P91, #11City Highway13.2 17.214.4 17.415.2 18.315.3 18.515.3 18.615.3 18.615.9 18.7

16 1916.1 19.216.2 19.416.2 19.416.7 20.616.8 21.1

Sum = 202.6 246Mean = 15.58 18.92Median = 15.9 18.7Mode = 15.3 18.6, 19.4

P. 97, #19 a,bAir index

284245484950555860

Range = 32IQR = 10

Q1 = P25

i=(25/100)(9)=2.25, round to 3Q1=45

Q3 = P75

i=(75/100)(9)=6.75, round to 7Q3=55

P. 97, #19 cAir index Dev. Dev. Sq.

28 -20.33 413.4442 -6.33 40.1145 -3.33 11.1148 -0.33 0.1149 0.67 0.4450 1.67 2.7855 6.67 44.4458 9.67 93.4460 11.67 136.11

Mean = 48.33Sum = 742Variance = 92.75Std. Dev. = 9.63

Variation in air quality in Anaheim higher, means similar

Last Time:

1

2

n

xxs i

N

xi2

Sample Standard Deviation

Population Standard Deviation

Using the Standard Deviation

•Chebyshev’s Theorem•Empirical Rule•Z scores

Chebyshev’s Theorem

At least [1 – (1/z)2] of the data values must be within z standard deviations of the mean, where z is any value greater than 1.

Since [1 – (½)2] = 1 – ¼ = ¾, 75% of the data values must lie within two standard deviations of the mean

Since [1 – (1/3)2] = 1 – 1/9 = 8/9, 88.9% of the data values must lie within three standard deviations of the mean

Application of Chebyshev

From 1926 to 2005 the average annual total return on large company stocks was 12.3%

The standard deviation of the annual returns was 20.2%

Source: A Random Walk Down Wall Street by Burton G. Malkiel, 2007 edition

Application of Chebyshev, cont.

Chebyshev’s Theorem states that there is at least a 75% chance that a randomly chosen year will have a return between -28.1% and 52.7%.

– 2(s) = 12.3% - 2(20.2%) = -28.1% + 2(s) = 12.3% + 2(20.2%) = 52.7%

Alternatively, there is up to a 25% chance the value will fall outside that range.

xx

Application of Chebyshev, cont.

From 1926 to 2005 the average annual total return on long-term government bonds was 5.8%

The standard deviation of the annual returns was 9.2%

What range would capture at least 75% of the values?

Source: A Random Walk Down Wall Street by Burton G. Malkiel, 2007 edition

Application of Chebyshev, cont.

– 2(s) = 5.8% - 2(9.2%) = -12.6% + 2(s) = 5.8% + 2(9.2%) = 24.2%

The corresponding range for U.S. Treasury Bills is -2.4% to 10%

xx

Source:http://disciplinedinvesting.blogspot.com/2007/02/stocks-versus-bonds.html

Source:http://disciplinedinvesting.blogspot.com/2007/02/stocks-versus-bonds.html

Empirical Rule

•Approximately 68% of the values will be within one standard deviation of the mean•Approximately 95% of the values will be within two standard deviations of the mean•Virtually all of the values will be within three standard deviations of the mean

The empirical rule applies when the values have a bell-shaped distribution.

Source: http://fisher.osu.edu/~diether_1/b822/riskret_2up.pdf

Z Score

s

xxz ii

The distance, measured in standard deviations, between some value and the mean. Also referred to as the “standardized value”

Z ScoreYear Annual return,

S&P 500Z Score

1980 32.5% 11990 -3.1%2000 -9.1%2007 5.5%

Z1980 = (32.5-12.3)/20.2 = 1Z1990 = (-3.1-12.3)/20.2 = -0.8Z2000 = (-9.1-12.3)/20.2 = -1.1Z2007 = (5.5-12.3)/20.2 = -0.3

Outliers

Observations with extremely small or extremely large values.

Values more than three standard deviations from the mean are typically considered outliers.

The S&P 500 index fell by 37% in 2008. Should it be considered an outlier?

Z2008 = (-37-12.3)/20.2 = -2.4

Distribution Shape - Skewness

A distribution is skewed when one side of a distribution has a longer tail than the other side.

