E_Statistics_chapter_4

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Transcript E_Statistics_chapter_4

© .S. B. Bhattacharjee
Ch 4_1
© .S. B. Bhattacharjee
Ch 4_2
What is meant by a measure of central
tendency?
An average is frequently referred to as a
measure of central tendency or central value.
This is a single value which is considered the
most representative or typical value for a given
set of data. It is the value around which data in
the set tend to cluster.
For example: The average starting salary for
social workers is TK.15,000 per Year and it gives
some idea of how much variety or heterogeneity
there is in the distribution )
© .S. B. Bhattacharjee
Ch 4_3
What are the objectives of averaging?
The following are two main objectives of the
study of average:
To get one single value that describes the characteristics
of the entire data. Measures of central value, by condensing
the mass of data in one single value, enable us to get an
idea of the entire data. Thus one value can represent
thousands, lakhs and even millions of values.
For example: It is impossible to remember the individual
incomes of millions of earning people of Bangladesh and
even if one could do it there is hardly any use. But if the
average income is obtained, we get one single value that
represents the entire population. Such a figure would throw
light on the standard of living of an average Bangladeshi.
© .S. B. Bhattacharjee
Ch 4_4
What are objectives of averaging?
To facilitate comparison. Measures of central value,
by reducing the mass of data in one single figure,
enable comparisons to be made. Comparison can be
made either at a point of time or over a period of time.
For example: The figure of average sales for December
may be compared with the sales figures of previous
months or with the sales figure of another competitive
firm.
© .S. B. Bhattacharjee
Ch 4_5
What should be the properties of a
good average?
Since an average is a single value representing a
group of values, it is desirable that such a value
satisfies the following properties:
It should be easy to understand: Since statistical methods
are designed to simplify complexity, it is desirable that an
average be such that can be readily understood, its use is
bound to be very limited.
It should be simple to compute: Not only an average
should be easy to understand but it also should be simple
to compute so that it can be used widely.
© .S. B. Bhattacharjee
Ch 4_6
What should be the properties of a good
average?
It should be based on all the observations: The average
should depend upon each and every observation so that if
any of the observation is dropped average itself is altered.
It should be rigidly defined: An average should be properly
defined so that it has one and only one interpretation. It
should preferably be defined by an algebraic formula so
that if different people compute the average from the same
figures they all get the same answer (Barring arithmetical
mistakes).
© .S. B. Bhattacharjee
Ch 4_7
What should be the properties of a good
average?
It should be capable of further algebraic treatment: We
should prefer to have an average that could be used for
further statistical computations.
For example: If we are given separately the figures of
average income and number of employees of two or more
companies we should be able to compute the combined
average.
© .S. B. Bhattacharjee
Ch 4_8
What should be the properties of a good
average?
It should have sampling stability: We should prefer to get
a value which has what the statisticians call ‘Sampling
stability’. This means that if we pick 10 different groups of
college students, and compute the average of each group,
we should expect to get approximately the same values.
It should not be unduly affected by the presence of
extreme values: Although each and every observation
should influence the value of the average, none of the
observations should influence it unduly. If one or two very
small or very large observations unduly affect the average,
i.e., either increase its value or reduce its value, the
average cannot be really typical of the entire set of data. In
other words, extremes may distort the average and reduce
its usefulness.
© .S. B. Bhattacharjee
Ch 4_9
What are various measures of central
tendency ?
The following are the measures of central
tendency which are generally used in Business:
Mean
Arithmetic mean
Geometric mean
Harmonic mean
Median
Mode
© .S. B. Bhattacharjee
Ch 4_10
How would you select a specific
measure of central tendency?
Selection of a measure of central tendency
largely depends on the nature of data.
Continued…….
© .S. B. Bhattacharjee
Ch 4_11
Nature of data
Figure:1
Measure of
Central tendency?
Yes
Yes
Nominal?
Mode
No
Yes
No
Mode
Ordinal?
No
Distribution
Skewed?
Yes
No
Mean
© .S. B. Bhattacharjee
Ch 4_12
What are various types of averages
or means?
Mean
Arithmetic mean
Geometric mean
Harmonic mean
Continued…….
© .S. B. Bhattacharjee
Ch 4_13
What is arithmetic mean?
The arithmetic mean, often simply referred to
as mean, is the total of the values of a set of
observations divided by their total number of
observations.
© .S. B. Bhattacharjee
Ch 4_14
What are the methods of computing
arithmetic mean?
For ungrouped data, arithmetic mean may be
computed by applying any of the following methods:
Direct method
Short-cut method
© .S. B. Bhattacharjee
Ch 4_15
X 1 , X 2 , ...... X N
What is direct method?
Thus, if X1 , X 2 , ...... X N represent the values of N
items or observations, the arithmetic mean
denoted by X is defined as:
N
X
X 1  X 2  ....... X N
© .S. B. Bhattacharjee
N

