Measure of central tendency

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Online statistical service
Measure of central tendency
and dispersion
Dr. I. Kannan Ph.D
Associate Professor of Microbiology
Tagore Medical College and Hospital
Chennai – 600127
dr.ikannan@tagoremch.com
statistics@tagoremch.com

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Measure of central tendency

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Introduction
•A measure of central tendency is a descriptive
statistic that describes the number or typical
value of a set of values that helps the researcher
to assume the overall distribution of values.
•There are three common measures of central
tendency:
•the mode
•the median
•the mean

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Mode
•Mode is the most frequently occurring value in
the set of data.
•Mode can be used both for categorical data and
numerical data.

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Mode example
• Marks obtained in physics (out of 50) by 25 students
of a class
•23, 21, 32, 44, 21, 24, 25, 32, 34, 21, 33, 40,
37, 38, 41, 21, 43, 21, 28, 30, 34, 36, 44, 21,
18
• 21 mark is the Mode of the above data as it occurs more
frequently (6 times).

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How to calculate mode?
• Mode is calculated by creating frequency table.
• Most frequently appearing value is taken as mode.
• Mode is 7 mark in the below example as it appears more
frequently in the data

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Types of modes
•Bimodal distribution: When a datum has
two modes (ie., two values has same high
frequency of distribution).
•Multimodal distribution: when a datum has
more than two modes.

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Mode - application
•It is rarely used by researchers as it is not an
useful measurement of central tendency.
•It is not sensitive and never predicts the exact
measure of central tendency of the data set.

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Median
•Median is the midpoint of the dataset
arranged either from highest to lowest or
lowest to highest.

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Calculation of median
•Arrange the data from highest to lowest
•Find the score in the middle
➢Find the middle by the formula (N + 1) / 2
where N is the number of scores or values in
the data.
➢If N is even number the median is the
average of the middle two scores

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Median example (when N is odd number)
• What is the median of the following values:
10 8 14 17 7 6 3 8 12 10 9
• Sort the values:
17 14 12 10 10 9 8 8 7 6 3
• Determine the middle value:
middle = (N + 1) / 2 = (11 + 1) / 2 = 6
• Middle value = median = 9

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Median example (when N is even number)
• What is the median of the following values:
28 18 19 44 16 11
• Sort the values:
44 28 19 18 16 11
• Determine the middle score:
middle = (N + 1) / 2 = (6 + 1) / 2 = 3.5
• Median = average of 3rd and 4th scores:
(19 + 18) / 2 = 18.5

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Median – application
•The median is used mostly when the distribution
of values is skewed (which will be discussed
later).

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Mean
• The mean is the arithmetic average of all the values
obtained by adding up all the values and dividing by
the total number of values.
n
X
X
=
ത𝑋 - Mean, ∑𝑋 is the sum of the values and n is the total number of values

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Mean – example
•Calculate the mean of the following data:
1 6 4 3 7
•Sum the scores (X):
1 + 6 + 4 + 3 + 6 = 20
•Divide the sum (X = 20) by the number of
scores (N = 5):
20 / 5 = 4
•Mean = ത𝑋 = 3

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Mean – application
•Mean is sensitive method and often used
method for the measurement of central
tendency provided the data should not be
skewed.
• It can be used for all the numerical data and
even for ordinal data.

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Measure of dispersion

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Dispersion
•Dispersion describes how a set of data is
spread out.
➢It measures the variation of the items
among themselves
➢It measures the variation around the
average.

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Methods to study dispersion
•Standard deviation
•Range
•Quartile Deviation
•Variance

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Standard deviation
•Standard deviation is a measure of how each
value in a data set varies or deviates from the
mean
•It is mandatory in all descriptive statistics done
in research that the mean value should
accompany with standard deviation

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Formula for standard deviation calculation

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Steps to calculate the standard deviation
•Find the mean of the set of data ( ത𝑋)
•Find the difference between each value and the
mean:
•Square the difference
•Find the average (mean) of these squares
•Take the square root to find the standard
deviation

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Calculation of standard deviation – example
X ഥ𝑿 X - ഥ𝑿 (X - ഥ𝑿)2
5 16.4 - 11.4 129.96
12 16.4 - 4.4 19.36
16 16.4 - 0.04 0.16
21 16.4 4.6 21.16
28 16.4 11.6 134.56
Σ (X - ഥ𝑿)2 305.2
Standard deviation = 305.2/4 = 76.3 = 8.73
Calculate standard deviation for 5 12 16 21 28

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Range
•Range is the simple measure of dispersion
•It is defined as the difference between the
largest value and the smallest value.

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Range - Example
•Find the range for the data set 4,6,10,
15, 20
➢Maximum Value 20
➢Minimum Value 4
➢Range = 20 - 4 = 16

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Range - limitations
•The range is a very crude measurement of the
spread of data.
•It is extremely sensitive to outliers (Outliers- a
data that differs significantly from other data
values).
•A single data value can greatly affect the value of
the range.

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Quartile deviation
•The Quartile Deviation(QD) is the product of half
of the difference between the upper and lower
quartiles.

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Quartile deviation - example
• Calculate quartile deviation of the data set : 22, 12, 14, 7, 18, 16, 11,
15, 12.
• First, arrange data in ascending order to find Q3 and Q1 and avoid
any duplicates.
• 7, 11, 12, 13, 14, 15, 16, 18, 22
• Q1 = ¼ (9 + 1) =¼ (10) - Q1=2.5 Term
• Q3 = ¾ (9 + 1) =¾ (10) - Q3= 7.5 Term
First Quartile Q1 = ¼ (n+1)th term and Third Quartile Q3 = ¾ (n+1)th term

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Quartile deviation – example – cont..
•Q1 = 2.5th term
= 2nd term + [0.5 × (3rd term – 2nd term)]
= 11 + [0.5 x (12 -11)] = 11 + (0.5 x 1)
= 11+ 0.5 = 11.5
•Q3 = 7.5th term
= 7th term + 0.5 × (8th term – 7th term)
= 16 + [0.5 x (18 -16)]
= 16 + (0.5 x 2) = 16 + 1 = 17

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Quartile deviation – example – cont..
•QD. = Q3 – Q1 / 2
(17-11.50) / 2
=5.5/2
QD=2.75

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Quartile deviation – uses and limitation
• QD depicts the extent to which the observations or the
values of the given dataset are spread out from the
mean.
• The Quartile deviation is used to study about the
dispersion of given data sets that lie in the main body of
the given series.
• The quartile deviation is good to use for descriptive
purposes especially when the data is highly skewed, or
multi-modal, or contains outliers
• However, it is not superior to standard deviation

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Box and Whisker Plot
• A Box and Whisker Plot (or Box Plot) is a convenient way of visually
displaying the data distribution through their quartiles.

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Variance
• Variance is known as the square of the standard
deviation.
• It is the average of the squared differences from the
Mean.

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Variance - limitations
•The mean and variance have a different unit
hence it is difficult to read the variance
along with the mean.
•We will have to calculate Standard Deviation
in order to have a proper understanding of
the dispersion of the data along the mean.

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Measure of central tendency

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