This document discusses measures of central tendency including the mean, median, and mode. It provides definitions and formulas for calculating the arithmetic mean, geometric mean, harmonic mean, median, and mode. Examples are given for calculating each measure. The merits and demerits of each measure are outlined. In conclusion, the mean is affected by outliers while the median and mode are robust to outliers, and the mode is easiest to calculate by counting frequencies.

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Prepared by: Basudev Sharma
A Term Paper on
Submitted to
Rajendra Pradhan
Adjunct Professor
HICAST, Kathmandu
Prepared by
Basudev Sharma
M.Sc.Ag. (ABM), 3rd
Semester
HICAST, Kathmandu
July, 2018

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Prepared by: Basudev Sharma
A Term Paper on
Measure of Central Tendency
Introduction:
According to Simpson and Kafka 'a measure of central tendency is
typical value around which other figures aggregate'. According to
Croxton and Cowden "An average is a single value within the
range of the data that is used to represent all the values in the
series. Since an average is somewhere within the range of data, it
is sometimes called a measure of central value‘. According to Prof
Bowley “Measures of central tendency (averages) are statistical
constants which enable us to comprehend in a single effort the
significance of the whole.”
In general terms, central tendency is a statistical measure that
determines a single value that accurately describes the center of
the distribution and represents the entire distribution of scores.
The goal of central tendency is to identify the single value that is
the best representative for the entire set of data.
The following are the commonly used average or central
tendency:
i. Mean
a. Arithmetic mean or simple mean
b. Geometric mean
c. Harmonic mean
ii. Median
iii. Mode

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Arithmetic mean, Geometric mean and Harmonic means are
usually called mathematical averages while Mode and Median are
called positional averages.
Mean: The mean is calculated by adding the value of each
individual item in a group and dividing it by the total number of
items in the group. For example, if we meet 10 people, and the
sum of the ages of all attendees is 420, the mean age of the
attendees is 420 divided by 10, or 42. Three types of mean are
generally found.
a) Arithmetic Mean: The arithmetic mean is the most widely
used measure of location. It requires the interval scale. Its
major characteristics are; all values are used, it is unique,
the sum of the deviations from the mean is 0, and it is
calculated by summing the values and dividing by the
number of values. Arithmetic mean is also found in two
types. i.e. population mean and sample mean. Formula for
arithmetic mean is;
=
Σx
n
b) The Geometric Mean: The geometric mean is the average of
a set of products, the calculation of which is commonly used
to determine the performance results of an investment or
portfolio. It is technically defined as "the nth
root product of

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'n' numbers." The geometric mean must be used when
working with percentages, which are derived from values,
while the standard arithmetic mean works with the values
themselves. The formula for the geometric mean is written
as;
c) Harmonic Mean: The harmonic mean is an average. It is
calculated by dividing the number of observations by the
reciprocal of each number in the series. Thus, the harmonic
mean is the reciprocal of the arithmetic mean of the
reciprocals. Formula for harmonic mean is;
Median: The median is the value that is the mid-point of a group
of values, having an equal number of items in the group above
and below it. For instance, in a room with five people aged 23, 25,
37, 44 and 87, the median age is 37, as there are an equal number
of persons older and younger than 37. It can be found by
arranging all values from lowest to highest and determining the
value in the middle
Mode: The mode is the most frequent data value. Mode is the
value of the variable which is predominant in the given data

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series. Thus in case of discrete frequency distribution, mode is the
value corresponding to maximum frequency. Sometimes there
may be no single mode if no one value appears more than any
other. There may also be two modes (bimodal), three modes
(trimodal), or more than three modes (multi-modal). Example, in
the series of data 12, 11, 15, 12, 12, 11, 14, 17, 15, 12, 13, the
mode is 12 which repeated 4 times.
Objectives:
The main objectives of Measure of Central Tendency are
1) To condense data in a single value.
2) To facilitate comparisons between data.
3) To describe measures of central tendency.
4) To calculate mean (arithmetic mean, geometric mean &
harmonic mean), mode and median.
5) To find out partition values like quartiles, deciles, percentiles
etc.
6) To know about measures of dispersion like range, semi-
inter-quartile range, mean deviation, standard deviation.
7) To calculate moments, Karls Pearsion’s β and γ
coefficients, skewness, kurtosis.
Assumption:
1. The mean value helps us find the “true value”