Xi

i 1
N
Ch 4_16
Example:
The monthly income (in Tk) of 10 employees working in
a firm is as follows:
4487
4489
4493 4502
4446 4475 4492 4572 4516 4468
Find the average monthly income. Applying the
formula we get:
 X  4487+4493+4502+4446+4475+4492+4572+4516+4468
+4489 = 44.949
X 
X
N
44940

 4494
10
Hence the average monthly income is Tk.4494.
© .S. B. Bhattacharjee
Ch 4_17
What is short cut method?
A short cut is one in which the arithmetic mean is
calculated by taking deviations from any arbitrary point .
The formula for computing mean by short cut method is
as follows:
d

X  A
d = (X – A )
Where ,
and
N
A = Arbitrary point (or assumed mean)
It should be noted that any value can be taken as
arbitrary point and the answer would be the same as
obtained by the direct method.
© .S. B. Bhattacharjee
Ch 4_18
Example: 2
Calculation of average monthly income by the short–
cut method from the following data. In this case 4460
is taken as the arbitrary point.
Calculation of average income
X
(X - 4460)
(TK)
(TK)
4487
+27
Assumed mean
4493
+33
= Tk.4460
4502
+42
4446
-14
4475
+15
4492
+32
4572
+112
4516
+56
4468
+8
4489
+29
© .S. B. Bhattacharjee
 = +340
Ch 4_19
Applying the formula we get:
d
340

X  A
 4460
 4460 34  Tk.4494
N
10
One may find that short-cut method takes more time as
compared to direct method. However, this is true only
for ungrouped data. In case of grouped data,
considerable saving in time is possible by adopting the
short-cut method.
© .S. B. Bhattacharjee
Ch 4_20
What are the methods of estimating
average from grouped data?
Direct method
Short-cut method
Continued…..
© .S. B. Bhattacharjee
Ch 4_21
Example 3
Compute the average from the following data
by direct method.
© .S. B. Bhattacharjee
Profits
(Tk. Lakhs)
No. of
Companies
200-400
500
400-600
300
600-800
280
800-1000
120
1,000-1,200
100
1.200-1,400
80
1.400-1,600
20
Ch 4_22
Direct Method
The formula for estimating average from grouped
data by direct method is:
fx

X
N
Where,
X = mid-point of various classes
f= the frequency of each class
N= the total frequency
Continued…….
© .S. B. Bhattacharjee
Ch 4_23
Calculate the average profits for all the companies.
Profits
(Tk. Lakhs)
Mid-points
X
No. of
Companies
f
fx
200-400
300
500
1,50,000
400-600
500
300
1,50,000
600-800
700
280
1,96,000
800-1000
900
120
1,08,000
1,000-1,200
1100
100
1,10,000
1.200-1,400
1300
80
1,04,000
1.400-1,600
1500
20
30,000
N=1,400
© .S. B. Bhattacharjee
Continued…….
 fx  8,48,000
Ch 4_24
fx 8,48,000

X

 605.71
N
1,400
Thus, the average profit is Tk. 605.71 lakhs.
© .S. B. Bhattacharjee
Ch 4_25
Short-cut Method
When short-cut method is used, the following
formula is applied.
fd

X  A
i
N
Where, A = Arbitrary point (assumed mean)
XA
d
i
and i = size of the equal class interval
© .S. B. Bhattacharjee
Ch 4_26
Example:
Mid-points
f
X  A
 d 
i
fd
19.5-29.5
24.5
2
-3.50
-7.00
29.5-39.5
34.5
12
-2.50
-30.00
39.5-49.5
44.5
15
-1.50
-22.50
49.5-59.5
54.5
20
-0.50
-10.00
59.5-69.5
64.5
18
+0.50
9.00
69.5-79.5
74.5
10
+1.50
15.00
79.5-89.5
84.5
9
+2.50
22.50
89.5-99.5
94.5
4
+3.50
14.00
Marks
N = 90
© .S. B. Bhattacharjee
Continued…….
 fd  9.0
Ch 4_27

fd

X  A
i
N

 9.0
 59.5 
 10
90
9.0
 59.5 
 10
90
 59.5  1  58.5
Here, assumed mean, A = 59.5
class-interval,
i =10
© .S. B. Bhattacharjee
Ch 4_28
What are the mathematical properties of
arithmetic mean?
The important mathematical properties of arithmetic
mean are:
1.The algebraic sum of the deviations of all the observations
from arithmetic mean is always zero, i.e.,
X  X  0 This
shall be clear from the following example:

X
10
20
30
40
50
 X 150
© .S. B. Bhattacharjee

X  X 
-20
- 10
0
+10
+20
X  X   0
Continued……
Ch 4_29
2. The sum of the squared deviations of all the
observations from arithmetic mean is minimum, that
is, less than the squared deviations of all the
observations from any other value than the mean.
The following example would clarify the point:
X
Here,
X 
X
N
20

