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Ch 4_1
Course: MBASubject: Quantitative Techniques
Unit:2
Ch 4_2
Ch 4_3
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 )
Ch 4_4
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.
Ch 4_5
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.
Ch 4_6
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.
Ch 4_7
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).
Ch 4_8
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.
Ch 4_9
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.
Ch 4_10
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
Ch 4_11
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…….
Ch 4_12
Measure of Central tendency?
Nominal?
Ordinal?
Distribution Skewed?
Mode
Mode
Mean
Yes Yes
Yes
Yes
No
No
No
No
Figure:1Nature of data
Ch 4_13
Mean Arithmetic mean Geometric mean Harmonic mean
What are various types of averages or means?
Continued…….
Ch 4_14
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.
Ch 4_15
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
Ch 4_16
What is direct method?
Thus, if represent the values of N items or observations, the arithmetic mean denoted by is defined as:
NXXX ......,, 21
NXXX ......,, 21
X
X
N
Xi
N
XXXX
N
iN
121 .......
Ch 4_17
Example:
The monthly income (in Tk) of 10 employees working in a firm is as follows:
4487 4493 4502 4446 4475 4492 4572 4516 4468 4489
Find the average monthly income. Applying the formula we get:
449410
44940
N
XX
4487+4493+4502+4446+4475+4492+4572+4516+4468+4489 = 44.949
X
Hence the average monthly income is Tk.4494.
Ch 4_18
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:
N
dAX
Where , d = (X – A )
and 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.
Ch 4_19
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.
X (TK)
(X - 4460)(TK)
4487449345024446447544924572451644684489
+27+33+42-14+15+32+112+56+8
+29
Calculation of average income
= +340
Assumed mean
= Tk.4460
Ch 4_20
4494.34446010
3404460 Tk
N
dAX
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.
Applying the formula we get:
Ch 4_21
What are the methods of estimating average from grouped data?
Direct method
Short-cut method
Continued…..
Ch 4_22
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
Example 3
Compute the average from the following data by direct method.
Ch 4_23
The formula for estimating average from grouped data by direct method is:
N
fxX
Where, X = mid-point of various classes
f= the frequency of each class
N= the total frequency
Continued…….
Direct Method
Ch 4_24
Calculate the average profits for all the companies.
000,48,8fx
Profits(Tk. Lakhs)
Mid-pointsX
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
Continued…….
Ch 4_25
71.605400,1
000,48,8
N
fxX
Thus, the average profit is Tk. 605.71 lakhs.
Ch 4_26
iN
fdAX
i
AXd
Where, A = Arbitrary point (assumed mean)
and i = size of the equal class interval
Short-cut MethodWhen short-cut method is used, the following formula is applied.
Ch 4_27
di
AX
Marks Mid-points f 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
Example:
0.9fdN = 90
Continued…….
Ch 4_28
5.5815.59
1090
0.95.59
1090
0.95.59
i
N
fdAX
Here, assumed mean, A = 59.5class-interval, i =10
Ch 4_29
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., This shall be clear from the following example:
0 XX
X
10 -20
20 - 10
30 0
40 +10
50 +20
150X 0 XX
Continued……
XX
Ch 4_30
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
2 -2 4
3 -1 1
4 0 0
5 +1 1
6 +2 4
20x
XX 2XX
0 XX
Continued……
102 XX
45
20
,
N
XXHere
Ch 4_31
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:
21
221112 NN
XNXNX
Continued……
Ch 4_32
12X = Combined mean of the two groups.
1X
2X
1N
2N
= Arithmetic mean of the first group.
= Arithmetic mean of the second group.
= Number of observations in the first group.
= Number of observations in the second group.
Where,
Continued………
Ch 4_33
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:
21
221112 NN
XNXNX
89.5538.180
99700080100
675080457010012
Tk
X
Ch 4_34
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……
Ch 4_35
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.
Ch 4_36
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.
Continued….
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.
Ch 4_37
wXwhere,
=The weighted arithmetic mean
X = The variable.
W = Weights attached to the variable X.
The formula for computing weighted arithmetic mean is given below:
,
W
WXwX
Ch 4_38
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., If we assume that 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
.70.30
3575100dayperTk
70.30
3507501000101010
3510751010010
Tk
X
Continued….
Ch 4_39
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…..
Ch 4_40
Wage per day (Tk)
X
No. of workers W
WX
100 20 2000
75 15 1125
35 5 175
W= 40 WX = 3300
50.8240
3300
W
WXwX
Hence the average wage per day paid by the contractor is Tk. 82.50.
Example:
Ch 4_41
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…….
Ch 4_42
How is harmonic mean computed?
