LECTURE N0 5.pdf

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MATH – 361
Introduction to Probability and Statistics
Lecture No. 05
Measures of Central Tendency
Reference: Ch # 1, Sec 1.2, Text Book
No. of Slides: 40

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After completing this lecture, students will be able to
➢ Compute different types of Mean for ungrouped data
➢ Compute different types of Mean for grouped data
Desired Learning Objectives

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Central Tendency
Measure of Central Tendency
A single figure tends to lie in the
middle of the data

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Mathematical Average
(Pythagorean Means)
1. Arithmetic Mean
2. Geometric Mean
3. Harmonic Mean
Measure of Central Tendency

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Definition
Geometric Mean is calculated as the nth root of the product of all n
values, where n is the number of values
For example, if the data contains only two values, the square root of
the product of the two values is the Geometric Mean. For three
values, the cube-root of the product of all three values is the GM,
and so on.
Geometric Mean (GM)

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Example
1, 3, 9, 27, 81, 243, 729…
Geometric Mean of our dataset is:
1 * 3 * 9 * 27 * 81 * 243 * 729 =
10,460,353,203
7th root of 10,460,353,203 = 27
Geometric Mean = 27
Geometric Mean (GM)

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Geometric Mean (GM)
➢ In this case, our Geometric Mean very much resembles the
middle value of our dataset
➢ For data in exponential growth rate i.e. having sort of
multiplicative relationship, the Geometric Mean will give a
closer ‘middle number’ than the Arithmetic Mean

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➢ The Geometric Mean is appropriate when the data contains
values with different units of measure, e.g. some measure are
height, some are dollars, some are miles, etc
➢ The Geometric Mean does not accept negative or zero values,
e.g. all values must be positive
Geometric Mean (GM)

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Real World Applications of the Geometric Mean
➢ Compound Interest
➢ Different Scales or Units
➢ Applications of the geometric mean are most common in
business and finance, where it is frequently used when dealing
with percentages to calculate growth rates and returns on a
portfolio of securities. It is also used in certain financial and stock
market indexes
Geometric Mean (GM)

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( )
1
log45 log32 log37 log46 log39 log36 log41 log48 log36
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+ + + + + + + +
Geometric Mean

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Example : Grouped Data
Find GM for the given grouped data
Classes Freq (f)
65 – 84 9
85 – 104 10
105 – 124 17
125 – 144 10
145 – 164 5
165 – 184 4
185 – 204 5
Geometric Mean

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Classes Class Marks (x) Freq (f) Log(x) f(log(x))
65 – 84 (65+84)/2 =74.5 9 Log(74.5) =1.8722 9 * 1.8722 =16.8498
85 – 104 94.5 10 1.9754 19.7540
105 – 124 114.5 17 2.0589 35.0013
125 – 144 134.5 10 2.1287 21.2870
145 – 164 154.5 5 2.1889 10.9445
165 – 184 174.5 4 2.2418 8.9672
185 – 204 194.5 5 2.2889 11.4445
Geometric Mean

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Applications of GM
➢ Growth Rates: The Geometric Mean is used in finance to
calculate average growth rates and is referred to as the
compounded annual growth rate
Geometric Mean

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Example : Growth Rates
➢ Consider a stock that grows by 10% in year one, declines by
20% in year two, and then grows by 30% in year three. If the
stock is at 100 in the starting. Find the Geometric Mean of the
growth rate
Geometric Mean

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1st year its growth of 10% on 100 i.e. the starting value
Growth at 10% = 10% of 100 = 10
Total = 100+10 =110
Relative Growth Rate = 110/100 =1.1
2nd year is decline of 20% so it will be calculated on first year’s
amount = 20% of 110 = 22
Amount remaining is due to decline =110-22
Relative Growth Rate = 88/110= 0.8
Geometric Mean

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3rd year is growth at 30% = 30% of 88 = 26.4
Total becomes = 88 + 26.4 = 114.4
Relative Growth Rate = 114.4/88 = 1.3
𝑮𝑴 = 𝒑𝒓𝒐𝒅𝒖𝒄𝒕 𝒐𝒇 𝒂𝒍𝒍 𝒓𝒆𝒍𝒂𝒕𝒊𝒗𝒆 𝒈𝒓𝒐𝒘𝒕𝒉 𝒓𝒂𝒕𝒆𝒔
ൗ
𝟏
𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒚𝒆𝒂𝒓𝒔
Geometric Mean

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ൗ
𝟏
𝑮𝑴 = 𝟏. 𝟏 ∗ 𝟎. 𝟖 ∗ 𝟏. 𝟑
𝟏
𝟑 = 1.046
Geometric Mean