5
4
X  X 
X  X 2 
2
-2
4
3
-1
1
4
0
0
5
+1
1
6
+2
4
2


X

X
x

20
X  X   0 

© .S. B. Bhattacharjee
Continued……
 10
Ch 4_30
3.If we have the arithmetic mean and number of
observations of two or more than two related
groups, we can compute combined average
of these groups by applying the following
formula:
X 12
N1 X 1  N 2 X 2

N1  N 2
Continued……
© .S. B. Bhattacharjee
Ch 4_31
Where,
X 12
= Combined mean of the two groups.
X1
= Arithmetic mean of the first group.
X2
= Arithmetic mean of the second group.
N1 = Number of observations in the first group.
N2
= Number of observations in the second group.
Continued………
© .S. B. Bhattacharjee
Ch 4_32
Example:
There are two branches of a company employing 100 and 80
employees respectively. If arithmetic means of the monthly
salaries paid by two branches are Tk. 4570 and Tk. 6750
respectively, find the arithmethtic mean of the salaries of the
employees of the company as a whole.
Applying the following formula, we get:
X 12
X 12
© .S. B. Bhattacharjee
N1 X 1  N 2 X 2

N1  N 2

100 4570  80  6750

100  80
997000

180
 Tk .5538.89
Ch 4_33
What are the merits of arithmetic mean?
Merits:
It possesses first six out of seven characteristics of
a good average.
The arithmetic mean is the most popular average in
practice.
It is a large number of characteristics. ???
Continued……
© .S. B. Bhattacharjee
Ch 4_34
What are the limitations of arithmetic
mean?
Limitations:
Arithmetic mean is unduly affected by the presence
of extreme values.
In opened frequency distribution, it is difficult to
compute mean without making assumption regarding
the size of the class-interval of the open-end classes.
The arithmetic mean is usually neither the most
commonly occurring value nor the middle value in a
distribution.
In extremely asymmetrical distribution, it is not a
good measure of central tendency.
© .S. B. Bhattacharjee
Ch 4_35
What is meant by weighted arithmetic mean?
A weighted average is an average estimated with due
weight or importance given to all the observations. The
terms ‘weight’ stands for the relative importance of the
different observations.
Problem: An important problem that arises while using
weighed mean is selection of weights. Weights may be
either actual or arbitrary, i.e., estimated.
Uses: Weighted mean is specially useful in problems
relating to the construction of index numbers and
standardized birth and death rates.
Continued….
© .S. B. Bhattacharjee
Ch 4_36
The formula for computing weighted arithmetic mean is
given below:
WX

Xw 
,
W
where,
Xw
=The weighted arithmetic mean
X = The variable.
W = Weights attached to the variable X.
© .S. B. Bhattacharjee
Ch 4_37
Example:
A contractor employs three types of workers – male,
female and children. To male worker he pays Tk. 100 per
day, to a female worker Tk. 75 per day and to a child
worker Tk. 35 per day. What is the average wage per day
paid by the contractor?
Solution: The simple average wage is not arithmetic
mean, i.e., 100  75  35  Tk . 70 per day . If we assume that
30
the number of male, female and child workers is the
same, this answer would be correct. For example, if we
take 10 workers in each case then the average wage
would be
10  100  10  75  10  35
X
© .S. B. Bhattacharjee


 
 
10  10  10
1000 750  350

 Tk .70
30
Ch 4_38
Continued….

Let us assume that the number of male, female
and child workers employed are 20, 15 and 5,
respectively. The average wage would be the
weighted mean calculated as follows:
Continued…..
© .S. B. Bhattacharjee
Ch 4_39
Example:
Wage per day
(Tk)
X
100
75
35
No. of workers WX
W
20
15
5
2000
1125
175
W= 40
WX = 3300
WX

Xw 
W
3300

 82.50
40
Hence the average wage per day paid by the contractor
is Tk. 82.50.
© .S. B. Bhattacharjee
Ch 4_40
What is meant by geometric mean?
The geometric mean (GM) is defined as Nth root of
the product of N observations of a given data. If
there are two observations, we take the square toot;
if there are three observations, the cube root; and so
on, The formula is:
G.M  N ( X 1)( X 2)( X 3)...(X N ) ,
where, X1, X2, X3….., XN refer to the various
observations of the data.
© .S. B. Bhattacharjee
Ch 4_41
How is geometric mean computed?
To simplify calculations logarithms are used.
Log G.M . 
log X 1  log X 2  ....log X N
N
  log X 

 G.M .  antilog


N


© .S. B. Bhattacharjee
log X


N
Ch 4_42
How is geometric mean calculated?
In ungrouped data, geometric mean is calculated with
the help of the following formula:
  log X
G.M .  Antilog

