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P91, #11. P. 97, #19 a,b. Q 1 = P 25 i =(25/100)(9)=2.25, round to 3 Q 1 =45. Q 3 = P 75 i =(75/100)(9)=6.75, round to 7 Q 3 =55. P. 97, #19 c. Variation in air quality in Anaheim higher, means similar. Last Time:. Sample Standard Deviation. Population Standard Deviation.

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p 97 19 a b
P. 97, #19 a,b

Q1 = P25

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

Q1=45

Q3 = P75

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

Q3=55

p 97 19 c
P. 97, #19 c

Variation in air quality in Anaheim higher, means similar

last time
Last Time:

Sample Standard Deviation

Population Standard Deviation

using the standard deviation
Using the Standard Deviation
  • Chebyshev’s Theorem
  • Empirical Rule
  • Z scores
chebyshev s theorem
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
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
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.

application of chebyshev cont1
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 cont2
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%

slide12

Source:

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

slide13

Source:

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

empirical rule
Empirical Rule

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

  • 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
slide15

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

z score
Z Score

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

z score1
Z Score

Z1980 = (32.5-12.3)/20.2 = 1

Z1990 = (-3.1-12.3)/20.2 = -0.8

Z2000 = (-9.1-12.3)/20.2 = -1.1

Z2007 = (5.5-12.3)/20.2 = -0.3

outliers
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
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.

distribution shape skewness1
Distribution Shape - Skewness

Mean = Median, Skewness =0

distribution shape skewness2
Distribution Shape - Skewness

Mean < Median, Skewness < 0

distribution shape skewness3
Distribution Shape - Skewness

Mean > Median, Skewness > 0

numerical measures of association
Numerical Measures of Association
  • Covariance
  • Correlation Coefficient
covariance
Covariance

Sample covariance:

Population covariance:

covariance s xy 0
Covariance , Sxy > 0

II

I

III

IV

Mean of X = 5.5, Mean of Y = 7.6

covariance s xy 01
Covariance , Sxy < 0

II

I

III

IV

Mean of X = 5.5, Mean of Y = 7.6

covariance s xy 02
Covariance, Sxy = 0

II

I

III

IV

Mean of X = 5.5, Mean of Y = 7.6

covariance cont
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
Correlation Coefficient

Pearson Product Moment Correlation Coefficient

Population:

Sample:

correlation coefficient cont
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
numerical example
Numerical Example

Mean of x = 3, Mean of y = 7

Sum of squared deviations, x = 14

Sum of squared deviations, y = 42

Sum of the product of deviations = -24

numerical example1
Numerical Example

Mean of x = 3, Mean of y = 5

Sum of squared deviations, x = 14

Sum of squared deviations, y = 18

Sum of the product of deviations = 13

example
Example

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

example1
Example

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

slide41

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