NXXXX
NMH
1....
111..
321
The formula for estimating harmonic mean is as follows:
Continued…….
Where number of observations is large, the computation of harmonic mean in the above manner becomes tedious.
Ch 4_43
X
NMH
1.For ungrouped data,=
For grouped data, =
Continued…….
To simplify calculations, we obtain reciprocals of the various observations and apply the following formulae:
.
1..
X
f
Nor
Xf
NMH
Ch 4_44
X
10 0.100
20 0.050
25 0. 04
40 0.025
50 0.020
X
1
23501
X
28212350
5
1.
X
NMH
Calculation of Harmonic Mean
Continued…….
Ch 4_45
Variable X f
0-10 5 8 1.600
10-20 15 15 1.000
20-30 25 20 0.800
30-40 35 4 0.114
40-50 45 3 0.067
Xf
1
5813
1
Xf
96135813
50
1..
Xf
NMH
Calculation of Harmonic Mean
Ch 4_46
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:
,.45
.20hourperkm
hours
km
Continued…….
Ch 4_47
Where X, the unit of the first term is an hour and the unit of the second term is a kilometer.
.,4
1
.20
5kmperhour
km
hours
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 six-hour day?
Continued…….
Ch 4_48
X
4 0.250
5 0.200
6 0.167
10 0.100
12 0.083
X
1
801
X
25680
5
1..
X
NMH
The average number of units per minutes will be obtained by calculating the harmonic mean.
Continued…….
Ch 4_49
Workers Productive rates
A 4 minutes per toy
B 6 minutes per toy
C 10 minutes per toy
D 15 minutes per toy
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:
Example:
Find the average minutes per toy by the group of workers.
Continued…….
Ch 4_50
If we assume that each of the four workers is assigned the same number of toys (constant value) to meet the order, or = 350 toys per worker, the arithmetic mean would give the correct answer.
4
400,1
4
38
4
354
151064
X
Continued…….
minutes per toy.
Ch 4_51
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
minutes4
38
400,1
250,12
Continued…….
Ch 4_52
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)
toyperminutes7
66
35
604
15
1
10
1
6
1
4
14
..
MH
minutes600,97
48400,1
Time required to complete 1,400 toys
Continued…….
Ch 4_53
minutes400,24
600,9 Each workers works for
Toys produced by D in 2400 minutes
Toys produced by B in 2400 minutes
Toys produced by C in 2400 minutes
Toys produced by A in 2400 minutes 6004
2400
4006
2400
24010
2400
16015
2400
Total = 1,400
Verification:
Ch 4_54
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.
Ch 4_55
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 smallest observations and as such is not a good representation of a statistical series.
It is in dealing with business problems harmonic mean is rarely used.
Ch 4_56
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.
Ch 4_57
Apply the formula : Median = Size of thN
2
1 observation.
From the following data of wages of 7 workers, compute the median wage:
Wages (in Tk.) 4600, 4650, 4580, 4690, 4660, 4606, 4640
Ch 4_58
S. No. Wages arranged in ascending order
1 4580
2 4600
3 4606
4 4640
5 4650
6 4660
7 4690
Calculation of Median from ungrouped data
Ch 4_59
nobservatiothnobservatiothN
ofSizeMedian 42
17
2
1
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 = 4.5th observation.
For finding out the value of 4.5th observation, we shall take the average of 4th and 5th observation. Hence the median shall be
46452
46504640
2
18
Ch 4_60
,...
2 if
fcpN
LMedian
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
Formula for calculation of median from grouped data
Continued…..
Ch 4_61
Calculation of Median Marks
Marks f c. f
19.5-29.5 2 2
29.5-39.5 12 14
39.5-49.5 15 29
49.5-59.5 20 49
59.5-69.5 18 67
69.5-79.5 10 77
79.5-89.5 9 86
89.5-99.5 4 90
Continued…..
Ch 4_62
2
N
theN
2
Median = Size of the observation
observation
= 45th observation
Hence median lies in the class 49.5-59.5
Continued…..
Ch 4_63
5.57
85.49
1020
165.49
1020
29455.49
..2
if
fcpN
LMedianHere, L = 49.5, N = 90, p.c.f = 29 , f =20 i = 10.
45
2
90
2
N
Ch 4_64
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.
Continued……
Ch 4_65
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
Continued……
Ch 4_66
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.
Ch 4_67
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……
Ch 4_68
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:
if
fcpjN
LQ j
...
4 for j = 1,2,3
if
fcpKN
LDk
...
10 for K = 1,2,…,9
Continued…….
Ch 4_69
for I = 1,2,…,99i
f
fcpIN
LP
...
