Measures of Central Tendency and Dispersion (Week-07).pptx

MEASURES OF
CENTERAL TENDENCY
AND DISTRIBUTION
WEEK # 07

MEASURE OF CENTRAL TENDENCY
 Central tendency may be defined as value of the variate which is
thoroughly representative of the series or the distribution as a whole.
It refers to the number used to represent the center or middle set of a
data set.
 The measures of central tendency describe a distribution in terms of
its most “frequent”, “typical” or “average” data value.
But there are different ways of representing or expressing the idea of
“typicality”. Most often used for this purpose are the Mean (the
average), the Mode (the most frequently occurring score) and the

CHARACTERISTICS OF
SATISFACTORY AVERAGE
Central Tendency can be considered of an average since the average is
the measure of how closely other observations cling to it. Following
are some of its characteristics:
1. It should be rigidly defined and not left to be estimated.
2. Its computation should be based on all the observations.
3. An average should be capable of being computed easily and
rapidly.
4. An average should be as little affected by the fluctuation of

TYPES OF AVERAGE-CENTRAL
TENDENCY
1. Arithmetic Mean
2. Median
3. Mode
4. Geometric Mean
5. Harmonic Mean
6. Weighted Mean

ARITHMETIC MEAN
This is the only form of average which is of practical
importance. Its value depends upon all of the observations.
If X1, X2, X3…………XN are the N observation, their will be
given by following all the expression:

ARITHMETIC MEAN-MERITS AND
DEMERITS
 It is rigidly defined.
 It is based on all the observation.
 It is also least affected by the fluctuations of sampling.
 It is very much affected by the values at extremes.
 Its value may not coincide with any of the given values.
 It can not be located on the frequency curve like median
and mode nor it can be obtained by inspection.

MEDIAN
When all the observation are arranged in ascending or
descending order of magnitude, the middle is known as the
median.

MEDIAN-MERITS AND DEMERITS
 It can be readily calculated and rigidly defined.
It can be easily and readily obtained even if the extreme values are
not known.
 Median always remains the same whatsoever method of
computation be applied.
 It fails to remain “satisfactory average” when there is great
variation among the item of population.
 It can not be precisely expressed when it falls between two
values.
 It is more likely to be affected by fluctuation of sampling.

MODE
This is that value of the variable which occurs most
frequently or whose frequency is maximum.
Also, if several samples are drawn from a population, the
important value which appear repeatedly in all the sample is
called the mode.
The mode is
6 as it
occurs the
most in the
given data
set.

MODE-MERITS AND DEMERITS
It can be obtained simply by inspection.
 Neither the extremes are needed in its computation nor it is
affected by them.
 As it is the item of the maximum frequency, the same item is
the mode in every sample of the population.
 In many cases, there is no single and well defined mode.
 When there are more than one mode in the series becomes difficult
and takes much time to compute it.
 It computation is not based on all the observation.
 It, when multiplied by the number of observation, does not give the
total of all the observation, as it is the case with arithmetic mean.

GEOMETRIC MEAN
 Arithmetic mean, although gives equal weightage to all the
items, has got a tendency towards the higher values.
 Sometimes we want an average having a tendency towards
the lower values. In such case we take the help of geometric
mean.
 Geometric mean of given series is always less than its
arithmetic mean. It is defined by the following relations.
If a1, a2 , a3, an are N individual of a certain data and (G.M)
their geometric mean,

GEOMETRIC MEAN
 For a large series, we can use log method to calculate the
geometric mean as the simple multiplication can hectic.
log (G.M) = 1/N . (f1. log (X1) + f2. log (X2) + … fn. log (Xn))
We will take anti-log to calculate the geometric mean.
Here f represents the frequency of occurrence of an item in

GEOMETRIC MEAN-MERITS AND
DEMERITS
This average is also rigidly defined and its computation is
based on all the observation.
 Its computation is not so easy as that of the arithmetic mean.
As it is of too abstract a mathematical character, it is not widely
used.
It is difficult in its computation.
 If any item of the series is zero, G.M. becomes zero, and if
there are certain items which are negative, it becomes

HARMONIC MEAN
Harmonic mean of a number of quantities is the reciprocal
of the arithmetic mean of their reciprocal.

HARMONIC MEAN-MERITS AND
DEMERITS
 It is rigidly defined and the calculation is based on all the
observation.
 This is also not much affected by fluctuations of sampling.
 Harmonic mean is neither easily calculated nor
comprehensible.
As it gives high weightage to smaller values it is not very useful
in the analysis of the economical data.
 It is not a good representative of any set of observation unless
the smaller values are to be given high weightage.

EXAMPLE
The data on the length (mm) of 20 types of Steel Rods are given
below:
Find the Arithmetic Mean, Geometric Mean, Harmonic Mean, Median
and Mode.
138, 138, 132, 149,164, 146, 147, 152,115, 168, 176, 154,132, 146,
147,
140,144, 161, 142, 145

EXAMPLE-SOLUTION
The data on the length (mm) of 20 types of Steel Rods are given below:
Find the Arithmetic Mean, Geometric Mean, Harmonic Mean, Median and
Mode.
138, 138, 132, 149,164, 146, 147, 152,150, 168, 176, 154,132, 146, 147,
140,144, 161, 142, 145

