DESCRIPTIVE STATISTICS:

MEASURES OF CENTRAL
TENDENCY

Week – 3
Dr. Muhammad Bilal
 Measures of Central Tendency
Central tendency is a characteristic of a data set that
relates to its average value. It is the central value in the
sense that it is located in the middle and the data
points cluster around it. Since it is the most
representative point of the data and a comparison
between two or more data sets may, therefore, be
made by their respective central points. In simple way,
it can be said that methods of measures of central
tendency are useful for the purpose of comparison of
two or more similar types of data sets. Most commonly
used measures are, arithmetic mean, median and
mode. Quartiles, deciles and percentiles are also
position indicators and useful for comprehensive
comparison of two or more sets of data.
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 Arithmetic Mean
Arithmetic mean or simply mean is most
commonly used measure of central tendency.
It has a very important property, when it is
subtracted from all the values of data, the
sum of the differences of mean from
observations is zero. It uses all observations
fully in its calculation.

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Arithmetic Mean

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Arithmetic Mean

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 Example
Suppose the weights of 14 sampled patients
are 62, 64, 65, 66, 68, 70, 70, 70, 70, 74, 74,
77, 77, 79 in kg, the mean for this data is:

62 + 64 + 65 + ... +79 1036

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 Properties of the Mean
The arithmetic mean possesses certain properties, some
desirable and some not so desirable. These properties
include the following:
1. Uniqueness. For a given set of data there is one and only
one arithmetic mean.
2. Simplicity. The arithmetic mean is easily understood and
easy to compute.
3. Since each and every value in a set of data enters into
the computation of the mean, it is affected by each value.
Extreme values, therefore, have an influence on the mean
and, in some cases, can so distort it that it becomes
undesirable as a measure of central tendency.

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 Example of Extreme Values
As an example of how extreme values may affect
the mean, consider the following situation.
Suppose the five physicians who practice in an
area are surveyed to determine their charges for
a certain procedure. Assume that they report
these charges: $75, $75, $80, $80, and $280. The
mean charge for the five physicians is found to be
$118, a value that is not very representative of
the set of data as a whole. The single atypical
value had the effect of inflating the mean.

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 Mean for Grouped Data
Given a grouped data, we first find the
midpoints of the groups, which are multiplied
by the corresponding frequencies of those
groups. All these products are added. This
sum is divided by the sum of all the
frequencies. Suppose the weights of 14
patients is given, the mean can be calculated
as:

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Example

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 Mean for Grouped Data
The mean obtained from grouped data may
be different from the mean obtained from
ungrouped data. This is because in grouped
data we assume that all the values in that
group is placed at the mid-value of the class
interval

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 Median
The median of a finite set of values is that value
which divides the set into two equal parts such
that the number of values equal to or greater
than the median is equal to the number of values
equal to or less than the median. If the number
of values is odd, the median will be the middle
value when all values have been arranged in
order of magnitude. When the number of values
is even, there is no single middle value. Instead
there are two middle values. In this case the
median is taken to be the mean of these two
middle values, when all values have been
arranged in the order of their magnitudes.
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 Median
The median observation of a data set is the
one when the observation have been ordered.
If, for example, we have 11 observations, the
median is (11+1)/2 = 6th ordered observation.
If we have 12 observations the median is
(12+1)/2 =6.5th ordered observation and is a
value halfway between the 6th and 7th ordered
observations.

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 Median: EXAMPLE
We wish to find the median age of the
subjects represented in the sample 38, 43, 50,
57, 57, 59, 61, 64, 65, 66

Since we have an even number of ages, there

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 Properties of the Median
1. Uniqueness. As is true with the mean, there
is only one median for a given set of data.
2. Simplicity. The median is easy to calculate.
3. It is not as drastically affected by extreme
values as is the mean

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 Mode
The mode of a set of values is that value which
occurs most frequently. If all the values are
different there is no mode; on the other hand, a
set of values may have more than one mode. The
mode may be used for describing qualitative
data. For example, suppose the patients seen in a
mental health clinic during a given year received
one of the following diagnoses: mental
retardation, organic brain syndrome, psychosis,
neurosis, and personality disorder. The diagnosis
occurring most frequently in the group of
patients would be called the modal diagnosis.
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 Mode Example
For example the scores of medical students in
a test are 2, 2, 2, 3, 5, 5, 5, 6, 6 in this case 2
and 5 are two modes.
A sample consisting of the data 6, 7,7, 7, 8, 8,
8, 8, 9, 9, 10, 11, 12, 12, 12, 12, 12, 13, 13,
and 14 mm would be said to have one mode:
at 12 mm.

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 Skewness
Data distributions may be classified on the
basis of whether they are symmetric or
asymmetric. If a distribution is symmetric, the
left half of its graph (histogram or frequency
polygon) will be a mirror image of its right
half. When the left half and right half of the
graph of a distribution are not mirror images
of each other, the distribution is asymmetric
or skewed.
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