1001
where the symbols have their usual meanings and interpretation.
Continued…….
Ch 4_70
Profits (Tk. lakhs) No. of companies
20-30 4
30-40 8
40-50 18
50-60 30
60-70 15
70-80 10
80-90 8
90-100 7
The profits earned by 100 companies during 2003-04 are given below:
Calculate Q1 , Median, d4 and P80 and interpret the values.
Continued…….
Ch 4_71
Calculation of Q1 , Q2, d4 and P80
Profits (Tk. lakhs)
f c.f.
20-30 4 4
30-40 8 12
40-50 18 30
50-60 30 60
60-70 15 75
70-80 10 85
80-90 8 93
90-100 7 100
Continued…….
Ch 4_72
22.4722.7401018
122540
...4
.5040
.254
1004/
1
1
1
if
fcpN
LQ
classtheinliesQHence
nobservatiothnobservatiothNofSizeQ
25 per cent of the companies earn an annual profit of Tk. 47.22 lakhs or less.
Continued…….
Ch 4_73
50 per cent of the companies earn an annual profit of Tk. 56.67.
6756
676501030
305050
..4
2
6050
.504
2
2
2
2
if
cfpN
LQ
classtheinliesQ
nobservatiothnobservatiothN
ofSizeQorMedian
Continued…….
Ch 4_74
Thus 40 per cent of the companies earn an annual profits of Tk. 53.33 lakhs or less.
Continued…….
.33.53
33.350
1030
304050
...10
4
6050
4010
4
4
4
4
if
fcpN
LD
clastheinliesD
nobservatiothnobservatiothN
ofSizeD
Ch 4_75
755701010
758070
...100
80
.8070
80100
10080
100
80
80
80
80
if
fcpN
LP
classtheinliesP
nobservatiothnobservatiotheN
ofSizeP
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.
Ch 4_76
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.
Ch 4_77
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.
Ch 4_78
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
Continued……
Ch 4_79
Size in inches Tally Frequency
12 5
20 8
24 13
29 4
Total 30
Calculation of modal Size
Since size 24 occurs the maximum number of times, the modal size of T.V. screen is 24 inches
Ch 4_80
Calculation of mode – grouped data
In the case of grouped data, the following formula is used for calculating mode:
iLM o
21
1
1
2
Where L = Lower limit of the modal class.
The difference between the frequency of the modal class and the frequency of the pre-modal 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.
Ch 4_81
ifff
ffLM
o
oo
21
1
2
Another form of this formula is:
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.
Ch 4_82
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.
)3
1ModeMeanmedianMean
A distribution containing more than one mode is called bimodal or multimodal. This cannot be determined by the said formula.
Ch 4_83
10
21820
51520
5.49
2
1
i
L
6456
1475497
50549
107
5549
1025
5549
,
21
1
iLM
knowWe
o
In the given Sum, mentioned in slide no ch 4-60
Ch 4_84
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.
Ch 4_85
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.
What are the limitations of mode?
Ch 4_86
What is the relationship among mean, median and mode?
A distribution in which the values of mean, median and mode coincide(match) 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:
Ch 4_87
UNDER PEAK
OF CURVE
DIVIDES AREA
IN HALVES
CENTRE OF
GRAVITY
XMe
Mo
Relationship among mean, median and mode
Ch 4_88
ModeMeanMedianMean 3
1
Karl Pearson has expressed this approximate relationship as follows:
Or Mode = 3 median – 2 Mean
and 3
2 ModeMeanMedian
If we know any of the two values out of the three, we can compute the third from these relationships.
Continued……
Ch 4_89
In a moderately asymmetrical distribution the Mode and Mean are 32.1 and 35.4 respectively. Calculate the Median.
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
Example:
Solution:
Ch 4_90
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:
,))...()()((. 321NNXXXXMG
where, X1, X2, X3….., XN refer to the various observations of the data.
Ch 4_91
To simplify calculations logarithms are used.
N
X
N
XXXMGLog N
loglog....loglog.. 21
How is geometric mean computed?
N
Xlogantilog..MG
Ch 4_92
How is geometric mean calculated?
In ungrouped data, geometric mean is calculated with the help of the following formula:
In grouped data, first midpoints are found out and then the following formula is used for calculating geometric mean :
X = midpoint
Where
N
XMG
log.. Antilog
,log
..
N
XfMG Antilog
Ch 4_93
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.
Continued…….
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.
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…….
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.
References
• Quantitative Techniques, by CR Kothari, Vikas publication
• Fundamentals of Statistics by SC Guta Publisher Sultan Chand
• Quantitative Techniques in management by N.D. Vohra Publisher: Tata Mcgraw hill