N





In grouped data, first midpoints are found out and then
the following formula is used for calculating geometric
mean :
  f log X
G.M .  Antilog

N


, Where


X = midpoint
© .S. B. Bhattacharjee
Ch 4_43
What are the applications of geometric
mean?
Geometric mean is specially useful in the following
cases:
The geometric mean is used to find the average per
cent increase in sales, production, population or other
economic or business data. For example, from 2002 to
2004 prices increased by 5%, 10% and 18%
respectively. The average annual increase is 11% as
given by the arithmetic average but it is 10.9% as
obtained by the geometric mean.
This average is also useful in measuring the growth
of population, because population increases in
geometric progression.
© .S. B. Bhattacharjee
Continued…….
Ch 4_44
Geometric mean is theoretically considered to be
the best average in the construction of index number.
It makes index numbers satisfy the time reversal test
and gives equal weights to equal ratio of change.
It is an average which is most suitable when large
weights have to be given to small values of
observations and small weights to large values of
observations, situations which we usually come
across in social and economic fields.
© .S. B. Bhattacharjee
Ch 4_45
What are the merits of geometric
mean?
Merits
Geometric mean is highly useful in averaging ratios
and percentages and in determining rates of increase
and decrease.
It is also capable of algebraic manipulation.
For example, if the geometric mean of two or more
series and their numbers of observations are known,
a combined geometric mean can easily be calculated.
Continued…….
© .S. B. Bhattacharjee
Ch 4_46
What are the limitations of geometric
mean?
Limitations
Compared to arithmetic mean, this average is
more difficult to compute and interpret.
Geometric mean cannot be computed when there
are both negative and positive values in a series
or more observations are having zero value.
© .S. B. Bhattacharjee
Ch 4_47
What is meant by harmonic mean?
The harmonic mean is based on the reciprocal of the
numbers averaged. It is defined as the reciprocal of
the arithmetic mean of the reciprocal of the
individual observation.
Continued…….
© .S. B. Bhattacharjee
Ch 4_48
How is harmonic mean computed?
The formula for estimating harmonic mean is
follows:
N
H .M . 
 1
1
1
1 




 .... 
X3
XN 
 X1 X 2
as
Where number of observations is large, the
computation of harmonic mean in the above manner
becomes tedious.
Continued…….
© .S. B. Bhattacharjee
Ch 4_49
To simplify calculations, we obtain reciprocals of
the various observations and apply the following
formulae:
N
For ungrouped data,= H .M 
1
  X 
N
For grouped data, =
N
H .M . 
or
1

 f 
  f  X    X . 
Continued…….
© .S. B. Bhattacharjee
Ch 4_50
Calculation of Harmonic Mean
X
10
20
25
40
50
N
H .M 
1
  X
© .S. B. Bhattacharjee
1
X
0.100
0.050
0. 04
0.025
0.020
1
 X  0  235
5

 21 28
 0  235


Continued…….
Ch 4_51
Calculation of Harmonic Mean
Variable
X
f
1
f 
X
0-10
10-20
5
15
8
15
1.600
1.000
20-30
30-40
40-50
25
35
45
20
4
3
0.800
0.114
0.067
1

  f  X   3  581
H .M . 
© .S. B. Bhattacharjee
N
1

  f  X 

50
 13  96
3  581
Ch 4_52
What are the applications of harmonic
mean?
The harmonic mean is restricted in its field of
applications. It is useful for computing the average rate
of increase of profits or average speed at which a
journey has been performed or the average price at
which an article has been sold. For example, if a man
walked 20 km., in 5 hours, the rate of his walking speed
can be expressed as follows:
20 km.
 4 km. per hour,
5 hours
Continued…….
© .S. B. Bhattacharjee
Ch 4_53
Where X, the unit of the first term is an hour and the
unit of the second term is a kilometer.
5 hours 1
 hour per km.,
20 km.
4
Example: In a certain factory a unit of work is completed
by A in 4 minutes, by B in 5 minutes, by C in 6 minutes,
by D in 10 minutes and by E in 12 minutes.
(a) What is the average number of units of work completed
per minute?
(b) At this rate how many units will they complete in a sixhour day?
Continued…….
© .S. B. Bhattacharjee
Ch 4_54
The average number of units per minutes will be
obtained by calculating the harmonic mean.
1
X
X
4
5
6
0.250
0.200
0.167
10
12
0.100
0.083

