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GEOMETRIC MEAN:HARMONIC MEAN, MID-QUARTILE RANGE

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MTH001 ­ Elementary Mathematics
LECTURE # 26:
·  Geometric mean
·  Harmonic mean
·  Relation between the arithmetic, geometric and harmonic means
·  Some other measures of central tendency
GEOMETRIC MEAN:
The geometric mean, G, of a set of n positive values X1, X2,...,Xn is defined as the positive
nth root of their product.
G = n X  1 X  2 ...X  n
(Where Xi > 0)
When n is large, the computation of the geometric mean becomes laborious as we have to
extract the nth root of the product of all the values.
The arithmetic is simplified by the use of logarithms.
Taking logarithms to the base 10, we get
1
[log X1 + log X  2 + ....+ log X  n ]
log G =
n
log
X
=
Hence
n
log X
G = anti log
  n
Example:
Find the geometric mean of numbers:
45, 32, 37, 46, 39,
36, 41, 48, 36.
Solution:
We need to compute the numerical value of
9
45× 32× 37 × 46× 39× 36× 41× 48× 36
=
But, obviously, it is a bit cumbersome to find the ninth root of a quantity. So we make use of
logarithms, as shown below:
X
log X
log X
45
1.6532
log G =
32
1.5052
n
37
1.5682
46
1.6628
14.3870
=
= 1.5986
39
1.5911
9
36
1.5563
41
1.6128
Hence G = anti log 1.5986
48
1.6812
= 39.68
36
1.5563
14.3870
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MTH001 ­ Elementary Mathematics
The above example pertained to the computation of the geometric mean in case of raw
data.
Next, we consider the computation of the geometric mean in the case of grouped data.
GEOMETRICMEAN
FOR GROUPED DATA:
In case of a frequency distribution having k classes with midpoints X1, X2,
...,Xk and the corresponding frequencies f1, f2, ..., fk (such that fi = n), the geometric
mean is given by
G = X11 Xf2 ....Xfk
f
n
2
k
Each value of X thus has to be multiplied by itself f times, and the whole procedure becomes
quite a formidable task!
In terms of logarithms, the formula becomes
1
[  f1 log X  1 + f  2 log X  2 + ... + f  k log X  k ]
log G =
n
f log X
=
n
Hence
f log X
G = anti og
l
n
Obviously, the above formula is much easier to handle.
Let us now apply it to an example.
Going back to the example of the EPA mileage ratings, we have:
No.
Class-mark
Mileage
log X
f log X
of
(midpoint)
Rating
Cars
X
30.0 - 32.9
2
31.45
1.4976
2.9952
33.0 - 35.9
4
34.45
1.5372
6.1488
36.0 - 38.9
14
37.45
1.5735
22.0290
39.0 - 41.9
8
40.45
1.6069
12.8552
42.0 - 44.9
2
43.45
1.6380
3.2760
30
47.3042
G = antilog
47.3042
30
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MTH001 ­ Elementary Mathematics
= antilog 1.5768 = 37.74
This means that, if we use the geometric mean to measures the central tendency of this
data set, then the central value of the mileage of those 30 cars comes out to be 37.74 miles
per gallon.
The question is, "When should we use the geometric mean?"
The answer to this question is that when relative changes in some variable quantity are
averaged, we prefer the geometric mean.
EXAMPLE:
Suppose it is discovered that a firm's turnover has increased during 4 years by the following
amounts:
Percentage
Compared
Year Turnover
With Year
Earlier
1958
£ 2,000
­
1959
£ 2,500
125
1960
£ 5,000
200
1961
£ 7,500
150
1962
£ 10,500
140
The yearly increase is shown in a percentage form in the right-hand column i.e. the turnover
of 1959 is 125 percent of the turnover of 1958, the turnover of 1960 is 200 percent of the
turnover of 1959, and so on. The firm's owner may be interested in knowing his average rate
of turnover growth.
If the arithmetic mean is adopted he finds his answer to be:
Arithmetic Mean:
125 + 200 + 150 + 140
4
= 153.75
i.e. we are concluding that the turnover for any year is 153.75% of the turnover for the
previous year. In other words, the turnover in each of the years considered appears to be
53.75 per cent higher than in the previous year.
If this percentage is used to calculate the turnover from 1958 to 1962 inclusive, we obtain:
153.75% of £ 2,000 = £ 3,075
153.75% of £ 3,075 = £ 4,728
153.75% of £ 4,728 = £ 7,269
153.75% of £ 7,269 = £ 11,176
Whereas the actual turnover figures were
Year Turnover
1958
£ 2,000
1959
£ 2,500
1960
£ 5,000
1961
£ 7,500
1962  £ 10,500
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MTH001 ­ Elementary Mathematics
It seems that both the individual figures and, more important, the total at the end of the
period, are incorrect. Using the arithmetic mean has exaggerated the `average' annual rate
of increase in the turnover of this firm. Obviously, we would like to rectify this false
impression. The geometric mean enables us to do so:
Geometric mean of the turnover figures:
(125 × 200 ×150 ×140)
4
4
= 525000000
= 151.37%
Now, if we utilize this particular value to obtain the individual turnover figures, we find that:
151.37% of £2,000 = £3,027
151.37% of £3,027 = £4,583
151.37% of £4,583 = £6,937
151.37% of £6,937 = £10,500
So that the turnover figure of 1962 is exactly the same as what we had in the original data.