The distribution is symmetric when the two sides of the distribution are mirror images of each other.

3

21

s

xx

nn

nSkewness i

Distribution Shape - Skewness

0 up to 20 20 up to 40 40 up to 60 60 up to 80 80 up to 10005

1015202530354045

Symmetric

Mean = Median, Skewness =0

Distribution Shape - Skewness

0 up to 20 20 up to 40 40 up to 60 60 up to 80 80 up to 10005

10152025303540

Skewed Left

Mean < Median, Skewness < 0

Distribution Shape - Skewness

0 up to 20 20 up to 40 40 up to 60 60 up to 80 80 up to 10005

10152025303540

Skewed Right

Mean > Median, Skewness > 0

Numerical Measures of Association

• Covariance• Correlation Coefficient

Covariance

Sample covariance:

1

n

yyxxs iixy

Population covariance:

N

yx yixixy

Covariance , Sxy > 0

0 2 4 6 8 10 120

2

4

6

8

10

12

III

III IV

Mean of X = 5.5, Mean of Y = 7.6

Covariance , Sxy < 0

0 2 4 6 8 10 120

2

4

6

8

10

12

III

III IV

Mean of X = 5.5, Mean of Y = 7.6

Covariance, Sxy = 0

0 2 4 6 8 10 120

2

4

6

8

10

12

III

III IV

Mean of X = 5.5, Mean of Y = 7.6

Covariance, cont.

The sign of the covariance indicates if the relationship is positive (direct) or negative (inverse).

However, the size of the covariance is not a good indicator of the strength of the relationship because it is sensitive to the units of measurement used.

Correlation Coefficient

Pearson Product Moment Correlation Coefficient

yx

xyxy ss

sr

yx

xyxy

Sample: Population:

Correlation Coefficient, cont.

Properties of the correlation coefficient:• Value is independent of the unit of measurement• Sign indicates whether relationship is positive or negative• Value can range from -1 to 1• A value of -1 or 1 indicates a perfect linear relationship

Correlation Coefficient, r=1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.50

5

10

15

20

25

30

Numerical Example

x y x- y- (x- )2 (y- )2(x- )(y- )

1 11 -2 4 4 16 -82 8 -1 1 1 1 -13 7 0 0 0 0 06 2 3 -5 9 25 -15

x y x y

Mean of x = 3, Mean of y = 7Sum of squared deviations, x = 14Sum of squared deviations, y = 42Sum of the product of deviations = -24

xy

Numerical Example, cont.

16.2

14

14

1

2

n

xxs ix

74.3

14

42

1

2

n

yys iy

8

14

24

1

n

yyxxs iixy

Numerical Example, cont.

99.074.316.2

8

yx

xyxy ss

sr

Numerical Example

x y x- y- (x- )2 (y- )2(x- )(y- )

1 2 -2 -3 4 9 62 4 -1 -1 1 1 13 7 0 2 0 4 06 7 3 2 9 4 6

x y x y

Mean of x = 3, Mean of y = 5Sum of squared deviations, x = 14Sum of squared deviations, y = 18Sum of the product of deviations = 13

xy

Numerical Example, cont.

16.2

14

14

1

2

n

xxs ix

45.2

14

18

1

2

n

yys iy

33.4

14

13

1

n

yyxxs iixy

Numerical Example, cont.

82.045.216.2

33.4

yx

xyxy ss

sr

Weighted Mean

i

ii

w

xwx

Example

Stock Price # of Shares$1 50$2 200$3 50Total value = $600

Fiji Stock Market, Day 1

Unweighted mean = (1+2+3)/3 = 2

Weighted mean = [(1)(50)+(2)(200)+(3)(50)]/(50+200+50)= 600/300=2

Example

Stock Price # of Shares$2 50$1 200$4 50Total value = $500

Fiji Stock Market, Day 2

Unweighted mean = (2+1+4)/3 = 2.33

Weighted mean = [(2)(50)+(1)(200)+(4)(50)]/(50+200+50)= 500/300=1.67

http://www.moneychimp.com/articles/volatility/standard_deviation.htm