LECTURE N0 5.pdf
- 2. 2 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
- 3. 3 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
- 4. 4 Central Tendency Measure of Central Tendency A single figure tends to lie in the middle of the data
- 5. 5 Mathematical Average (Pythagorean Means) 1. Arithmetic Mean 2. Geometric Mean 3. Harmonic Mean Measure of Central Tendency
- 6. 6 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)
- 7. 7 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)
- 8. 8 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
- 9. 9 ➢ 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)
- 10. 10 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)
- 12. 12 ( ) 1 log45 log32 log37 log46 log39 log36 log41 log48 log36 9 + + + + + + + + Geometric Mean
- 15. 15 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
- 16. 16 Example : Grouped Data 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
- 18. 18 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
- 19. 19 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
- 20. 20 Example : Growth Rates 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
- 21. 21 Example : Growth Rates 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
- 22. 22 𝑮𝑴 = 𝒑𝒓𝒐𝒅𝒖𝒄𝒕 𝒐𝒇 𝒂𝒍𝒍 𝒓𝒆𝒍𝒂𝒕𝒊𝒗𝒆 𝒈𝒓𝒐𝒘𝒕𝒉 𝒓𝒂𝒕𝒆𝒔 ൗ 𝟏 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒚𝒆𝒂𝒓𝒔 𝑮𝑴 = 𝟏. 𝟏 ∗ 𝟎. 𝟖 ∗ 𝟏. 𝟑 𝟏 𝟑 = 1.046 Geometric Mean
- 23. 23 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
- 24. 24 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
- 25. 25 Geometric Mean 𝑀𝑒𝑎𝑛 𝐺𝑟𝑜𝑤𝑡ℎ = 𝐺𝑀 = 1.1 ∗ 0.7272 ∗ 1.875 Τ 1 3 = 1.48885 Τ 1 3 𝑮𝑴 = 𝟏. 𝟏𝟒𝟒𝟕 𝑮𝑴 = 𝒑𝒓𝒐𝒅𝒖𝒄𝒕 𝒐𝒇 𝒂𝒍𝒍 𝒓𝒆𝒍𝒂𝒕𝒊𝒗𝒆 𝒈𝒓𝒐𝒘𝒕𝒉 𝒓𝒂𝒕𝒆𝒔 ൗ 𝟏 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒚𝒆𝒂𝒓𝒔
- 26. 26 ➢ 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)
- 27. 27 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)
- 28. 28 ➢ 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)
- 29. 29 ➢ 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)
- 32. 32 Example : Grouped Data 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)
- 34. 34 ➢ 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?
- 35. 35 ➢ 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 How to Choose the Correct Mean?
- 36. 36 ➢ 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 How to Choose the Correct Mean?
- 37. 37 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
- 38. 38 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
- 39. 39 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
- 40. 40 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