Interpretation:
If the turnover of this company were to increase annually at a constant rate, then the annual
increase would have been 51.37 percent.(On the average, each year's turnover is 51.37%
higher than that in the previous year.) The above example clearly indicates the significance
of the geometric mean in a situation when relative changes in a variable quantity are to be
averaged.
But we should bear in mind that such situations are not encountered too often, and
that the occasion to calculate the geometric mean arises less frequently than the arithmetic
mean.(The most frequently used measure of central tendency is the arithmetic mean.)
The next measure of central tendency that we will discuss is the harmonic mean.
HARMONIC MEAN;
The harmonic mean is defined as the reciprocal of the arithmetic mean of the
reciprocals of the values. HARMONIC MEAN
In case of raw data:
n
H .M . =
1
X
⎝  ⎠
In case of grouped data (data grouped into a frequency distribution):
n
H .M . =
1
f⎜  ⎟
X
(where X represents the midpoints of the various classes).
EXAMPLE:
Suppose a car travels 100 miles with 10 stops, each stop after an interval of 10
miles. Suppose that the speeds at which the car travels these 10 intervals are 30, 35, 40,
40, 45, 40, 50, 55, 55 and 30 miles per hours respectively.
What is the average speed with which the car traveled the total distance of 100 miles?
If we find the arithmetic mean of the 10 speeds, we obtain:
Arithmetic mean of the 10 speeds:
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MTH001 ­ Elementary Mathematics
30 + 35 + .... + 30
10
420
miles per hour.
=
= 42
10
But, if we study the problem carefully, we find that the above answer is incorrect.
By definition, the average speed is the speed with which the car
would have traveled the 100 mile distance if it had maintained a constant speed throughout
the 10 intervals of 10 miles each.
Total distance travelled
Average speed =
Total time taken
Now, total distance traveled = 100 miles.
Total time taken will be computed as shown below:
Distance
Distance
Interval
Distance
Speed =
Time =
Time
Speed
1
10 miles
30 mph
10/30 = 0.3333 hrs
2
10 miles
35 mph
10/35 = 0.2857 hrs
3
10 miles
40 mph
10/40 = 0.2500 hrs
4
10 miles
40 mph
10/40 = 0.2500 hrs
5
10 miles
45 mph
10/45 = 0.2222 hrs
6
10 miles
40 mph
10/40 = 0.2500 hrs
7
10 miles
50 mph
10/50 = 0.2000 hrs
8
10 miles
55 mph
10/55 = 0.1818 hrs
9
10 miles
55 mph
10/55 = 0.1818 hrs
10
10 miles
30 mph
10/30 = 0.333 hrs
Total =
100 miles
Total Time = 2.4881 hrs
Hence
100
Average speed =
= 40.2 mph
2.4881
which is not the same as 42 miles per hour.
Let us now try the harmonic mean to find the average speed of the car.
n
H .M . =
1
X
⎝  ⎠
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MTH001 ­ Elementary Mathematics
where n is the no. of terms)
We have:
X
1/X
30
1/30 = 0.0333
n
H.M. =
35
1/35 = 0.0286
1
40
1/40 = 0.0250
X
10
40
1/40 = 0.0250
=
45
1/45 = 0.0222
0.2488
40
1/40 = 0.0250
= 40.2 mph
50
1/50 = 0.0200
55
1/55 = 0.0182
Hence it is clear that the
55
1/55 = 0.0182
harmonic mean gives the
30
1/30 = 0.0333
totally correct result.
1
= 0.2488
X
The key question is, "When should we compute the harmonic mean of a data set?"
The answer to this question will be easy to understand if we consider the following rules:
RULES
1.
When values are given as x per y where x is constant and y is variable, the Harmonic
Mean is the appropriate average to use.
2.
When values are given as x per y where y is constant and x is variable, the
Arithmetic Mean is the appropriate average to use.
3.
When relative changes in some variable quantity are to be averaged, the geometric
mean is the appropriate average to use.
We have already discussed the geometric and the harmonic means.
Let us now try to understand Rule No. 1 with the help of an example:
EXAMPLE:
If 10 students have obtained the following marks (in a test) out of 20:
13, 11, 9, 9, 6,
5, 19, 17, 12, 9
Then the average marks (by the formula of the arithmetic mean) are:
13 + 11 + 9 + 9 + 6 + 5 + 19 + 17 + 12 + 9
10
110
=
= 11
10
This is equivalent to
13  11  9
9
6
5  19  17  12  9
+
+
+
+
+
+
+
+
+
20  20  20  20  20  20  20  20  20  20
10
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MTH001 ­ Elementary Mathematics
110
110
11
= 20 =
=
(i.e. the average10arks of×his grou0 of students are 11 out of 20).
