econ10005 lecture 3

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ECON10005

Quantitative Methods 1

Topic 2:What Do Skydiving & Stock Prices

Have In Common?

Lecture 3:Measures of Location & Variation

Reading: Chapters 4 (4.1, 4.2), 6 (249-50)

ECON10005 Measures of Location & Variation Lecture 3 - Slide 2

Today: Measures of Location & Variation

Measures oflocation, variationand association

Measures ofLocation

Lecture 3

Lecture 4 Measures ofAssociation

Mean, Median &Mode

Location &Skewness

Measures ofVariation

Variance &Standard Deviation

The Coefficient ofVariation

Covariance

Correlation


Measures ofLocation

Lecture 3


Mean, Median & Mode


Mean, Median &Mode

Location &Skewness




Covariance

Correlation

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Different Measures of Location

The term location refers to the centre of a data set. Thereare different ways of measuring this.

If a sample of size n is taken from a population X, with theobserved values x1, x2, , xn, the sample mean is theaverage of those values:

The median is the middle observation in an ordered array(or the average of the two middle observations if there is nosingle middle observation).

The mode is the most frequently occurring observation;there may more than one mode in a dataset.

ECON10005 Measures of Location & Variation

=+ + +

= =

n

i1 2 n i 1

xx x ... x

xn n

Lecture 3 -Slide 4

A Quick Example (from SSK)

7 research assistance have salaries (in $000s) of 42, 45,40, 46, 44, 40 and 43.

The mean is:

To find the median, order the observations from lowest tohighest: 40, 40, 42, 43, 44, 45, 46

The median is the middle observation: 43

The mode is the most frequent or common value: 40


ix 42 45 40 46 44 40 43x 42.857

n 7

+ + + + + += = =

Lecture 3 -Slide 5

The Population Mean

If we have a population of size N and we know all thepopulation data x1, x2,, xN, the population meanis:

A sample mean x is an estimate of the population mean .

Furthermore, x gives us the sampling error.

We expect that as n increases, the sample becomes morerepresentative of the population, and x converges on .


= =

N

ii 1

x

N

Lecture 3 -Slide 6

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Comparing Measures of Location

The mean is easy to calculate and is often used formaking inferences, but is sensitive to extremeobservations.

The median is not sensitive to extreme observations,and is sometimes used when the mean is an unhelpfulmeasure of location.

The mode is only really used when we are interested inknowing the most common outcome.



Measures ofLocation

Lecture 3


Location & Skewness


Mean, Median &Mode

Location &Skewness




Covariance

Correlation

Symmetric Distributions A Quick Example

Consider the following relative frequency distribution:

This distribution is symmetricabout its mean:

For this distribution, mean = median = mode = 4

The distribution is plotted on the next slide.


Observations 1 2 3 4 5 6 7

Frequencies 0.05 0.12 0.17 0.32 0.17 0.12 0.05

Lecture 3 -Slide 9

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One Possible Symmetric Distribution


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1 2 3 4 5 6 7

Frequency

Observation

Lecture 3 - Slide 10

Another Possible Symmetric Distribution


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

1 2 3 4 5 6 7

Frequency

Observation

This is a uniform distribution

Skewed Distributions - Summary

If a distribution is not symmetric then it is skewed.

If the right tail of a distribution is longer than the left then itis skewed to the right (right-skewed or positively skewed).

If the left tail is longer the distribution is skewed to the left(left-skewed or negatively skewed).

If a distribution is uni-modal then we can show that it is: Symmetrical if mean = median = mode

Right-skewed if mean > median > mode

Left-skewed if mode > median > mean


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Skewness in Household Income

For the household income data, we had mean > median >

mode, which implies the data was positively skewed.A histogram of the data generated using the steps fromLecture 2 supports this:


0

50

100

150

200

250

300

350

400

450

500

10000 30000 50000 70000 90000 110000130000150000170000190000 More

Frequency

Bins (Upper Limits)

Household Income , 2005, in dollars


Measures ofLocation

Lecture 3


Variance & Standard Deviation


Mean, Median &Mode

Location &Skewness




Covariance

Correlation

Measures of Variation

Observations may be widely dispersed about the averagevalue, or they may exhibit low variation and cluster tightlyabout the average value.

Consider the following sample data on the percentagereturns to different stocks in two portfolios, A and B:

The mean return for A is 14.

The mean return for B is also 14, but the returns are muchmore spread out around the mean value.


A 12 13 14 15 16

B 3 9 14 19 25


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Population Variance & Standard Deviation

Population variance measures an average of the squared

deviations between each observation and the populationmean.

For a population of size N with population mean , thepopulation variance, denoted 2, is:

The square root is known as the population standarddeviation:


( )N

22i

i 1

1x

N = =

( )N

22

ii 1

1x

N = = =


Sample Variance & Standard Deviation

If we do not have population data, we cannot determine thedeviation of each observation from the population mean.

We can take a sample of n observations (x1, x2,, xn) fromthe population and find the sample mean.

We can then measure the deviation of an observation from

the sample mean: (xi x).

We can then construct a sample statistic for variance andstandard deviation: estimates of the population parameters.


