice Cream # of people
Vanilla 1
Chocolate 4Strawberry 4
Coffee 1
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Disadvantages of Mode (cont'd) May not even exist in a meaningful way on
continuous data. Consider life expectancy data:
Country Life ExpectancyAlgeria 68Egypt 66
Libya 73
Morocco 67
Nigeria 41
Sudan 55Tunisia 72
One could say that every value is a mode, or that
none is.
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Disadvantages of Mode (cont'd) May not lie near the center of the data at all in
ordered data. Consider our answers about how
many hours of television people watched lastnight:
0, 0, 0, .5, 1, 1.5, 2, 2, 3, 4
The modal response is not a typical one.
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Geometric Mean Used when looking at growth rates.
For example, economic growth, interest rates,
population growth.
Asks what growth rate, if it were constant each
year, would get you from starting value to ending
value
We won't use it in this class, but keep it in mind ifyou're working with time-series data such as
financial data