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

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