Sample Variance & Standard Deviation

Sample variance measures the average of the squareddeviations between each observation and the samplemean.

For a sample of size n with sample mean x, the samplevariance, denoted s2, is:

The square root is known as the sample standarddeviation:


( )n

22

ii 1

1s x x

n 1 ==

( )n

22i

i 1

1s s x x

n 1 == =


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Calculating Measures of Dispersion

Consider the data for portfolio B.

The mean is 14, so the deviations from the mean are:

These sum to zero. This is why we use squared deviations.


Observation 3 9 14 19 2 5

Deviation -11 -5 0 5 11

Squared Deviation 121 25 0 25 121

( )n

22

ii 1

1 121 25 0 25 121And so, s x x 73

n 1 4=

+ + + += = =

And so, s 73 8.54= =




Measures ofLocation

Lecture 3


The Coefficient of Variation


Mean, Median &Mode

Location &Skewness



Covariance

Correlation

Coefficient of Variation

The coefficient of variation measures the variation in asample (given by its standard deviation) relative to thatsamples mean.

Denoted CV, it is expressed as a percentage, providing aunit-free measure:

This lets us compare the relative variation in differentsamples without having to worry about differences in unitsor scale.


sCV 100 %

x

=


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Using Measures of Variation

Consider again the data on returns on two portfolios:

How might knowing the sample variance, standarddeviation and coefficients of variation for each portfoliohelp you decide which portfolio you would want toinvest in?


A 12 13 14 15 16

B 3 9 14 19 25


Comparing Coefficients of Variation

With a mean of 14 and a standard deviation of 8.54, forportfolio B, the coefficient of variation is:

You should be able to show that portfolio A has a meanof 14, a variance of 2.5, a standard deviation of 1.58and a coefficient of variation of around 11%.

How might this information help you decide whichportfolio you would want to invest in?


8.54CV 100 % 100(0.61)% 61%

14

= = =


Objectives for This Lecture

After attending this lecture and completing all the relatedreadings and questions you should be able to:

Distinguish between, calculate and interpret thepopulation and sample mean

Explain the relative merits of the mean and the median.

Use the relative position of the mean, median and modeto determine whether a distribution is symmetrical,positively skewed or negatively skewed

Distinguish between, calculate and interpret populationand sample variance and standard deviation

Explain the relative merits of the sample standarddeviation and the coefficient of variation

Calculate any of these measures using Excel

Quant itative Methods for Business Measures of Location & Variation Lecture 3 - S lide 24

8

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Measures ofLocation

Lecture 3


Next Time...

Measures oflocation, variation

and association

Mean, Median &Mode

Location &Skewness



Covariance

Correlation


Appendix: Some Tips on Using Excel

The lecture notes will often include some tips on using theAnalysis Tools in Excel. They are provided to help you getstarted with the computing and so will typically not bediscussed in lectures. They are best read when at acomputer so that you can try things yourself.

Because the Analysis Toolbox is not available in the Macversion of Excel, Microsoft recommend using the freelyavailable StatPlus. A primer for StatPlus is available on the

subjects LMS page.

Both programs produce similar output but users of StatPlusshould ensure that they know how to interpret the output ofExcel as seen in both the Lecture Notes and the textbook.

Measures of Location for Household Income

Recall the dataset from the last lecture for Australianhouseholds.

Suppose we want to calculate the mean, median andmode for household income (in column D).


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Excel Commands for Measures of Location

To obtain the mean, pick an empty cell and type the

following command:=AVERAGE(D2:D2001)

Press enter, and Excel will return the value 71,927.77

To obtain the median, pick an empty cell and type thefollowing command:

=MEDIAN(D2:D2001)

Press enter, and Excel will return the value 58,852

To obtain the mode in Excel, pick an empty cell andtype the following command:

=MODE(D2:D2001)

Press enter, and Excel will return the value 12,220


Measures of Location in Excels Toolpak

In the Data Analysis Toolpak, select DescriptiveStatistics.


Input Range and Summary Statistics

Give Excel the input range and check the box markedsummary statistics.


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Excel Output

Excel generates a range ofdescriptive statistics.

These include the mean,median and mode, whichare identical to the valuesreturned from entering directcommands.

Note this output alsocontains measures ofvariation such as the samplevariance and standarddeviation.


Excel and Measures of Variation

For the data on household income:

To obtain the sample variance, pick an empty cell andtype the following command:

=VAR(D2:D2001)

Press enter, and Excel will return the value3,851,046,073.33 (or 3.85E+09)

To obtain the samplestandard deviation, pick an emptycell and type the following command:

=STDEV(D2:D2001)

Press enter, and Excel will return the value 62,056.80

If you have population data and want the populationvalues, use the commands VARP or STDEVP instead.


Excel and the Coefficient of Variation

Excel doesnt generate the coefficient of variation aspart of its standard set of summary statistics.

However, as it gives both the mean and samplestandard deviation, it is still easy to calculate.

For the data on household income, the mean was71,927.77, and the standard deviation was 62,056.797.

This gives a coefficient of variation of approximately86.28%.


econ10005 lecture 3

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