A useful mean in the physical sciences (such as voltage) is the
quadratic mean (QM), which is found by taking the square root of
the average of the squares of each value. The formula is:
QM
=
r
∑
x
2
n
The
QM
is also used in Forestry, e.g. to compute the mean dbh for
trees.
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Measures of Central Tendecny - The Mean
In some situations, the observations may not have equal importance.
In such situations, the observations are given differential weights - e.g.
when computing a grade point average. Since courses vary in their
credit value, the number of credits must be used as weights.
The measure of central tendency of such data is known as
the
weighted mean
.It is given by the formula:
WM
=
∑
n
i
=1
w
i
x
i
∑
n
i
=1
w
i
where,
w
i
s are the weights and
x
i
s are the data values for variable
X
.
The mean is a useful measure of the central value of a set of
measurements, but it is subject to distortion due to the presence of
one or more extreme values in the set.
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Measures of Central Tendecny - The Mean
In these situations, the extreme values (called
outliers
) pull the mean
in the direction of the outliers to find the balancing point, thus
distorting the mean as a measure of the central value.
A variation of the mean, called a
trimmed mean
, drops the highest
and lowest extreme values and averages the rest.
For example, a 5% trimmed mean drops the highest 5% and the
lowest 5% of the measurements and averages the rest. Similarly, a
10% trimmed mean drops the highest and the lowest 10% of the
measurements and averages the rest.
Consider the following example:
Example 4
The numbers of wood ants in seven pitfall traps set overnight in deciduous
forest woodland are:
25, 4, 12, 9, 15, 18, 202
Calculate the mean number of ants per trap.
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Measures of Central Tendecny - The Mean
Solution for Example 4
¯
x
=
∑
n
i
=1
x
i
n
=
287
7
= 40
.
7
ants
40.7 is larger than six observations and almost five times smaller than
202!
Since the mean takes into account the values of all observations in a
sample; it can be greatly distorted by a single exceptional value!
When a few exceptional values distort the mean in this way, a
resistant measure of the average, namely the
median
may be more
appropriate.
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Measures of Central Tendecny - The Median
The second measure of central tendency we consider is the median.
Definition of Median
The
median
of a set of measurements is defined to be the middle value
when the measurements are arranged from lowest to highest. When the
data set is ordered, it is called a
data array
. Thus, the median is the
midpoint of the data array
The median either will be a specific value in the data set or will fall
between two values, as shown in examples 5 and 6:
Example 5
Find the median for the set of measurements 2, 9, 11, 5, 6.

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- Fall '18
- F. TAILOKA
- Standard Deviation, Mean, Central Tendecny