© .S. B. Bhattacharjee
1
 08
X
N
H .M . 
1
  X
Continued…….
5

 6  25
 08


Ch 4_55
Example:
A toy factory has assigned a group of 4 workers to
complete an order of 1, 400 toys of certain type. The
productive rates of the four workers are given below:
Workers
A
B
Productive rates
4 minutes per toy
6 minutes per toy
C
D
10 minutes per toy
15 minutes per toy
Find the average minutes per toy by the group of
workers.
© .S. B. Bhattacharjee
Continued…….
Ch 4_56
If we assume that each of the four workers is
assigned the same number of toys (constant value) to
1,400
meet the order, or
= 350 toys per worker, the
4
arithmetic mean would give the correct answer.
4  6  10  15
X 
4
35

4
3
8
minutes per toy.
4
Continued…….
© .S. B. Bhattacharjee
Ch 4_57
Verification
Time required by A to complete 350 toys ×4 =1,400 minutes
Time required by B to complete 350 toys ×6 =2,100 minutes
Time required by C to complete 350 toys ×10 =3,500 minutes
Time required by D to complete 350 toys ×15 =5,250 minutes
12,250 minutes.
In 12,250 minutes, 1,400 toys will be completed.
Hence, in completing one toy time taken will be
12,250
3
 8 minutes
1,400
4
© .S. B. Bhattacharjee
Continued…….
Ch 4_58
However, if we assume that each worker works the
same amount of time but produces different number of
toys, harmonic mean would be more appropriate. This
assumption is more true in practice (people working
same amount of time but having different output)
4  60
6
H .M  .

 6 minutes per toy
1 1 1
1
35
7
  
4 6 10 15
4
Time required to complete 1,400 toys
1,400  48

 9,600 minutes
7
Continued…….
© .S. B. Bhattacharjee
Ch 4_59
Verification:
Each workers works for
9,600
 2,400 minutes
4
2400
 600
Toys produced by A in 2400 minutes
4
2400

 400
Toys produced by B in 2400 minutes
6
Toys produced by C in 2400 minutes  2400  240
10
2400
Toys produced by D in 2400 minutes

 160

15
Total = 1,400
© .S. B. Bhattacharjee
Ch 4_60
What are merits of harmonic mean?
Merits
The harmonic mean, like the arithmetic mean and
geometric
mean,
is
computed
from
all
observations.
It is useful in special cases for averaging rates.
© .S. B. Bhattacharjee
Ch 4_61
What are the limitations of harmonic
mean?
Limitations
Harmonic mean cannot be computed when there
are both positive and negative observations or one
or more observations have zero value.
It also gives largest weight to
observations and as such is not
representation of a statistical series.
smallest
a good
It is in dealing with business problems harmonic
mean is rarely used.
© .S. B. Bhattacharjee
Ch 4_62
What is meant by median ?
Median is a point in a distribution of scores above and
below which exactly half of the cases fall. This is a
value which appears in the middle of ordered sequence
of values. This is also known as positional average.
The term ‘position’ refers to the place of a value in a
series.
Example: If the income of five persons is Tk.7000,
7200,7500,7600,7800, then the median income would be
Tk.7500.
© .S. B. Bhattacharjee
Ch 4_63
Apply the formula : Median = Size of
N 1
th
2
observation.
From the following data of wages of 7 workers, compute
the median wage:
Wages (in Tk.) 4600, 4650, 4580, 4690, 4660, 4606, 4640
© .S. B. Bhattacharjee
Ch 4_64
Calculation of Median from ungrouped
data
S. No.
Wages arranged in
ascending order
1
2
3
4
4580
4600
4606
4640
5
6
7
4650
4660
4690
© .S. B. Bhattacharjee
Ch 4_65
N 1
7 1
Median  Size of
th observatio n 
 4th observatio n
2
2
Value of 4th observation is 4640. Hence median wages = 4640.
In the above illustration, the number of observations was odd
and, therefore, it was possible to determine value of 4th
observation. When the number of observations are 8 the median
would be the value of 8  1 = 4.5th observation.
2
For finding out the value of 4.5th observation, we shall take the
average of 4th and 5th observation. Hence the median shall be
4640  4650
 4645
2
© .S. B. Bhattacharjee
Ch 4_66
Formula for calculation of median from grouped data
N
 p.c. f .
Median L  2
 i,
f
Where L = Lower level of median class i.e.
the class in which the middle observation
in the distribution lies
p.c.f.= Preceding cumulative frequency to the
median class.
i = The class-interval of the median class
Continued…..
© .S. B. Bhattacharjee
Ch 4_67
Calculation of Median Marks
Marks
19.5-29.5
f
2
c. f
2
29.5-39.5
39.5-49.5
49.5-59.5
59.5-69.5
12
15
20
18
14
29
49
67
69.5-79.5
79.5-89.5
89.5-99.5
10
9
4
77
86
90
Continued…..
© .S. B. Bhattacharjee
Ch 4_68
Median = Size of
N
2
N

the
2
the observation
observation
= 45th observation
Hence median lies in the class 49.5-59.5
Continued…..
© .S. B. Bhattacharjee
Ch 4_69
N
 p.c. f
Here, L = 49.5, N =
2
 Median L 
i
90, p.c.f = 29 , f =20
f
i = 10.
45  29
 49.5 
 10
20
 N 90

