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PROBABILITY AND COUNTING RULES IN STATISTICS

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VincentVallejo3

The document provides a detailed overview of probability and counting rules, including techniques like combinatorial analysis, probability distributions, and various counting methods such as permutations and combinations. It covers fundamental concepts such as mean, variance, standard deviation, and introduces different means such as weighted and geometric mean, alongside central tendency measures like median and mode. Additionally, it presents practical examples to illustrate these concepts in various contexts.

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Statistics and Probability
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PROBABILITY AND COUNTING RULES INTRODUCTION COUNTING RULES SAMPLE SPACES AND PROBABILITY THE ADDITION RULES AND MULTIPLICATION RULES FOR PROBABILITY BINOMIAL, AND POISSON PROBABILITY DISTRIBUTION MEAN, VARIANCE, STANDARD DEVIATION, MATHEMATICAL EXPECTATION RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTION MARGINAL AND CONDITIONAL PROBABILITIES
INTRODUCTION  In this chapter we will deal with some counting techniques without enumeration of the number of possible outcomes of a particular set. Such techniques are sometimes called combinatorial analysis.
INTRODUCTION  Also we will deal with probability theory. These include the topics such as probability distribution, mathematical expectation, binomial distribution, and Poisson distribution.
COUNTING RULES  Fundamental Counting Rules Sum Rule Product Rule  Factorial Notation Permutation Permutation with repeated elements. Circular Permutation Combination Combination of different things taken any number at a time.
COUNTING RULES  Sum Rule  Suppose that an event can be performed by either two different procedures, with m possible outcomes for the first procedures and n possible outcomes for the second. If the two sets of possible outcomes are disjoint, then the number of possible outcomes for the event is; m + n
COUNTING RULES  Example:  A scholarship is available, and the professor to receive this scholarship must be chosen from the Education, Criminology or Information Technology Department. How many different choices are there for this scholarship if there are 15 qualified professors from Education Department, 50 qualified professors from Criminology Dept. and 26 qualified professors form IT Dept.?
COUNTING RULES  Example:  First thing, we need to do, is to analyze first what is asked, and then try to list down the following givens. We have: 15 qualified professors from Education 50 qualified professors from Criminology 26 qualified professors from IT. 15 + 50 + 26 = 91 Therefore, there are 91 possible choices towards the scholarship.
COUNTING RULES  Product Rule  In a sequence of n events in which the first has n1 possibilities and the second has n2, and the third is n3, and so forth, the total of possibilities of sequence will be; n1(n2)(n3)…(nk)
COUNTING RULES  Example:  A student has a choice of 5 sandwiches and 6 juices. In how many ways he can choose 1 sandwich and 1 juice?  Solution: He can choose a sandwich in 5 ways, and with each of these choices there are 6 ways of choosing a juice. Hence, the required number of ways = 5(6) = 30 ways
COUNTING RULES  Factorial Notation  Factorial notation n! (which read as n factorial) is the product of the first n consecutive natural numbers. 0! is defined to be 1. n! = n(n-1)(n-2)(n-3)…(3)(2)(1) or 3! = (3)(2)(1) = 6
COUNTING RULES  Example:  Evaluate the following: 1. 1! = 2. 5! = 3. 6! – 4! = 4. 2!(3!) = 5. 10!/5! =
COUNTING RULES  Example:  Evaluate the following: 1. 1! = 1 2. 5! = (5)(4)(3)(2)(1) = 120 3. 6! – 4! = [(6)(5)(4)(3)(2)(1)] – [(4)(3)(2)(1)] = 720 – 24 = 696 4. 2!(3!) = (2)(1) [(3)(2)(1)] = (2)(6) = 12 5. 10!/5! = (10)(9)(8)…(2)(1)/ (5)(4)(3)(2)(1) = 3, 628, 800/120 = 30, 240
COUNTING RULES  Permutation  A permutation is an arrangement of all or part of a number or things (or objects) in a definite order. The number of permutation n of objects taken r at a time is given by; P(n, r) = n Pr = n!/(n-r)!, 0 ≤ r ≤ n.
COUNTING RULES  Permutation with repeated Elements  It often happens that objects which are virtually identical get arrange. Our inability to distinguish between these items reduces the number of possible permutation by the number of ways these identical items themselves can be arrange; Pn= n! /n! (n2!) (n3!)…
COUNTING RULES  Circular Permutation  When things are arranged in places along a closed curve or a circle, in which any place may regarded as the first or last place, they can for a circular permutation. Thus with n distinguishable objects we have (n-1) arrangement; Pc= (n – 1)!
COUNTING RULES  Example:  Evaluate the following: 1. P(4, 0) 2. P(5, 2) 3. P (7, 7)
