econ10005 lecture 3
Post on 03-Apr-2018
213 Views
Preview:
TRANSCRIPT
-
7/28/2019 ECON10005 Lecture 3
1/11
1
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
ECON10005 Measures of Location & Variation Lecture 3 - Slide 3
Measures ofLocation
Lecture 3
Lecture 4 Measures ofAssociation
Mean, Median & Mode
Measures oflocation, variationand association
Mean, Median &Mode
Location &Skewness
Measures ofVariation
Variance &Standard Deviation
The Coefficient ofVariation
Covariance
Correlation
-
7/28/2019 ECON10005 Lecture 3
2/11
2
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
ECON10005 Measures of Location & Variation
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 .
ECON10005 Measures of Location & Variation
= =
N
ii 1
x
N
Lecture 3 -Slide 6
-
7/28/2019 ECON10005 Lecture 3
3/11
3
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.
ECON10005 Measures of Location & Variation Lecture 3 - Slide 7
ECON10005 Measures of Location & Variation Lecture 3 - Slide 8
Measures ofLocation
Lecture 3
Lecture 4 Measures ofAssociation
Location & Skewness
Measures oflocation, variationand association
Mean, Median &Mode
Location &Skewness
Measures ofVariation
Variance &Standard Deviation
The Coefficient ofVariation
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.
ECON10005 Measures of Location & Variation
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
-
7/28/2019 ECON10005 Lecture 3
4/11
4
One Possible Symmetric Distribution
ECON10005 Measures of Location & Variation
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
ECON10005 Measures of Location & Variation Lecture 3 - Slide 11
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
ECON10005 Measures of Location & Variation Lecture 3 - Slide 12
-
7/28/2019 ECON10005 Lecture 3
5/11
5
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:
ECON10005 Measures of Location & Variation Lecture 3 - Slide 13
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
ECON10005 Measures of Location & Variation Lecture 3 - Slide 14
Measures ofLocation
Lecture 3
Lecture 4 Measures ofAssociation
Variance & Standard Deviation
Measures oflocation, variationand association
Mean, Median &Mode
Location &Skewness
Measures ofVariation
Variance &Standard Deviation
The Coefficient ofVariation
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.
ECON10005 Measures of Location & Variation
A 12 13 14 15 16
B 3 9 14 19 25
Lecture 3 - Slide 15
-
7/28/2019 ECON10005 Lecture 3
6/11
6
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:
ECON10005 Measures of Location & Variation
( )N
22i
i 1
1x
N = =
( )N
22
ii 1
1x
N = = =
Lecture 3 - Slide 16
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.
ECON10005 Measures of Location & Variation Lecture 3 - Slide 17
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:
ECON10005 Measures of Location & Variation
( )n
22
ii 1
1s x x
n 1 ==
( )n
22i
i 1
1s s x x
n 1 == =
Lecture 3 - Slide 18
-
7/28/2019 ECON10005 Lecture 3
7/11
7
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.
ECON10005 Measures of Location & Variation
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= =
Lecture 3 - Slide 19
Variance &Standard Deviation
ECON10005 Measures of Location & Variation
Measures ofLocation
Lecture 3
Lecture 4 Measures ofAssociation
The Coefficient of Variation
Measures oflocation, variationand association
Mean, Median &Mode
Location &Skewness
Measures ofVariation
The Coefficient ofVariation
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.
ECON10005 Measures of Location & Variation
sCV 100 %
x
=
Lecture 3 - Slide 21
-
7/28/2019 ECON10005 Lecture 3
8/11
8
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?
ECON10005 Measures of Location & Variation
A 12 13 14 15 16
B 3 9 14 19 25
Lecture 3 - Slide 22
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?
ECON10005 Measures of Location & Variation
8.54CV 100 % 100(0.61)% 61%
14
= = =
Lecture 3 - Slide 23
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
-
7/28/2019 ECON10005 Lecture 3
9/11
9
Variance &Standard Deviation
ECON10005 Measures of Location & Variation Lecture 3 - Slide 25
Measures ofLocation
Lecture 3
Lecture 4 Measures ofAssociation
Next Time...
Measures oflocation, variation
and association
Mean, Median &Mode
Location &Skewness
Measures ofVariation
The Coefficient ofVariation
Covariance
Correlation
ECON10005 Measures of Location & Variation Lecture 3 - Slide 26
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).
ECON10005 Measures of Location & Variation Lecture 3 - Slide 27
-
7/28/2019 ECON10005 Lecture 3
10/11
10
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
ECON10005 Measures of Location & Variation Lecture 3 - Slide 28
Measures of Location in Excels Toolpak
In the Data Analysis Toolpak, select DescriptiveStatistics.
ECON10005 Measures of Location & Variation Lecture 3 - Slide 29
Input Range and Summary Statistics
Give Excel the input range and check the box markedsummary statistics.
ECON10005 Measures of Location & Variation Lecture 3 - Slide 30
-
7/28/2019 ECON10005 Lecture 3
11/11
11
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.
ECON10005 Measures of Location & Variation Lecture 3 - Slide 31
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.
ECON10005 Measures of Location & Variation Lecture 3 - Slide 32
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 Measures of Location & Variation Lecture 3 - Slide 33
top related