16
    45
 49.5 
 10
20
2 2

 49.5  8
 57.5
© .S. B. Bhattacharjee
Ch 4_70
What are merits of median?
Merits
The median is superior to arithmetic mean in certain
respects.
It is especially useful in case of open–end
distribution and also it is not influenced by the
presence of extreme values.
In fact when extreme values are present in the data,
the median is a more satisfactory measure of central
tendency than the mean.
© .S. B. Bhattacharjee
Continued……
Ch 4_71
Merits
The sum of the deviations of observations from median
(ignoring signs) is minimum. In other words, the absolute
deviation of observations from the median is less than
from any other value in the distribution
© .S. B. Bhattacharjee
Continued……
Ch 4_72
What are the limitations of median?
Limitations
The median is not capable of algebraic treatment.
Median cannot be used for determining the estimation
purposes since it is more affected by sampling
fluctuations.
The median tends to be rather unstable value if the
number of observations is small.
© .S. B. Bhattacharjee
Ch 4_73
What are positional measures ?
Positional measures are those that are estimated by
dividing a series into a equal number of parts.
Important amongst these are quartiles, deciles and
percentiles.
Quartiles are those values of the variate which divide
the total frequency into four equal parts, deciles divide
the total frequency in 10 equal parts and the
percentiles divide the total frequency in 100 equal
parts.
Continued……
© .S. B. Bhattacharjee
Ch 4_74
How are quartiles, deciles and
percentiles computed?
The procedure for computing quartiles, deciles, etc., is
the same as for median. For grouped data, the following
formulae are used for quartiles, deciles and percentiles:
jN
 p.c. f .
Qj  L  4
i
f
for j = 1,2,3
KN
 p.c. f .
Dk  L  10
i
f
© .S. B. Bhattacharjee
Continued…….
for K = 1,2,…,9
Ch 4_75
IN
 p.c. f .
P1  L  100
i
f
for I = 1,2,…,99
where the symbols have their usual meanings and
interpretation.
© .S. B. Bhattacharjee
Continued…….
Ch 4_76
The profits earned by 100 companies during 2003-04
are given below:
Profits (Tk. lakhs)
20-30
No. of companies
4
30-40
40-50
50-60
60-70
8
18
30
15
70-80
80-90
90-100
10
8
7
Calculate Q1 , Median, d4 and P80 and interpret the
values.
© .S. B. Bhattacharjee
Continued…….
Ch 4_77
Calculation of Q1 , Q2, d4 and P80
Profits (Tk.
lakhs)
f
c.f.
20-30
30-40
40-50
50-60
4
8
18
30
4
12
30
60
60-70
70-80
80-90
15
10
8
75
85
93
90-100
7
100
© .S. B. Bhattacharjee
Continued…….
Ch 4_78
100
Q1  Size of N / 4th observation 
 25th observation.
4
Hence Q1 lies in the class 40  50.
N
 p.c. f .
Q1  L  4
i
f
25  12
 40 
 10  40  7.22  47.22
18
25 per cent of the companies earn an annual profit of Tk.
47.22 lakhs or less.
© .S. B. Bhattacharjee
Continued…….
Ch 4_79
2N
Medianor Q2  Size of
th observation  50th observation.
4
Q2 lies in the class 50  60
2N
 p.cf .
Q2  L  4
i
f
50  30
 50 
 10  50  6  67
30
 56  67
50 per cent of the companies earn an annual
profit of Tk. 56.67.
© .S. B. Bhattacharjee
Continued…….
Ch 4_80
4N
D4  Size of
th observation  40 th observation
10
D4 lies in the clas 50  60
4N
 p.c. f .
D4  L  10
i
f
40  30
 50 
 10
30
 50  3.33
 53.33.
Thus 40 per cent of the companies earn an annual
profits of Tk. 53.33 lakhs or less.
© .S. B. Bhattacharjee
Continued…….
Ch 4_81
80N
80  100
P80  Size of
the observation 
 80th observation
100
100
P80 lies in the class 70  80.
80N
 p.c. f .
P80  L  100
i
f
80  75
 70 
 10  70  5  75
10
This means that 80 per cent of the companies earn an
annual profit of Tk. 75 lakhs or less and 20 per cent of
the companies earn an annual profit of more than Tk. 75
lakhs.
© .S. B. Bhattacharjee
Ch 4_82
What is meant by Mode?
Mode refers to the most common value in a distribution
or the largest category of variable. It may also defined
as the value which occurs the maximum number of
times, i.e. having the maximum frequency.
© .S. B. Bhattacharjee
Ch 4_83
How is mode calculated?
It involves fitting mathematically some appropriate type
of frequency curve to the grouped data and the
determination of the value on the X-axis below the peak
of the curve. However, there are several elementary
methods of estimating the mode.
Method for ungrouped
Method for grouped data.
© .S. B. Bhattacharjee
Ch 4_84
Calculation of mode- ungrouped data
The following figures relate to the preferences with
regard to size of screen (in inches) of T.V. sets of 30
persons selected at random from a locality. Find the
modal size of the T.V. screen.
12
20
12
24
29
20
12
20
29
24
24
20
12
20
24
29
24
24
20
24
24
20
24
24
12
24
20
29
24
24
© .S. B. Bhattacharjee
Continued……
Ch 4_85
Calculation of modal Size
Size in inches
Tally
Frequency
12
5
20
24
8
13
29
4
Total 30
Since size 24 occurs the maximum number of times,
the modal size of T.V. screen is 24 inches
© .S. B. Bhattacharjee
Ch 4_86
Calculation of mode – grouped data
In the case of grouped data, the following formula is
used for calculating mode:
Where
1
Mo  L 
i
1   2
L = Lower limit of the modal class.
1 
2 
The difference between the frequency of the
