1.2
Describing Distributions with Numbers
Measures of Center
The Mean
x
The mean
x
of a set of observations is equal to the sum of their values divided by the
number of observations.
If the
n
observations are x
1
, x
2
, …, x
n
, their mean is:
n
n
2
1
x
x
x
+
…
+
+
or, more compactly:
n
1
∑
i
x
The
∑
(capital Greek sigma) means “add them all up.”
The weakness of the mean is that it is not a resistant measure
of center:
it is not resistant
to the influence of a few extreme observations.
In the example below, note that
decreasing the minimum value significantly has a large impact on the mean.
x
:
80, 90, 90, 100
x
= 90
y
:
20, 90, 90, 100
y
= 75
The Median M
The median M
is the midpoint of a distribution.
To find the median of a distribution:
1) Arrange all the observations in order from smallest to largest.
2) If the number of observations
n
is odd, the median is the center observation.
The
location of the median is (
n
+1)/2 observations up from the bottom of the list.
3) If the number of observations n is even, the median is the mean of the two center
observations.
The location of the median is (
n
+1)/2 observations up from the bottom of
the list.
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a
:
1, 4, 8, 14, 20
[
n
+1]/2= 6/2= 3
The third observation is the median.
The median is 8.
b
:
1, 4, 8, 14, 20, 21
[
n
+1]/2= 7/2= 3.5
The median is the average of the third and fourth observations.
The median is 11.
c
:
1, 4, 8, 14, 20, 100
[
n
+1]/2=7/2= 3.5
The median is the average of the third and fourth observations.
The median is 11.
The strength of the median is that it is a resistant measure of center:
it is resistant to the
influence of a few extreme observations.
In the example above, note that increasing the
maximum value significantly from
b
to
c
did not change the median at all.
The Mode
The mode
is the number that occurs most frequently in a set of data values.
x
:
80, 90, 90, 100
The mode is 90.
Measures of Spread
While we care about measures of center, we also care about measures of spread.
For
instance, median income might be the same in country
A
and country
B
.
If, however,
everyone has the same income in country
A
while there is extreme inequality in country
B
, then country
A
and country
B
are very different countries.
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 Spring '08
 ABDUS,S.
 Standard Deviation, xnew, M M M M M M M M M M M M M City

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