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Chapter 2
Describing distributions with numbers
Chapter Outline
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1. Measuring center: the mean
2. Measuring center: the median
3. Comparing the mean and the median
4. Measuring spread: the quartiles
5. The five-number summary and boxplots
6. Measuring spread: the standard deviation
7. Choosing measures of center and spread
Measuring center: the mean
Notation: x
 It is simply the ordinary arithmetic
average.
 Suppose that we have n observations
(data size, number of individuals).
 Observations are denoted as x1, x2, x3,
…xn.
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Measuring center: the mean
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How to get
x
?
x1  x2  x3  ...  xn
x

n
Example 2.1 (P.33)
x
i
n
Measuring center: the median
Notation: M
 Median M is the midpoint of a
distribution half the observations are
smaller than M and the other half are
larger than M.
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Measuring center: the median
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How to find M?
– 1. Sort all observations in increasing order
(This step is important!!!)
n 1
(
– 2. If n is odd, 2 )th observation is M. if n is
even, average of two center values is M.
n 1
(
Note that 2 ) is the location of the
median in the ordered list, not the
median value.
Measuring center: the median
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Examples
Case 1. 11, 21, 13, 24, 15, 26, 17
 Case 2. 11, 21, 13, 24, 15, 26
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Example 2.2, 2.3 (P.35)
Mean vs. Median
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Median is more resistant than the mean.
 The mean and median of a symmetric
distribution are close together. If the
distribution is exactly symmetric, the mean
and median are exactly the same. In a
skewed distribution, the mean is farther out in
the long tail than is the median.
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Example
1, 2, 3, 4, 5, 6, 10000
Inference :
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Strongly skewed distributions are
reported with median than the mean.
Measuring Spread: The Quartiles
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The quartiles mark out the middle half of
the distribution.
Calculating the Quartiles :
– Step1.
Arrange the observations in increasing order and
locate the median M in the ordered list of
observations.
– Step2.
The first quartile Q1 is the median of the
observations whose position in the ordered list is
to the left of the location of the overall median.
– Step3.
The third quartile Q3 is the median of the
observations whose position in the ordered list is
to the right of the location of the overall median.
Measuring spread: the quartiles
Example 2.4 (P. 37)
 Example 2.5 (P. 38)
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Note:
(1) It is important to sort data first before
we try to find quartiles!
(2) Quartiles are resistant.
The five-number summary and boxplots
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The five-number summary:
Minimum, Q1, M, Q3, Maximum.
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Boxplot is a graph of five number
summary.
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Boxplots are most useful for side-by-side
comparison of several distributions.
Boxplot
1. A boxplot is a graph of the fivenumber summary
 2. A central box spans the quartiles
 3. A line in the box marks the median
 4. Lines extended from the box out to
the minimum and maximum
 5. Range = maximum - minimum
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The five-number summary and boxplot
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Figure 2.2(P.39): side-by-side boxplots
comparing the distributions of earning
for two levels of education.
The five-number summary and boxplots
Inference :
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Boxplot also gives an indication of the
symmetry or skewness of a distribution.
-- In a symmetric distribution Q1 and Q3
are equally distant from the median,
but in case of right skewed one the
third quartile would be further above the
median than the first quartile bellow it.
Measuring spread: the standard deviation
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It says how far the observations are from their mean.
The variance s2 of a set of observations is an average of the
squares of the deviations of the observations from their mean.
Notation: s2 for variance and s for standard deviation
2
2
2
(
x

x
)

(
x

x
)

...

(
x

x
)
2
n
s2  1
n 1
2
(
x

x
)
 i

n 1
( xi2 )  n( x ) 2

n 1
Why (n-1) ?
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As the sum of the deviations ( xi  x )
always equals 0, so the knowledge of (n-1) of
them determines the last one.
--- Only (n-1) of the squared deviations are
variable but not the last one, so we average
by dividing the total by (n-1).
The number (n-1) is called the degrees of
freedom of the variance or standard deviation
Measuring spread: the standard deviation
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To find the variance and the standard
deviation
– 1. Find the mean of the data set
– 2. Subtract the mean from each number (we call
that deviation)
– 3. Square each result
– 4. Sum all the square
– 5. Divide the sum of square by n-1, where n is the
number of all observations. Now you get variance
– 6. Standard deviation is just the positive square
root of the variance.
Measuring spread: the standard deviation
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Example 2.6 (P.42)
Properties of
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2
s
and s
s measures spread about the mean and
should be used only when the mean is
chosen as the measure of center.
 s  0 and s=0 only when each of the
observation values does not differ from each
other.
 S is not resistant.
Choosing measures of center and spread
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With a skewed distribution or with a
distribution with extreme outliers, fivenumber summary is better.
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With a symmetric distribution (without
outliers), mean and standard deviation
are better.