modal class and the frequency of the premodal class, i.e., preceding class.
The difference between the frequency of the
modal class and the post-modal class, i.e.,
succeeding class.
i = The size of the modal class.
© .S. B. Bhattacharjee
Ch 4_87
Another form of this formula is:
f1  f o
Mo  L 
i
2 f1  f o  f 2
where,
L = Lower limit of the modal class
f1 = Frequency of the modal class
fo = Frequency of the class preceding the
modal class.
f2 = Frequency of the class succeeding the
modal class.
© .S. B. Bhattacharjee
Ch 4_88
A distribution containing more than one mode is called
bimodal or multimodal. This cannot be determined by the
said formula.
When mode is ill-defined, its value (value of mode)
may be ascertained by the following approximate
formula based upon the relationship between mean,
median and mode.
1
Mean  median  Mean  Mode ) 
3
© .S. B. Bhattacharjee
Ch 4_89
In the given Sum, mentioned in slide no ch 4-68
L  49.5
1  20  15  5
 2  20  18  2
i  10
We know,
1
Mo  L 
i
1   2
5
 10
52
5
 49  5   10
7
50
 49  5 
7
 49  5  7  14
 56  64
 49  5 
© .S. B. Bhattacharjee
Ch 4_90
How would you locate mode
graphically?
In a frequency distribution the value of mode can be
determined graphically. The steps in calculation are :
Draw a histogram of the given data.
Draw two lines diagonally on the inside of the modal
class bar, starting from each upper corner of the bar to
the upper corner of the adjacent bar.
Draw a perpendicular line from the intersection of the
two diagonal lines to the X-axis (horizontal scale) which
gives modal value.
© .S. B. Bhattacharjee
Ch 4_91
The daily profits in Tk. of 100 shops are given us
follows:
Profits
No. of shops
0-100
12
100-200
18
200-300
27
300-400
20
400-500
17
500-600
6
Draw the histogram and thence find the modal value.
Check this value by direct calculation.
© .S. B. Bhattacharjee
Ch 4_92
Mode lies in the class 200-300
1
Mo  L 
i
1   2
9
 200 
 100
97
Here,
L  200
1  27  18  9
 2  27  20  7
i  100
900
 200 
16
 200  56  25
 256 25
© .S. B. Bhattacharjee
Ch 4_93
Locating Mode graphically
30
27
No. of shops
25
18
20
15
0-100
20
100-200
17
200-300
12
10
300-400
6
5
400-500
500-600
0
Mode=Tk.258 profits (in Tk. Lakhs)
© .S. B. Bhattacharjee
Ch 4_94
From the diagram, the modal value is also 256. Hence
by both the method the same value of mode is
obtained.
Mode can also be determined from frequency polygon
in which case perpendicular is drawn on the base from
the apex of the polygon and the point where it meets
the base gives the modal value.
However, graphic method of determining mode can be
used only where there is one class containing the
highest frequency. If two or more classes have the
same highest frequency, mode cannot be determined
graphically.
© .S. B. Bhattacharjee
Ch 4_95
What are the merits of mode?
Merits
Like median, the mode is not affected by extreme values
and its value can be obtained in open-end distributions
without ascertaining the class limits.
Mode can be easily used to describe qualitative
phenomenon.
For example, when we want to compare the consumer
preferences for different types of products, say, soap,
toothpastes, are etc., of different media of advertising,
we should compare the modal preferences.
In such distributions where there is an outstanding large
frequency, mode happens to be meaningful as an
average.
© .S. B. Bhattacharjee
Ch 4_96
What are the limitations of mode?
Limitations
Mode is not a rigidly defined measure as there are
several formulae for calculating the mode, all of
which usually give somewhat different answers.
The value of mode cannot always be computed,
such as ,in case of bimodal distributions.
© .S. B. Bhattacharjee
Ch 4_97
What is the relationship among mean,
median and mode?
A distribution in which the values of mean, median and
mode coincide is known as symmetrical distribution.
Conversely stated, when the values of mean, median
and mode are not equal, the distribution is known as
asymmetrical or skewed. In moderately skewed or
asymmetrical
distributions,
a
very
important
relationship exists among mean, median and mode. In
such distributions, the distance between the mean and
the median is approximately one-third of the distance
between the mean and mode as will be clear from the
following diagram:
© .S. B. Bhattacharjee
Ch 4_98
Relationship among mean, median and mode
DIVIDES AREA
IN HALVES
UNDER PEAK
OF CURVE
CENTRE
OF
GRAVITY
X
Mo
Me
© .S. B. Bhattacharjee
Ch 4_99
Karl Pearson has expressed
relationship as follows:
Or
and
this
approximate
1
Mean  Median  Mean  Mode 
3
Mode = 3 median – 2 Mean
2 Mean  Mode
Median 
3
If we know any of the two values out of the three, we
can compute the third from these relationships.
Continued……
© .S. B. Bhattacharjee
Ch 4_100
Example:
In a moderately asymmetrical distribution the Mode and
Mean are 32.1 and 35.4 respectively. Calculate the
Median.
Solution:
Mode = 3 Median – 2Mean
Here, Mode = 32.1, Mean =35.4
Substitution the values,
32.1 =3 Median – 2×35.4
Or
32.1= 3 Median – 70.8
Or,
3 Median =32.1+70.8
Or
3 Median = 102.9

