chap 3-1 ef 507 quantitative methods for economics and finance fall 2008 chapter 3 describing data:...

Chap 3-1

EF 507

QUANTITATIVE METHODS FOR ECONOMICS AND FINANCE

FALL 2008

Chapter 3

Describing Data: Numerical

Chap 3-2

Describing Data Numerically

Arithmetic Mean

Median

Mode

Describing Data Numerically

Variance

Standard Deviation

Coefficient of Variation

Range

Central Tendency Variation

Chap 3-3

Measures of Central Tendency

Central Tendency

Mean Median Mode

n

xx

n

1ii

Overview

Midpoint of ranked values

Most frequently observed value

Arithmetic average

Chap 3-4

Arithmetic Mean

The arithmetic mean (mean) is the most common measure of central tendency

For a population of N values:

For a sample of size n:

Sample sizen

xxx

n

xx n21

n

1ii

Observed

values

N

xxx

N

xμ N21

N

1ii

Population size

Population values

Chap 3-5

Arithmetic Mean

The most common measure of central tendency Mean = sum of values divided by the number of values Affected by extreme values (outliers)

(continued)

0 1 2 3 4 5 6 7 8 9 10

Mean = 3

0 1 2 3 4 5 6 7 8 9 10

Mean = 4

35

15

5

54321

4

5

20

5

104321

Chap 3-6

Median

In an ordered list, the median is the “middle” number (50% above, 50% below)

Not affected by extreme values

0 1 2 3 4 5 6 7 8 9 10

Median = 3

0 1 2 3 4 5 6 7 8 9 10

Median = 3

Chap 3-7

Finding the Median

The location of the median:

If the number of values is odd, the median is the middle number If the number of values is even, the median is the average of

the two middle numbers

Note that is not the value of the median, only the

position of the median in the ranked data

dataorderedtheinposition2

1npositionMedian

2

1n

Chap 3-8

Mode

A measure of central tendency Value that occurs most often Not affected by extreme values Used for either numerical or categorical data There may be no mode There may be several modes

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mode = 9

0 1 2 3 4 5 6

No Mode

Chap 3-9

Five houses on a hill by the beach

Review Example

$2,000 K

$500 K

$300 K

$100 K

$100 K

House Prices:

$2,000,000 500,000 300,000 100,000 100,000

Chap 3-10

Review Example:Summary Statistics

Mean: ($3,000,000/5)

= $600,000

Median: middle value of ranked data = $300,000

Mode: most frequent value = $100,000

House Prices:

$2,000,000 500,000 300,000 100,000 100,000

Sum 3,000,000

Chap 3-11

Mean is generally used, unless extreme values (outliers) exist

If outliers exist, then median is often used, since the median is not sensitive to extreme values. Example: Median home prices may be

reported for a region – less sensitive to outliers

Which measure of location is the “best”?

Chap 3-12

Shape of a Distribution

Describes how data are distributed Measures of shape

Symmetric or skewed

Mean = Median Mean < Median Median < Mean

Right-SkewedLeft-Skewed Symmetric

Chap 3-13

Same center, different variation

Measures of Variability

Variation

VarianceStandard Deviation

Coefficient of Variation

Range

Measures of variation give information on the spread or variability of the data values.

Chap 3-14

Range

Simplest measure of variation Difference between the largest and the smallest

observations:

Range = Xlargest – Xsmallest

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Range = 14 - 1 = 13

Example:

Chap 3-15

Ignores the way in which data are distributed

Sensitive to outliers

7 8 9 10 11 12

Range = 12 - 7 = 5

7 8 9 10 11 12

Range = 12 - 7 = 5

Disadvantages of the Range

1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5

1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120

Range = 5 - 1 = 4

Range = 120 - 1 = 119

Chap 3-16

Quartiles

Quartiles split the ranked data into 4 segments with an equal number of values per segment

25% 25% 25% 25%

The first quartile, Q1, is the value for which 25% of the observations are smaller and 75% are larger

Q2 is the same as the median (50% are smaller, 50% are larger)

Only 25% of the observations are greater than the third quartile

Q1 Q2 Q3

Chap 3-17

Average of squared deviations of values from the mean (Why “N-1”?)

Population variance:

Population Variance

1-N

μ)(xσ

N

1i

2i

2

Where = population mean

N = population size

xi = ith value of the variable x

μ

Chap 3-18

Average (approximately) of squared deviations of values from the mean

Sample variance:

Sample Variance

1-n

)x(xs

n

1i

2i

2

Where = arithmetic mean

n = sample size

Xi = ith value of the variable X

X

Chap 3-19

Population Standard Deviation

Most commonly used measure of variation Shows variation about the mean Has the same units as the original data

Population standard deviation:

1-N

μ)(xσ

N

1i

2i

Chap 3-20

Sample Standard Deviation

Most commonly used measure of variation Shows variation about the mean Has the same units as the original data

Sample standard deviation:

1-n

)x(xS

n

1i

2i

Chap 3-21

Calculation Example:Sample Standard Deviation

Sample Data (xi) : 10 12 14 15 17 18 18 24

n = 8 Mean = x = 16

4.24267

126

18

16)(2416)(1416)(1216)(10

1n

)x(24)x(14)x(12)X(10s

2222

2222

A measure of the “average” scatter around the mean

Chap 3-22

Measuring variation

Small standard deviation

Large standard deviation

Chap 3-23

Comparing Standard Deviations

Mean = 15.5 s = 3.338 11 12 13 14 15 16 17 18 19 20 21

11 12 13 14 15 16 17 18 19 20 21

Data B

Data A

Mean = 15.5 s = 0.926

11 12 13 14 15 16 17 18 19 20 21

Mean = 15.5 s = 4.570

Data C

Chap 3-24

Advantages of Variance and Standard Deviation

Each value in the data set is used in the calculation

Values far from the mean are given extra weight (because deviations from the mean are squared)

Chap 3-25

For any population with mean μ and standard deviation σ , and k > 1 , the percentage of observations that fall within the interval

[μ + kσ] Is at least

Chebyshev’s Theorem

)]%(1/k100[1 2

Chap 3-26

Regardless of how the data are distributed, at least (1 - 1/k2) of the values will fall within k standard deviations of the mean (for k > 1)

Examples:

(1 - 1/12) = 0% ……..... k=1 (μ ± 1σ)

(1 - 1/22) = 75% …........ k=2 (μ ± 2σ)

(1 - 1/32) = 89% ………. k=3 (μ ± 3σ)

Chebyshev’s Theorem

withinAt least

(continued)

Chap 3-27

If the data distribution is bell-shaped, then the interval:

contains about 68% of the values in the population or the sample

The Empirical Rule

1σμ

μ

68%

1σμ

Chap 3-28

contains about 95% of the values in the population or the sample

contains about 99.7% of the values in the population or the sample

The Empirical Rule

2σμ

3σμ

3σμ

99.7%95%

2σμ

Chap 3-29

Coefficient of Variation

Measures relative variation Always in percentage (%) Shows variation relative to mean Can be used to compare two or more sets of

data measured in different units

100%x

sCV

Chap 3-30

Comparing Coefficient of Variation

Stock A: Average price last year = $50 Standard deviation = $5

Stock B: Average price last year = $100 Standard deviation = $5

Both stocks have the same standard deviation, but stock B is less variable relative to its price

10%100%$50

$5100%

x

sCVA

5%100%$100

$5100%

x

sCVB

Chap 3-31

Using Microsoft Excel

Descriptive Statistics can be obtained from Microsoft® Excel

Use menu choice:

tools / data analysis / descriptive statistics

Enter details in dialog box

Chap 3-32

Using Excel

Use menu choice:

tools / data analysis /

descriptive statistics

Chap 3-33

Enter dialog box details

Check box for summary statistics

Click OK

Using Excel(continued)

Chap 3-34

Excel output

Microsoft Excel

descriptive statistics output,

using the house price data:

House Prices:

$2,000,000 500,000 300,000 100,000 100,000

Chap 3-35

Weighted Mean

The weighted mean of a set of data is

Where wi is the weight of the ith observation

Use when data is already grouped into n classes, with wi values in the ith class

i

nn2211

n

1iii

w

xwxwxw

w

xwx

Chap 3-36

The Sample Covariance The covariance measures the strength of the linear relationship

between two variables

The population covariance:

The sample covariance:

Only concerned with the strength of the relationship No causal effect is implied

N

))(y(xy),(xCov

N

1iyixi

xy

1n

)y)(yx(xsy),(xCov

n

1iii

xy

Chap 3-37

Covariance between two variables:

Cov(x,y) > 0 x and y tend to move in the same direction

Cov(x,y) < 0 x and y tend to move in opposite directions

Cov(x,y) = 0 x and y are independent

Interpreting Covariance

Chap 3-38

Coefficient of Correlation

Measures the relative strength of the linear relationship between two variables

Population correlation coefficient (rho):

Sample correlation coefficient (r):

YX ss

y),(xCovr

YXσσ

y),(xCovρ

Chap 3-39

Features of Correlation Coefficient, r

Unit free (difference with covariance)

Ranges between –1 and 1

The closer to –1, the stronger the negative linear

relationship

The closer to 1, the stronger the positive linear

relationship

The closer to 0, the weaker any positive linear

relationship

Chap 3-40

Scatter Plots of Data with Various Correlation Coefficients

Y

X

Y

X

Y

X

Y

X

Y

X

r = -1 r = -0.6 r = 0

r = +0.3r = +1

Y

Xr = 0

Chap 3-41

Using Excel to Find the Correlation Coefficient

Select Tools/Data Analysis

Choose Correlation from the selection menu

Click OK . . .

Chap 3-42

Using Excel to Find the Correlation Coefficient

Input data range and select appropriate options

Click OK to get output

(continued)

Chap 3-43

Interpreting the Result

r = 0.733

There is a relatively strong positive linear relationship between test score #1 and test score #2

Students who scored high on the first test tended to score high on second test

Scatter Plot of Test Scores

70

75

80

85

90

95

100

70 75 80 85 90 95 100

Test #1 ScoreT

est

#2 S

core

Chap 3-44

Obtaining Linear Relationships

An equation can be fit to show the best linear relationship between two variables:

Y = β0 + β1X

Where Y is the dependent variable and X is the independent variable (Ch 12)