MesurMean, median, mode: typical values.

SALVADOR D. ARQUILITA,BS Bio,MMEM
School Principal III
Central Tendency
ED 703 (Multivariate Analysis)
Measure of

Measure of Central Tendency
Measures of Central Tendency
Measures of central tendency describe a set of data
by identifying the central position in the data set as a
single representative value. There are generally three
measures of central tendency, commonly used in
statistics- mean, median, and mode. Mean is the
most common measure of central tendency used to
describe a data set.

Measure of Central Tendency
We come across new data every day. We
find them in newspapers, articles, in our bank
statements, mobile and electricity bills. Now
the question arises whether we can figure out
some important features of the data by
considering only certain representatives of
the data. This is possible by using measures
of central tendency. In the following sections,
we will look at the different measures of
central tendency and the methods to calculate

What are Measures of Central Tendency?
Measures of central tendency are the values that
describe a data set by identifying the central
position of the data. There are 3 main measures of
central tendency - Mean, Median and Mode.
•Mean- Sum of all observations divided by the total
number of observations.
•Median- The middle or central value in an ordered
set.
•Mode- The most frequently occurring value in a
data set.

Measures of Central Tendency Definition
The central tendency is defined as the
statistical measure that can be used to
represent the entire distribution or a
dataset using a single value called a
measure of central tendency. Any of the
measures of central tendency provides an
accurate description of the entire data in
the distribution.

Measures of Central Tendency Example
Let us understand the concept of the measures
of central tendency using an example. The
monthly salary of an employee for the 5 months
is given in the table below.

Suppose, we want to express the salary of the
employee using a single value and not 5 different
values for 5 months. This value that can be used to
represent the data for salaries for 5 months here
can be referred to as the measure of central
tendency. The three possible ways to find the
central measure of the tendency for the above
data are,
Mean: The mean salary of the given salary can be
used as on of the measures of central tendency,
i.e., = (105 + 95 + 105 + 105 + 100)/5 = $102.
x
̄

Mode: If we use the most frequently occurring
value to represent the above data, i.e., $105,
the measure of central tendency would be
mode.
Median: If we use the central value, i.e., $105
for the ordered set of salaries, given as, $95,
$100, $105, $015, $105, then the measure of
central tendency here would be median.

We can use the following table for reference to
check the best measure of central tendency
suitable for a particular type of variable:

Mean as a Measure of Central Tendency
The mean (or arithmetic mean) often called the average is
most likely one of the measures of central tendency that
you are most familiar with. It is also known as average.
Mean is simply the sum of all the components in a group
or collection, divided by the number of components.
We generally denote the mean of a given data-set by ,
x
̄
pronounced “x bar”. The formula to calculate the mean
for ungrouped data to represent it as the measure is
given as,

For a set of observations:
Mean = Sum of the terms/Number of terms
For a set of grouped data: Mean, = fx/ f
x
̄ Σ Σ
where,
• = the mean value of the set of given data.
x
̄
•f = frequency of each class
•x = mid-interval value of each class

Example:
The weights of 8 boys in kilograms: 45, 39, 53,
45, 43, 48, 50, 45. Find the mean weight for the given
set of data.
Therefore, the mean weight of the group:
Mean = Sum of the weights/Number of boys
= (45 + 39 + 53 + 45 + 43 + 48 + 50 + 45)/8
= 368/8
= 46
Thus, the mean weight of the group is 46 kilograms.

When Not to Use the Mean as the Measure of
Central Tendency?
Using mean as the measure of central
tendency brings out one major disadvantage, i.e.,
mean is particularly sensitive to outliers. This is for
the case when the values in a data are unusually
larger or smaller compared to the rest of the data.

Median as a Measure of Central Tendency
Median, one of the measures of central
tendency, is the value of the given data-set that is the
middle-most observation, obtained after arranging
the data in ascending order is called the median of
the data. The major advantage of using the median as
a central tendency is that it is less affected by outliers
and skewed data. We can calculate the median for
different types of data, grouped data, or ungrouped
data using the median formula.

For ungrouped data: For odd number of
observations, Median = [(n + 1)/2]th term. For
even number of observations, Median = [(n/2)th
term + ((n/2) + 1)th term]/2
For grouped data: Median = l + [((n/2) - c)/f] × h
where,
l = Lower limit of the median class
c = Cumulative frequency
h = Class size
n = Number of observations
Median class = Class where n/2 lies
Let us use the same example given above to find
the median now.

