Measures of Variability
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Same center,
different variation
Variation
Variance Standard
Deviation
Coefficient of
Variation
Range Interquartile
Range
Measures of variation give
information on the spread
or variability of the data
values.
Ch. 2-1
2.2
Range
Simplest measure of variation
Difference between the largest and the smallest
observations:
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Range = Xlargest – Xsmallest
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Range = 14 - 1 = 13
Example:
Ch. 2-2
Disadvantages of the Range
Ignores the way in which data are distributed
Sensitive to outliers
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7 8 9 10 11 12
Range = 12 - 7 = 5
7 8 9 10 11 12
Range = 12 - 7 = 5
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5
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
Ch. 2-3
Interquartile Range
Can eliminate some outlier problems by using the interquartile range (IQR)
Eliminate high- and low-valued observations and calculate the range of the middle 50% of the data
Interquartile range = 3rd quartile – 1st quartile
IQR = Q3 – Q1
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Interquartile Range
Copyright © 2013 Pearson Education Ch. 2-5
The interquartile range (IQR) measures the
spread in the middle 50% of the data
Defined as the difference between the
observation at the third quartile and the
observation at the first quartile
IQR = Q3 - Q1
Box-and-Whisker Plot
Copyright © 2013 Pearson Education Ch. 2-6
A box-and-whisker plot is a graph that describes the
shape of a distribution
Created from the five-number summary: the
minimum value, Q1, the median, Q3, and the
maximum
The inner box shows the range from Q1 to Q3, with a
line drawn at the median
Two “whiskers” extend from the box. One whisker is
the line from Q1 to the minimum, the other is the line
from Q3 to the maximum value
Box Plot
Copyright © 2013 Pearson Education Ch. 2-7
Population Variance
Average of squared deviations of values from
the mean (Karl Pearson 1893)
Population variance:
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N
μ)(x
σ
N
1i
2
i
2
Where = population mean
N = population size
xi = ith value of the variable x
μ
Ch. 2-8
Sample Variance
Average (approximately) of squared deviations
of values from the mean
Sample variance:
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1-n
)x(x
s
n
1i
2
i
2
Defect: Not in the same unit of original data values.
Ch. 2-9
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:
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N
μ)(x
σ
N
1i
2
i
Ch. 2-10
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:
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1-n
)x(x
S
n
1i
2
i
Ch. 2-11
Calculation Example:Sample Standard Deviation
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Sample
Data (xi) : 10 12 14 15 17 18 18 24
n = 8 Mean = x = 16
4.30957
130
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
Ch. 2-12
Measuring variation
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Small standard deviation
Large standard deviation
Ch. 2-13
Comparing Standard Deviations
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s = 3.338(compare to the two
cases below)
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
s = 0.926 (values are concentrated
near the mean)
11 12 13 14 15 16 17 18 19 20 21s = 4.570(values are dispersed far
from the mean)Data C
Ch. 2-14
Mean = 15.5 for each data set
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)
Copyright © 2013 Pearson Education Ch. 2-15
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
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100%x
sCV
Ch. 2-16
100%μ
σCV
Population coefficient of
variation:
Sample coefficient of
variation:
Comparing Coefficient of Variation
Stock A:
Average price last year = $50
Standard deviation = $5
Stock B:
Average price last year = $100
Standard deviation = $5
Copyright © 2013 Pearson Education
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
Ch. 2-17
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
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Chebychev’s Theorem
)]%(1/k100[12
Ch. 2-18
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 - 1/1.52) = 55.6% ……... k = 1.5 (μ ± 1.5σ)
(1 - 1/22) = 75% …........... k = 2 (μ ± 2σ)
(1 - 1/32) = 89% …….…... k = 3 (μ ± 3σ)
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Chebychev’s Theorem
withinAt least
(continued)
Ch. 2-19
If the data distribution is bell-shaped, then
the interval:
contains about 68% of the values in
the population or the sample
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The Empirical Rule
1σμ
μ
68%
1 σμ
Ch. 2-20
contains about 95% of the values in
the population or the sample
contains almost all (about 99.7%) of
the values in the population or the sample
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The Empirical Rule
2 σμ
3 σμ
3 σμ
99.7%95%
2 σμ
Ch. 2-21
(continued)
A z-score shows the position of a value
relative to the mean of the distribution.
indicates the number of standard deviations a
value is from the mean.
A z-score greater than zero indicates that the value is
greater than the mean
a z-score less than zero indicates that the value is
less than the mean
a z-score of zero indicates that the value is equal to
the mean.
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z-Score
Ch. 2-22
If the data set is the entire population of data
and the population mean, µ , and the population
standard deviation, σ, are known, then for each
value, xi, the z-score associated with xi is
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z-Score
Ch. 2-23
σ
μ-xz
i
(continued)
If intelligence is measured for a population
using an IQ score, where the mean IQ score
is 100 and the standard deviation is 15, what
is the z-score for an IQ of 121?
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z-Score
Ch. 2-24
1.415
100- 121
σ
μ-xz
i
(continued)
A score of 121 is 1.4 standard
deviations above the mean.