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2. The mean value helps eliminate noise (imperfections) in
data
3. When the mean value is not reliable, it is because of
methodological flaws
4. The noise in data represents the effects of variables
unrelated to the one being measured
5. Median refers to position when data is put in an array not
the actual number.
Application or Uses:
The mean, median and mode are measures of central tendency
within a distribution of numerical values. The mean is more
commonly known as the average. The median is the mid-point in
a distribution of values among cases, with an equal number of
cases above and below the median. The mode is the value that
occurs most often in the distribution.
Mean, median and mode reveal different aspects of the data. Any
one will give a general idea, but may mislead as well; having all
three will give a more complete picture. For example, for the
data: 5, 7, 6, 127, we get a mean of 36.25, a number that fits the
arithmetic but seems a little out of place. The median, 6.5, may
have more relevance to the series, but says nothing about the
outlier. Since the series has no repeated numbers, it has no
mode; this also reveals valuable information about the data.

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As a consumer of information, it is important that we can make
decisions about which measures are most useful. Just because we
can use mean, median and mode in the real world doesn't mean
that each measure applies to any situation. For example, if we
wish to find the average grade on a test for our class but one
student fell asleep and scored a 0, the mean would show a much
lower average because of one low grade, while the median would
show how the middle group of students scored. Using these
measures in everyday life involves not only understanding the
differences between them, but also which one is appropriate for
a given situation.
No single average can be regarded as the best or most suitable
under all circumstances. Each average has its merits and demerits
and its own particular field of importance and utility. A proper
selection of an average depends on the 1) nature of the data and
2) purpose of enquiry or requirement of the data.
Arithmetic mean satisfies almost all the requisites of a good
average and hence can be regarded as the best average but it
cannot be used
1) In case of highly skewed data.
2) In case of uneven or irregular spread of the data.
3) In open end distributions.
4) When average growth or average speed is required.

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5) When there are extreme values in the data.
Except in these cases arithmetic mean is widely used in practice.
Median is the best average in open end distributions or in
distributions which give highly skew or j or reverse j type
frequency curves. In such cases A.M. gives unnecessarily high or
low value whereas median gives a more representative value. But
in case of fairly symmetric distribution there is nothing to choose
between mean, median and mode, as they are very close to each
other.
Mode is especially useful to describe qualitative data. According
to Freunel and Williams, consumer preferences for different kinds
of products can be compared using modal preferences as we
cannot compute mean or median. Mode can best describe the
average size of shoes or shirts.
Geometric mean is useful to average relative changes, averaging
ratios and percentages. It is theoretically the best average for
construction of index number. But it should not be used for
measuring absolute changes.
Harmonic mean is useful in problems where values of a variable
are compared with a constant quantity of another variable like
time, distance travelled within a given time, quantities purchased
or sold over a unit.

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In general we can say that arithmetic mean is the best of all
averages and other averages may be used under special
circumstances.
Calculation Procedure
Mean (Arithmetic Mean)
To calculate the arithmetic mean of a set of data we must first
add up (sum) all of the data values (x) and then divide the result
by the number of values (n). Since Σis the symbol used to indicate
that values are to be summed we obtain the following formula for
the mean ( ).
=
Σx
n
Example: Find the mean of: 6, 8, 11, 5, 2, 9, 7, 8
=
Σx
=
6+8+11+5+2+9+7+8
=
56
= 7
n 8 8
Mean (Geometric Mean): Calculate the geometric mean of set of
data 4, 8, 3, 9 and 17, first multiply the numbers together and
then take the 5th root because there are 5 numbers.
GM = (4*8*3*9*17)(1/5) = 6.81
Actually as a mathematical rule, the geometric mean will always
be equal to or less than the arithmetic mean.