Example: The weights of 8 boys in kilograms: 45,
39, 53, 45, 43, 48, 50, 45. Find the median.
Solution:
Arranging the given data set in ascending
order: 39, 43, 45, 45, 45, 48, 50, 53
Total number of observations = 8
For even number of observation, Median = [(n/2)th
term + ((n/2) + 1)th term]/2
Median = (4th term + 5th term)/2 = (45 + 45)/2
⇒
= 45

Mode as a Measure of Central Tendency
Mode is one of the measures of the central
tendency, defined as the value which appears most
often in the given data, i.e. the observation with
the highest frequency is called the mode of data.
The mode for grouped data or ungrouped data
can be calculated using the mode formulas given
below,
Mode for ungrouped data: Most recurring
observation in the data set.

Mode for grouped data:
•L is the lower limit of the modal class
•h is the size of the class interval
•fm is the frequency of the modal class
•f1 is the frequency of the class preceding the
modal class
•f2 is the frequency of the class succeeding the
modal class

Example:
The weights of 8 boys in kilograms: 45, 39, 53,
45, 43, 48, 50, 45. Find the mode.
Solution:
Since the mode is the most occurring
observation in the given set.
Mode = 45

Empirical Relation Between Measures of Central
Tendency
The three measures of central tendency i.e. mean,
median, and mode are closely connected by the
following relations (called an empirical relationship).
2Mean + Mode = 3Median
For instance, if we are asked to calculate the mean,
median, and mode of continuous grouped data, then
we can calculate mean and median using the formulae
as discussed in the previous sections and then find
mode using the empirical relation.

Example: The median and mode for a given data
set are 56 and 54 respectively. Find the
approximate value of the mean for this data set.
2Mean + Mode = 3Median
2Mean = 3Median - Mode
2Mean = 3 × 56 - 54
2Mean = 168 - 54 = 114
Mean = 57

Measures of Central Tendency and Type of Distribution
Any data set is a distribution of 'n' number of
observations. The best measure of the central tendency of
any given data depends on this type of distribution. Some
types of distributions in statistics are given as,
•Normal Distribution
•Skewed Distribution
Let us understand how the type of distribution can affect
the values of different measures of central tendency.

Measures of Central Tendency for Normal
Distribution
Here is the frequency distribution table for a set of
data:

We can observe the histogram for the above-given
symmetrical distribution as shown below,

The above histogram displays a symmetrical
distribution of data. Finding the mean, median, and mode
for this data-set, we observe that the three measures of
central tendency mean, median, and mode are all located in
the center of the distribution graph. Thus, we can infer that
in a perfectly symmetrical distribution, the mean and the
median are the same. The above-given example had one
mode, i.e, it is a unimodal set, and therefore the mode is the
same as the mean and median. In a symmetrical distribution
that has two modes, i.e. the given set is bimodal, the two
modes would be different from the mean and median.

Measures of Central Tendency for Skewed
Distribution
For skewed distributions, if the distribution of
data is skewed to the left, the mean is less than the
median, which is often less than the mode. If the
distribution of data is skewed to the right, then the
mode is often less than the median, which is less
than the mean. Let us understand each case using
different examples.

Measures of Central Tendency for Right-Skewed
Distribution
Consider the following data-set and plot the
histogram for the same to check the type of distribution.

We observe the given
data set is an example of a
right or positively skewed
distribution. Calculating the
three measures of central
tendency, we find mean = 10,
median = 9, and mode = 9.
We, therefore, infer that if the
distribution of data is skewed
to the right, then the mode is,
lesser than the mean. And
median generally lies between
the values of mode and mean.

Measures of Central Tendency for Left-Skewed Distribution
Consider the following data-set and plot the histogram for the
same to check the type of distribution.
We observe the given data set is an example of left or
negatively skewed distribution. Calculating the three measures of
central tendency, we find mean = 15.75, median = 18, and mode
= 21. We, therefore, infer that if the distribution of data is skewed
to the left, then the mode is, greater than the median, which is
greater than the mean.

Let us summarize the above observations using the
graphs given below.

Important Notes on Measures of Central Tendency:
•The three most common measures of central tendency are mean,
median, and mode.
•Mean is simply the sum of all the components in a group or
collection, divided by the number of components.
•The value of the middle-most observation obtained after
arranging the data in ascending order is called the median of the
data.
•The value which appears most often in the given data i.e. the
observation with the highest frequency is called the mode of
data.
•The three measures of central tendency i.e. mean, median and
mode are closely connected by the following relations (called an

MesurMean, median, mode: typical values.

MesurMean, median, mode: typical values.

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