10 t 20  2 p
m
In the above example, the point to be noted was that all the marks were expressible
as x per y where the denominator y was constant i.e. equal to 20, and hence, it was
appropriate to compute the arithmetic mean.
Let us now consider a mathematical relationship exists between these
three measures of central tendency.
RELATION BETWEEN ARITHMETIC, GEOMETRIC
AND HARMONIC MEANS:
Arithmetic Mean > Geometric Mean >Harmonic Mean
We have considered the five most well-known measures of central tendency i.e. arithmetic
mean, median, mode, geometric mean and harmonic mean.
It is interesting to note that there are some other measures of central tendency as well.
Two of these are the mid range, and the mid quartile range.
Let us consider these one by one:
MID-RANGE:
If there are n observations with x0 and xm as their smallest and largest observations
respectively, then their mid-range is defined as
x0 + xm
mid - range =
2
It is obvious that if we add the smallest value with the largest, and divide by 2, we will get a
value which is more or less in the middle of the data-set.
MID-QUARTILE RANGE:
If x1, x2... xn are n observations with Q1 and Q3 as their first and
third quartiles respectively, then their mid-quartile range is defined as
Q1 + Q3
mid - quartile range =
2
Similar to the case of the mid-range, if we take the arithmetic mean of the upper and lower
quartiles, we will obtain a value that is somewhere in the middle of the data-set.
The mid-quartile range is also known as the mid-hinge.
Let us now revise briefly the core concept of central tendency:
Masses of data are usually expressed in the form of frequency tables
so that it becomes easy to comprehend the data.
Usually, a statistician would like to go a step ahead and to compute a number that will
represent the data in some definite way.
Any such single number that represents a whole set of data is called `Average'.
Technically speaking, there are many kinds of averages (i.e. there are several ways to
compute them). These quantities that represent the data-set are called "measures of central
tendency".
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Table of Contents:
  1. Recommended Books:Set of Integers, SYMBOLIC REPRESENTATION
  2. Truth Tables for:DE MORGAN’S LAWS, TAUTOLOGY
  3. APPLYING LAWS OF LOGIC:TRANSLATING ENGLISH SENTENCES TO SYMBOLS
  4. BICONDITIONAL:LOGICAL EQUIVALENCE INVOLVING BICONDITIONAL
  5. BICONDITIONAL:ARGUMENT, VALID AND INVALID ARGUMENT
  6. BICONDITIONAL:TABULAR FORM, SUBSET, EQUAL SETS
  7. BICONDITIONAL:UNION, VENN DIAGRAM FOR UNION
  8. ORDERED PAIR:BINARY RELATION, BINARY RELATION
  9. REFLEXIVE RELATION:SYMMETRIC RELATION, TRANSITIVE RELATION
  10. REFLEXIVE RELATION:IRREFLEXIVE RELATION, ANTISYMMETRIC RELATION
  11. RELATIONS AND FUNCTIONS:FUNCTIONS AND NONFUNCTIONS
  12. INJECTIVE FUNCTION or ONE-TO-ONE FUNCTION:FUNCTION NOT ONTO
  13. SEQUENCE:ARITHMETIC SEQUENCE, GEOMETRIC SEQUENCE:
  14. SERIES:SUMMATION NOTATION, COMPUTING SUMMATIONS:
  15. Applications of Basic Mathematics Part 1:BASIC ARITHMETIC OPERATIONS
  16. Applications of Basic Mathematics Part 4:PERCENTAGE CHANGE
  17. Applications of Basic Mathematics Part 5:DECREASE IN RATE
  18. Applications of Basic Mathematics:NOTATIONS, ACCUMULATED VALUE
  19. Matrix and its dimension Types of matrix:TYPICAL APPLICATIONS
  20. MATRICES:Matrix Representation, ADDITION AND SUBTRACTION OF MATRICES
  21. RATIO AND PROPORTION MERCHANDISING:Punch recipe, PROPORTION
  22. WHAT IS STATISTICS?:CHARACTERISTICS OF THE SCIENCE OF STATISTICS
  23. WHAT IS STATISTICS?:COMPONENT BAR CHAR, MULTIPLE BAR CHART
  24. WHAT IS STATISTICS?:DESIRABLE PROPERTIES OF THE MODE, THE ARITHMETIC MEAN
  25. Median in Case of a Frequency Distribution of a Continuous Variable
  26. GEOMETRIC MEAN:HARMONIC MEAN, MID-QUARTILE RANGE
  27. GEOMETRIC MEAN:Number of Pupils, QUARTILE DEVIATION:
  28. GEOMETRIC MEAN:MEAN DEVIATION FOR GROUPED DATA
  29. COUNTING RULES:RULE OF PERMUTATION, RULE OF COMBINATION
  30. Definitions of Probability:MUTUALLY EXCLUSIVE EVENTS, Venn Diagram
  31. THE RELATIVE FREQUENCY DEFINITION OF PROBABILITY:ADDITION LAW
  32. THE RELATIVE FREQUENCY DEFINITION OF PROBABILITY:INDEPENDENT EVENTS