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Mean (Harmonic Mean): Calculation of harmonic mean of the
data 1, 5, 8, & 10 is given as;
HM =
4
=
4
=
4
=2.807018
1
+
1
+
1
+
1 57 1.425
1 5 8 10 40
Arithmetic mean is always greater than the harmonic mean. The
harmonic mean cannot be made arbitrarily large by changing
some values to bigger ones
Median
The median value of a set of data is the middle value of the
ordered data. That is, the data must be put in numerical order
first.
Worked examples, Find the median of the following:
a) 11, 4, 9, 7, 10, 5, 6
Ordering the data gives 4, 5, 6, 7 , 9, 10, 11 and the middle value
is 7.
b) 1, 3, 0.5, 0.6, 2, 2.5, 3.1, 2.9
Ordering the data gives 0.5, 0.6, 1, 2, 2.5, 2.9, 3, 3.1

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n+1
2
(2+2.5)
2
Here there is a middle pair 2 and 2.5. The median is between
these 2 values i.e. the mean of them = 2.25
In general the median is at the th value.
Mode
The modal value of a set of data is the most frequently occurring
value. Worked example is, find the mode for: 2, 6, 3, 9, 5, 6, 2, 6
It can be seen that the most frequently occurring value is 6.
(There are 3 of these).
Merits and demerits of the central tendency
Merits of Mean
1. It is rigidly defined.
2. It is easy to understand & easy to calculate.
3. It is based upon all values of the given data.
4. It is capable of further mathematical treatment.
5. It is not much affected by sampling fluctuations.
Demerits of Mean
1. It cannot be calculated if any observations are missing.
2. It cannot be calculated for the data with open end classes.
3. It is affected by extreme values.
4. It cannot be located graphically.
5. It may be number which is not present in the data.
6. It can be calculated for the data representing qualitative
characteristic.

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Merits of Median
1. It is rigidly defined.
2. It is easy to understand & easy to calculate.
3. It is not affected by extreme values.
4. Even if extreme values are not known median can be calculated.
5. It can be located just by inspection in many cases.
6. It can be located graphically.
7. It is not much affected by sampling fluctuations.
8. It can be calculated for data based on ordinal scale.
Demerits of Median
1. It is not based upon all values of the given data.
2. For larger data size the arrangement of data in the
increasing order is difficult process.
3. It is not capable of further mathematical treatment.
4. It is insensitive to some changes in the data values.
Merits of Mode
1. It is easy to understand & easy to calculate.
2. It is not affected by extreme values or sampling fluctuations.
3. Even if extreme values are not known mode can be calculated.
4. It can be located just by inspection in many cases.
5. It is always present within the data.
6. It can be located graphically.
7. It is applicable for both qualitative and quantitative data.

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Demerits of Mode
1. It is not rigidly defined.
2. It is not based upon all values of the given data.
3. It is not capable of further mathematical treatment.
Conclusions
The arithmetic mean is the only measure of central tendency
where the sum of the deviations of each value from the mean is
zero! It is easily affected by extremes, such as very big or small
numbers in the set (non-robust). Extreme numbers relative to the
rest of the data is called outliers!
The Median is the midpoint of the values after they have been
ordered from the smallest to the largest. Equivalently, the median
is a number which divides the data set into two equal parts, each
item in one part is no more than this number, and each item in
another part is no less than this number. If the total number of
items n is an odd number, then the number on the (n+1)/2
position is the median; If n is an even number, then the average
of the two numbers on the n/2 and n/2+1 positions is the median.
It is easy to calculate but does not allow easy mathematical
treatment. It is not affected by extremely large or small numbers
(robust).
Mode is the number that has the highest frequency. It is easy to
calculate just by counting the repeated number and mode is also
not affected by extremely large or small numbers.

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References
 https://www.slideshare.net/raiuniversity/mba-i-qt-
unit2measures-of-central-tendency?from_action=save
 http://reflectd.co/2013/08/10/how-mean-is-the-mean/
 https://www.jcu.edu.au/__data/assets/pdf_file/0018/115830/B
asic-Statistics-3_Describing-Data_Measures-of-Central-
Tendency.pdf
 https://www.investopedia.com/terms/g/geometricmean.asp
 https://www.investopedia.com/terms/h/harmonicaverage.asp
 https://sciencing.com/do-mode-mean-average-everyday-
8752223.html
 https://sciencing.com/uses-mean-median-mode-6323388.html
 https://www.slideshare.net/HardikAgarwal3/applications-of-
central-tendency?from_action=save
 https://www.slideshare.net/Aeijaz/statistical-analysis-and-its-
applications?utm_source=slideshow&utm_medium=ssemail&ut
m_campaign=download_notification
 http://www.lboro.ac.uk/media/wwwlboroacuk/content/mlsc/do
wnloads/mean_median_mode.pdf