This document provides information on measures of central tendency and dispersion in statistics. It defines the mode, median and mean as common measures of central tendency, and how to calculate each one. For measures of dispersion, it discusses standard deviation, range, quartile deviation and variance. It provides examples of calculating each measure and their appropriate uses and limitations. The document is from an online statistics service that provides statistical analysis and calculations.
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Measure of central tendency
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Measure of central tendency
and dispersion
Dr. I. Kannan Ph.D
Associate Professor of Microbiology
Tagore Medical College and Hospital
Chennai ā 600127
dr.ikannan@tagoremch.com
statistics@tagoremch.com
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Introduction
ā¢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.
ā¢There are three common measures of central
tendency:
ā¢the mode
ā¢the median
ā¢the mean
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Mode example
ā¢ Marks obtained in physics (out of 50) by 25 students
of a class
ā¢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
ā¢ 21 mark is the Mode of the above data as it occurs more
frequently (6 times).
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How to calculate mode?
ā¢ Mode is calculated by creating frequency table.
ā¢ Most frequently appearing value is taken as mode.
ā¢ Mode is 7 mark in the below example as it appears more
frequently in the data
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Types of modes
ā¢Bimodal distribution: When a datum has
two modes (ie., two values has same high
frequency of distribution).
ā¢Multimodal distribution: when a datum has
more than two modes.
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Mode - application
ā¢It is rarely used by researchers as it is not an
useful measurement of central tendency.
ā¢It is not sensitive and never predicts the exact
measure of central tendency of the data set.
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Calculation of median
ā¢Arrange the data from highest to lowest
ā¢Find the score in the middle
ā¢Find the middle by the formula (N + 1) / 2
where N is the number of scores or values in
the data.
ā¢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)
ā¢ What is the median of the following values:
10 8 14 17 7 6 3 8 12 10 9
ā¢ Sort the values:
17 14 12 10 10 9 8 8 7 6 3
ā¢ Determine the middle value:
middle = (N + 1) / 2 = (11 + 1) / 2 = 6
ā¢ Middle value = median = 9
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Median example (when N is even number)
ā¢ What is the median of the following values:
28 18 19 44 16 11
ā¢ Sort the values:
44 28 19 18 16 11
ā¢ Determine the middle score:
middle = (N + 1) / 2 = (6 + 1) / 2 = 3.5
ā¢ Median = average of 3rd and 4th scores:
(19 + 18) / 2 = 18.5
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Mean
ā¢ 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
ļ„=
ą“¤š - Mean, āš is the sum of the values and n is the total number of values
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Mean ā example
ā¢Calculate the mean of the following data:
1 6 4 3 7
ā¢Sum the scores (ļX):
1 + 6 + 4 + 3 + 6 = 20
ā¢Divide the sum (ļX = 20) by the number of
scores (N = 5):
20 / 5 = 4
ā¢Mean = ą“¤š = 3
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Mean ā application
ā¢Mean is sensitive method and often used
method for the measurement of central
tendency provided the data should not be
skewed.
ā¢ It can be used for all the numerical data and
even for ordinal data.
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Standard deviation
ā¢Standard deviation is a measure of how each
value in a data set varies or deviates from the
mean
ā¢It is mandatory in all descriptive statistics done
in research that the mean value should
accompany with standard deviation
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Steps to calculate the standard deviation
ā¢Find the mean of the set of data ( ą“¤š)
ā¢Find the difference between each value and the
mean:
ā¢Square the difference
ā¢Find the average (mean) of these squares
ā¢Take the square root to find the standard
deviation
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Calculation of standard deviation ā example
X ą“„šæ X - ą“„šæ (X - ą“„šæ)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
Ī£ (X - ą“„šæ)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 - limitations
ā¢The range is a very crude measurement of the
spread of data.
ā¢It is extremely sensitive to outliers (Outliers- a
data that differs significantly from other data
values).
ā¢A single data value can greatly affect the value of
the range.
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Quartile deviation - example
ā¢ Calculate quartile deviation of the data set : 22, 12, 14, 7, 18, 16, 11,
15, 12.
ā¢ First, arrange data in ascending order to find Q3 and Q1 and avoid
any duplicates.
ā¢ 7, 11, 12, 13, 14, 15, 16, 18, 22
ā¢ Q1 = Ā¼ (9 + 1) =Ā¼ (10) - Q1=2.5 Term
ā¢ Q3 = Ā¾ (9 + 1) =Ā¾ (10) - Q3= 7.5 Term
First Quartile Q1 = Ā¼ (n+1)th term and Third Quartile Q3 = Ā¾ (n+1)th term
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Quartile deviation ā example ā cont..
ā¢Q1 = 2.5th term
= 2nd term + [0.5 Ć (3rd term ā 2nd term)]
= 11 + [0.5 x (12 -11)] = 11 + (0.5 x 1)
= 11+ 0.5 = 11.5
ā¢Q3 = 7.5th term
= 7th term + 0.5 Ć (8th term ā 7th term)
= 16 + [0.5 x (18 -16)]
= 16 + (0.5 x 2) = 16 + 1 = 17
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Quartile deviation ā uses and limitation
ā¢ QD depicts the extent to which the observations or the
values of the given dataset are spread out from the
mean.
ā¢ The Quartile deviation is used to study about the
dispersion of given data sets that lie in the main body of
the given series.
ā¢ The quartile deviation is good to use for descriptive
purposes especially when the data is highly skewed, or
multi-modal, or contains outliers
ā¢ However, it is not superior to standard deviation
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Box and Whisker Plot
ā¢ 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 - limitations
ā¢The mean and variance have a different unit
hence it is difficult to read the variance
along with the mean.
ā¢We will have to calculate Standard Deviation
in order to have a proper understanding of
the dispersion of the data along the mean.