Weighted Mean and Measures of Grouped Data
The weighted mean of a set of data is
Where wi is the weight of the ith observation
and
Use when data is already grouped into n classes, with wi values in the ith class
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n
xwxwxw
n
xw
xnn2211
n
1i
ii
Ch. 2-25
i
wn
2.3
Approximations for Grouped Data
Suppose data are grouped into K classes, with
frequencies f1, f2, . . ., fK, and the midpoints of the
classes are m1, m2, . . ., mK
For a sample of n observations, the mean is
Copyright © 2013 Pearson Education
n
mf
x
K
1i
ii
K
1i
ifnwhere
Ch. 2-26
Approximations for Grouped Data
Suppose data are grouped into K classes, with
frequencies f1, f2, . . ., fK, and the midpoints of the
classes are m1, m2, . . ., mK
For a sample of n observations, the variance is
Copyright © 2013 Pearson Education Ch. 2-27
1n
)x(mf
s
K
1i
2
ii
2
Measures of Relationships Between Variables
Two measures of the relationship between variable are
Covariance a measure of the direction of a linear relationship
between two variables
Correlation Coefficient a measure of both the direction and the strength of a
linear relationship between two variables
Copyright © 2013 Pearson Education Ch. 2-28
2.4
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
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N
))(y(x
y),(xCov
N
1i
yixi
xy
1n
)y)(yx(x
sy),(xCov
n
1i
ii
xy
Ch. 2-29
Interpreting Covariance
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
Copyright © 2013 Pearson Education Ch. 2-30
Coefficient of Correlation
Measures the relative strength of the linear relationship between two variables
Population correlation coefficient:
Sample correlation coefficient:
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YXss
y),(xCovr
YXσσ
y),(xCovρ
Ch. 2-31
Features of Correlation Coefficient, r
Unit free
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
Copyright © 2013 Pearson Education Ch. 2-32
Scatter Plots of Data with Various Correlation Coefficients
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Y
X
Y
X
Y
X
Y
X
Y
X
r = -1 r = -.6 r = 0
r = +.3r = +1
Y
Xr = 0
Ch. 2-33
Interpreting the Result
r = .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
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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
co
re
Ch. 2-34
Covariance
Let X and Y be discrete random variables with means
μX and μY
The expected value of (X - μX)(Y - μY) is called the
covariance between X and Y
For discrete random variables
An equivalent expression is
x y
yxYXy))P(x,μ)(yμ(x)]μ)(YμE[(XY)Cov(X,
x y
yxyxμμy)xyP(x,μμE(XY)Y)Cov(X,
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EducationCh. 4-35
Correlation
The correlation between X and Y is:
-1 ≤ ρ ≤ 1
ρ = 0 no linear relationship between X and Y
ρ > 0 positive linear relationship between X and Y when X is high (low) then Y is likely to be high (low)
ρ = +1 perfect positive linear dependency
ρ < 0 negative linear relationship between X and Y when X is high (low) then Y is likely to be low (high)
ρ = -1 perfect negative linear dependency
YXσσ
Y)Cov(X,Y)Corr(X,ρ
Copyright © 2013 Pearson
EducationCh. 4-36
Covariance and Independence
The covariance measures the strength of the
linear relationship between two variables
If two random variables are statistically
independent, the covariance between them
is 0
The converse is not necessarily true
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EducationCh. 4-37
Example: Investment Returns
Return per $1,000 for two types of investments
P(xiyi) Economic condition Passive Fund X Aggressive Fund Y
.2 Recession - $ 25 - $200
.5 Stable Economy + 50 + 60
.3 Expanding Economy + 100 + 350
Investment
E(x) = μx = (-25)(.2) +(50)(.5) + (100)(.3) = 50
E(y) = μy = (-200)(.2) +(60)(.5) + (350)(.3) = 95
Copyright © 2013 Pearson
EducationCh. 4-38
Computing the Standard Deviation for Investment Returns
P(xiyi) Economic condition Passive Fund X Aggressive Fund Y
0.2 Recession - $ 25 - $200
0.5 Stable Economy + 50 + 60
0.3 Expanding Economy + 100 + 350
Investment
43.30
(0.3)50)(100(0.5)50)(50(0.2)50)(-25σ 222
X
193.71
(0.3)95)(350(0.5)95)(60(0.2)95)(-200σ 222
y
Copyright © 2013 Pearson
EducationCh. 4-39
Covariance for Investment Returns
P(xiyi) Economic condition Passive Fund X Aggressive Fund Y
.2 Recession - $ 25 - $200
.5 Stable Economy + 50 + 60
.3 Expanding Economy + 100 + 350
Investment
8250
95)(.3)50)(350(100
95)(.5)50)(60(5095)(.2)200-50)((-25Y)Cov(X,
Copyright © 2013 Pearson
EducationCh. 4-40
Portfolio Example
Investment X: μx = 50 σx = 43.30
Investment Y: μy = 95 σy = 193.21
σxy = 8250
Suppose 40% of the portfolio (P) is in Investment X and 60% is in Investment Y:
The portfolio return and portfolio variability are between the values for investments X and Y considered individually
77)95()6(.)50(4.E(P)
04.133
8250)2(.4)(.6)((193.21))6(.(43.30)(.4)σ2222
P
Copyright © 2013 Pearson
EducationCh. 4-41
Interpreting the Results for Investment Returns
The aggressive fund has a higher expected return, but much more risk
μy = 95 > μx = 50
but
σy = 193.21 > σx = 43.30
The Covariance of 8250 indicates that the two investments are positively related and will vary in the same direction
Copyright © 2013 Pearson
EducationCh. 4-42