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Applications of GM
➢ Portfolio Returns: The Geometric Mean is commonly used to
calculate the annual return on portfolio of securities as well
➢ Consider a portfolio of stocks that goes up from $100 to $110 in
year one, then declines to $80 in year two and goes up to $150 in
year three. The return on portfolio is then calculated as
Geometric Mean

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Example : Portfolio Returns
Stock starting value 100
➢ 1st year it reached to 110
➢ Rate of growth = 110/100 =1.1
➢ 2nd year stocked declined to 80
➢ Rate of growth = 80/110 = 0.7272
➢ 3rd year stock again increased to 150
➢ Rate of growth = 150/80 =1.875
Geometric Mean

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Geometric Mean
𝑀𝑒𝑎𝑛 𝐺𝑟𝑜𝑤𝑡ℎ = 𝐺𝑀 = 1.1 ∗ 0.7272 ∗ 1.875 Τ
1
3 = 1.48885 Τ
1
3
𝑮𝑴 = 𝟏. 𝟏𝟒𝟒𝟕
ൗ
𝟏

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➢ As Arithmetic Mean requires addition & the Geometric Mean
employs multiplication, the Harmonic Mean utilizes reciprocals
➢ It is defined for non-zero positive values as the reciprocal of the
Arithmetic Mean of the reciprocals of the values
Harmonic Mean (HM)

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Harmonic Mean
➢ The Harmonic Mean is calculated as the number of
values n divided by the sum of the reciprocal of the values
➢ The Harmonic Mean is the appropriate mean if the data comprise
ratios & rates
Harmonic Mean (HM)

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➢ In certain situations, especially many situations involving rates
and ratios, the Harmonic Mean provides the truest average
➢ Example: Suppose in first test a typist types 400 words in 50
minutes, in second test he types the same words (400) in 40
minutes and in third test he takes 30 minutes to type the 400
words. Then average time of typing can be calculated by
Harmonic Mean
Harmonic Mean (HM)

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➢ Example: If a vehicle travels a certain distance d at a speed x (60
km/h) and then the same distance again at a speed y (40 km/h),
then its average speed is the Harmonic Mean of x and y (48
km/h)
Harmonic Mean (HM)

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Classes Class Marks (x) Freq (f) f/x
65 – 84 (65+84)/2 =74.5 9 9/74.5 = 0.12081
85 – 104 94.5 10 0.10582
105 – 124 114.5 17 0.14847
125 – 144 134.5 10 0.07435
145 – 164 154.5 5 0.03236
165 – 184 174.5 4 0.02292
185 – 204 194.5 5 0.02571
Harmonic Mean (HM)

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➢ The Arithmetic Mean is the most commonly used mean, although
it may not be appropriate in some cases
➢ The exceptions are if the data contains negative or zero values,
then the Geometric and Harmonic Means cannot be used directly
➢ To average compound rate changes over consistent periods: use
the Geometric Mean
How to Choose the Correct Mean?

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➢ To average rates over different periods or lengths: use
the Harmonic Mean
➢ If your data displays an additive structure: the Arithmetic Mean is
used
➢ If your data reveals a multiplicative structure and / or has large
outliers: the Geometric or Harmonic Mean might be more
appropriate

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➢ There are pitfalls & tradeoffs to any decision
✓ loss of meaningful scale or units when using the Geometric
Mean
✓ Datasets with 0’s cannot be used with
the Geometric or Harmonic Means, & datasets with negative
numbers also rule out the Geometric Mean

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Practice problem 1
Calculate Geometric Mean, Harmonic Mean from the following
grouped data
Classes Frequency
2-4 3
5-7 7
8-10 9
11-13 5
14-16 4
17-19 6
Central Tendency

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Practice problem 2
Calculate Arithmetic, Geometric & Harmonic Mean from the
following data
42,36,45,33,54,46,27,38,51,49,29,32
Central Tendency

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Practice problem 3
A man travels from Lahore to Islamabad by a car and takes 4 hours
to cover the whole distance. In the first hour he travels at a speed
of 50 km/hr, in the second hour his speed is 64 km/hr, in third hour
his speed is 80 km/hr and in the fourth hour he travels at the speed
of 55 km/hr. Find the average speed of the motorist
Central Tendency

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Practice problem 4
If a strain of bacteria increases its population by 20% in the first
hour, 30% in the next hour and 50% in the next hour, find out an
estimate of the mean percentage growth in population. Starting
with the population of 100 bacteria
Central Tendency

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Study Links
https://www.youtube.com/watch?v=jXKYI7wyqp0
https://www.youtube.com/watch?v=kfEuqcA6vYw

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