Median = 34.3
© .S. B. Bhattacharjee
Ch 4_101
What are the important features influencing
selection of progressive average?
This average is based upon the arithmetic mean. The
important features of this average are:
It is a cumulative average. In the calculation of this
average all previous figures are added and no previous
figure is left as is done in the case of moving average.
The progressive average of the first year would remain
the same; the
progressive average for the second year
ab
is equal to
; for the third year a  b  c , for the
2
3
fourth year a  b  c  d ; and so on.
4
Continued…….
© .S. B. Bhattacharjee
Ch 4_102
What are the important features influencing
selection of progressive average?
The average value can be obtained for all the years.
The moving average, on the other hand, cannot be
computed for all the years. The longer the period of
moving average, the greater the number of years for
which the moving average cannot be computed.
This average is generally used during the early years
of the working of a business. For example, the figures
of sales, profits or production of each successive
year may be compared with the respective figures for
the entire previous period in other to find out how a
business is growing.
© .S. B. Bhattacharjee
Ch 4_103
What are the important features
influencing selection of progressive
average?
© .S. B. Bhattacharjee
Ch 4_104
What are the situations unfavorable to
using arithmetic mean?
In the following cases, arithmetic mean should
not be used:
In highly-skewed distributions
In distributions with open-end intervals
When the distributions
is unevenly spread,
concentration being small or large at irregular
points.
When an average rate of growth or change over a
period of time is required.
Continued…….
© .S. B. Bhattacharjee
Ch 4_105
What are the situations unfavorable to
using arithmetic mean?
When the observations form
progression, i.e., 1, 2, 4, 8, 16, etc.
a
geometric
When averaging of rates (i.e., speed, fluctuations
in the prices of articles, etc.) is needed.
When there are very large and very small values of
observations arithmetic mean would be seriously
misleading on account of undue influence of
extreme values.
© .S. B. Bhattacharjee
Ch 4_106
What are the critical points one has to
keep in mind while using an average?
Since an average is a single value representing a
group of values, it must be properly interpreted,
otherwise, there is every possibility of jumping to
wrong conclusion.
An average may give us a value that does not exist in
the data. For example, The arithmetic mean of 100,
300, 250, 50, 100
Continued……..
© .S. B. Bhattacharjee
Ch 4_107
What are the critical points one has to
keep in mind while using an average?
At times an average may give absurd results. For
example, if we calculate average size of a family we
may get a value 4.8. But this is impossible as persons
cannot be in fractions.
Measures of central value fail to give us any idea
about the formation of the series. Two or more series
may have the same central value but may differ
widely in composition.
© .S. B. Bhattacharjee
Continued…..
Ch 4_108
Concluding Remarks
We must remember that an average is a
measure of central tendency. Hence unless
the data show a clear-cut concentration of
observations an average may
not be
meaningful at all. This evidently precludes
the use of any average to typify a bimodal or
a U-shaped or a J-shaped distribution.
© .S. B. Bhattacharjee
Ch 4_109
© .S. B. Bhattacharjee
Ch 4_110