S/R Length Log (X) 1/X
1 138 2.14 0.0072
2 138 2.14 0.0072
3 132 2.12 0.0076
4 149 2.17 0.0067
5 164 2.21 0.0061
6 146 2.16 0.0068
7 147 2.17 0.0068
8 152 2.18 0.0066
9 150 2.18 0.0067
10 168 2.23 0.0060
11 176 2.25 0.0057
12 154 2.19 0.0065
13 132 2.12 0.0076
14 146 2.16 0.0068
15 147 2.17 0.0068
16 140 2.15 0.0071
17 144 2.16 0.0069
18 161 2.21 0.0062
19 142 2.15 0.0070
20 145 2.16 0.0069
Total 2971 43.41 0.1354

Measures of Central Tendency and Dispersion (Week-07).pptx

The data in the ascending order of magnitude is 132, 132, 138, 138,
140, 142, 144, 145, 146,146,147, 147, 149, 150, 152, 154, 161,
164,168,176

The given set of data is polymodal type: 132,138,146 and
147 are Repeated twice. Hence there are four modes.:
Mode:
132
138
146
147
EXAMPLE-SOLUTION

MEASURE OF DISPERSION
It is quite obvious that for studying a series, a study of the
extent of scatter of the observation of dispersion is also
essential along with the study of the central tendency in
order to throw more light on the nature of the series.
Simply dispersion (also called variability, scatter, or spread)
is the extent to which a distribution is stretched or
squeezed.

DIFFERENT MEASURES OF
DISPERSION
1. Range
2. Mean Deviation
3. Standard Deviation
4. Variance
5. Quartile Deviation
6. Coefficient of Variation
7. Standard Error

RANGE
 Range is the simplest measure of dispersion.
 It is the difference the between highest and the lowest
terms of a series of observations.
Range = XH -XL
XH = Highest Variate Value
XL = Lowest Variate Value

RANGE-PROPERTIES
 Its value usually increases with the increase in the size of the
sample.
 It is usually unstable in repeated sampling experiments of the
same size and large ones.
It is very rough measure of dispersion and is entirely
unsuitable for precise and accurate studies.
 The only merits possessed by ‘Range’ are that it is (i) simple,

MEAN DEVIATION
The deviation without any sign
convention are known as
absolute deviations.
The mean of these absolute
deviations is called the mean
deviation.
If the deviations are calculated
from the mean, the measure of
dispersion is called mean
deviation about the mean.

RANGE-PROPERTIES
1. A notable characteristic of mean deviation is that it is the least
when calculated about the Median .
2. Standard deviation is not less than the mean deviation in a
discrete, i.e., it is either to or greater than the M.D. about Mean
3. When a greater accuracy is required, standard deviation is used as
a measure of dispersion.
4. When an average other than the A.M. Is calculated as a measure of
central tendency M. D. about that average is the only suitable
measure of dispersion.

STANDARD DEVIATION
 Its calculation is also based on the deviations from the
arithmetic mean.
 In case of mean deviation the difficulty we faced was that
the sum of the deviations from the arithmetic mean always
coming zero, is solved by taking these deviation
irrespective of sign in standard deviation.
 That difficulty is solved by squaring them and taking the
square root of their average.

STANDARD DEVIATION
Standard Deviation (S.D)
Where,
xi = An observation or variate value
μ = Arithmetic mean of the population
N = Number of given observations

STANDARD DEVIATION-
PROPERTIES
 It is rigidly defined.
 Its computation is based on all the observation.
 If all the variate values are the same, S.D.=0
 S.D. is least affected by fluctuations of sampling.
 It is used in computing different statistical quantities like,
regression coefficients, correlation coefficient, etc.

VARIANCE
 Variance is the square of the standard deviation.
Variance= (S. D.)2
 This term is now being used very extensively in
 The statistical analysis of the results from experiments.
 The variance of a population is generally represented by the
symbol σ² and its unbiased estimate calculated from the
sample, by the symbol s².

QUARTILE DEVIATION
This measure of dispersion is expressed in terms of
quartiles and known quartile deviation or semi-inter-
quartile range.
where,
Q1 = Lower Quartile
Q3 = Lower Quartile

QUARTILE DEVIATION
This measure of dispersion is expressed in terms of
quartiles and known quartile deviation or semi-inter-
quartile range.
where,
Q1 = Lower Quartile
Q3 = Lower Quartile
Qi= [i * (n + 1) /4] th observation

COEFFICIENT OF VARIATION
This is also a relative measure of dispersion and it is
especially important on account of the widely used measure
of central tendency and dispersion i.e., Arithmetic Mean and
Standard deviation.
C.V =
It is expressed in percentage, and used to compare the

STANDARD ERROR
The term ‘Standard error’ of any estimate is used for a measure of
the average magnitude of the difference between the sample
estimate and the population parameter taken over all possible
samples of the same size, from the population.
This term is applied for the standard deviation of the sampling
distribution of any estimate.
If S be the standard deviation of the sample size N, the estimate of

EXAMPLE
Find Range, Quartile Deviation, Mean Deviation about x and
Standard Deviation and their relative measure for the
following data:
1.3, 1.1, 1.0, 2.0, 1.7, 2.0, 1.9, 1.8, 1.6, 1.5
On arranging the above data in ascending order, we get:
1.0, 1.1, 1.3, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.0
Range: H - L= 2.0 - 1.0 = 1.0

S/R Xi Xi-µ (Xi-µ)^2
1 1.0 0.59 0.3481
2 1.1 0.49 0.2401
3 1.3 0.29 0.0841
4 1.5 0.09 0.0081
5 1.6 0.01 0.0001
6 1.7 0.11 0.121
7 1.8 0.21 0.441
8 1.9 0.31 0.961
9 2.0 0.41 0.1681
10 2.0 0.41 0.1681
Total 15.9 2.92 2.5397

MEAN DEVIATION
STANDARD DEVIATION

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