P91, #11

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# P91, #11 - PowerPoint PPT Presentation

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

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

Variation in air quality in Anaheim higher, means similar

Last Time:

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.

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%

Source:

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

Source:

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

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

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

Z Score

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

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

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.

Distribution Shape - Skewness

Mean = Median, Skewness =0

Distribution Shape - Skewness

Mean < Median, Skewness < 0

Distribution Shape - Skewness

Mean > Median, Skewness > 0

Numerical Measures of Association
• Covariance
• Correlation Coefficient
Covariance

Sample covariance:

Population covariance:

Covariance , Sxy > 0

II

I

III

IV

Mean of X = 5.5, Mean of Y = 7.6

Covariance , Sxy < 0

II

I

III

IV

Mean of X = 5.5, Mean of Y = 7.6

Covariance, Sxy = 0

II

I

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

Population:

Sample:

Correlation Coefficient, cont.
• Properties of the correlation coefficient:
• Value is independent of the unit of measurement
• Value can range from -1 to 1
• A value of -1 or 1 indicates a perfect linear relationship
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 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

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

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