COUNTING RULES  Example:  Evaluate the following: 1. P(4, 0) = 4!/4! = (4)(3)(2)(1)/(4)(3)(2)(1) = 24/24 = 1 2. P(5, 2) = 5!/(5-2)! = (5)(4)(3)(2)(1)/(3)(2)(1) = 120/6 = 20 3. P (7, 7) = 7!/(7-7)! = 7!/0! = (7)(6)…(2)(1)/1 = 5, 040
COUNTING RULES  Example:  How many different ways can a manager and a supervisor can be selected for a company branch in Manila if there are 8 employees available?
COUNTING RULES  Example:  How many different ways can a manager and a supervisor can be selected for a company branch in Manila if there are 8 employees available? P(8, 2) = 8!/(8-2)! = 8!/6! = (8)(7)(6)…(2)(1)/(6)(5)…(2)(1) = 40, 320/720 = 56. Hence, there would be 56 ways to select a manager and a supervisor.
COUNTING RULES  Example:  In how many ways can 4 students seated at a round table?
COUNTING RULES  Example:  In how many ways can 4 students seated at a round table? Solution: Let one of them seated anywhere. Then the 3 remaining can be seated in 3! ways. Thus, there are Pc = (n-1)! = 3! = 6 ways of arranging 4 persons in a circle.
COUNTING RULES  Example:  There are 4 copies of Statistics book, 5 copies of Probability book, and 3 copies of Forecasting book. In how many ways can they be arranged on shelf?
COUNTING RULES  Example:  Solution: There are 4 + 5 + 3 = 12 books. The arrangement of a book is Pn = n!/n1! n2! n3! = 12!/4!(5!)(3!) = 27, 720 So, the book can be arrange in 27,720 ways in a shelf.
COUNTING RULES  Combination  A combination is a grouping or selection of all or part of a number of things (or objects) without reference to the arrangement of the things selected. The number of combinations of n objects taken r at a time is given by ; C(n, r) = n Pr = n!/(n-r)! r!, 0 ≤ r ≤ n.
COUNTING RULES  Combination of different things taken any number at a time.  The total number of combinations Cn of n different things taken 1, 2, 3…, n at a time is; Cn = 2n - 1
COUNTING RULES  Example:  Evaluate the following: 1. C(5, 0) 2. C(4, 4) 3. C(6, 2)
COUNTING RULES  Example:  In how many ways can 4 board members be selected out of 15 board members of a company to represent the body in the stockholders meeting? Solution: Since this is a combination problem, the answer is; C (15, 4) = 15!/(15-4)4! = 15!/11!(4!) = 1,365 Hence, there would be 1, 365 ways to select a committee to represent the body.
COUNTING RULES  Example:  In how many ways can 4 board members be selected out of 15 board members of a company to represent the body in the stockholders meeting?
WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE  The weighted mean is found by multiplying each value by it’s corresponding weight and dividing by the sum of the weights.
WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE = weighted mean = corresponding weight = value of any particular observation or measurements
WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE At the mathematics Dept. of PCC there are 18 instructors, 12 assistant professors, 7 associate professors, and 3 professors. Their monthly salary are 30,500, 33,700, 38, 600, and 45, 000. What is weighted mean salary? EXAMPLE
WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE = (18)(30,500) + (12)(33,700) + (7)(38,600) + (3)(45,000) /18 + 12 + 7 + 3 = 1, 358, 600/40 = 33, 965 weighted mean salary
WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE  The geometric mean of a set of n positive numbers is defined as the nth root of the product of the n numbers.
WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE  There are two application of geometric mean: The first one is to average, percents, indexes, and relatives. The second one is to establish the average percent increase in production, sales, or other business transaction or economic series from one period of time to another.
WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE
COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Suppose the profits earned by the MSS Construction Company on five projects were 5, 6, 4, 8, and 10 percent, respectively. What is geometric mean profit? EXAMPLE GEOMETRIC MEAN WEIGHTED MEAN
COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE EXAMPLE GEOMETRIC MEAN WEIGHTED MEAN
COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Badminton as a sport grew rapidly in 2008. From January and December 2008 the number of badminton clubs in Manila increased from 20 to 115. Compute the mean monthly percent increase in the number of badminton clubs. EXAMPLE GEOMETRIC MEAN WEIGHTED MEAN
COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE EXAMPLE GEOMETRIC MEAN WEIGHTED MEAN
WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE  Note: The geometric cannot be computed if one of the numbers is zero or negative.
WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE  The combined mean is the grand mean of all the values in all groups when two or more groups are combined.
WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE = Combined mean = Sample mean = sample size
COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE A study comparing the typical household incomes for 3 districts in the City of Manila was initiated to see where differences in household incomes lie across districts. The mean household incomes for a sample of 45 different families in three districts of Manila are shown in the following table. Calculate a combined mean to obtain the average household income for all 45 families in the Manila sample EXAMPLE WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN
COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE EXAMPLE WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN District 1 District 2 District 3 = 30,400 = 27, 300 = 42, 000 = 12 = 18 = 15 = 30, 400 (12) + 27, 300 ( 18) + 42, 500 (15) / 12 + 18 + 15 = 1, 493, 700/45 = 33, 193.33 combined mean in 3 districts of Manila
COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE The median is the midpoint of the data array. When the data set is ordered whether ascending or descending, it is called a data array. WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN
COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Median is an appropriate measure of central tendency for data that are ordinal or above, but is more valuable in an ordinal type of data. WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN
COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Properties of Median: 1. The median is unique, there is only one median for a set of data. 2. The median is found by arranging the set of data from lowest to highest or vice versa and getting the value of the middle observation. 3. Median is not affected by the extreme small or large values. WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN
COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Properties of Median: 4. Median can be computed for an open-ended frequency distribution. 5. Median can be applied for ordinal, interval, and ratio data. WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN
COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Median for Ungrouped Data: To determine the value of median for ungrouped data we need to consider two rules. 1. If n is odd, the median is the middle ranked. 2. If n is even, then the median is the average of two middle ranked values. Median (Rank Value) = n + 1/2 n = the population/sample size. WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN
COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Find the median of ages of 9 middle-management employees of a certain company. The ages are 53, 45, 59, 48, 54, 46, 51, 58, and 55. WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN EXAMPLE
COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE The daily rates of a sample of eight employees at GMS Inc. are 550, 420, 560, 500, 700, 670, 860, 480. Find the median daily rate of employees. WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN EXAMPLE
COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Median for Grouped Data: Take note that the median is located in the middle value of frequency distribution. It is the value that separates the upper half of the distribution from lower half. WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN
COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Median for Grouped Data: It is also obvious to note that it is a measure of central tendency because it is the exact center of the scores in a distribution. Median (Ranked Value) = N/2 WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN
COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Median for Grouped Data: WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN
COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Determine the median of the frequency distribution on the ages 50 people taking travel tours. Given the table WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN EXAMPLE Class Limits f 18-26 3 27-35 5 36-44 9 45-53 14 54-62 11 63-71 6 72-80 2
WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE The mode is the value in a data set that appears most frequently.
WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Properties of Mode: 1. The mode is found by locating the most frequently value. 2. The mode is the easiest average to compute.
WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Properties of Mode: 3. There can be more than one mode or even no mode in any given data set. 4. Mode is not affected by the extreme small or large values. 5. Mode can be applied for nominal, interval, and ratio data.
WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE EXAMPLE

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PROBABILITY AND COUNTING RULES IN STATISTICS

  • 1. PROBABILITY AND COUNTING RULES INTRODUCTION COUNTING RULES SAMPLE SPACES AND PROBABILITY THE ADDITION RULES AND MULTIPLICATION RULES FOR PROBABILITY BINOMIAL, AND POISSON PROBABILITY DISTRIBUTION MEAN, VARIANCE, STANDARD DEVIATION, MATHEMATICAL EXPECTATION RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTION MARGINAL AND CONDITIONAL PROBABILITIES
  • 2. INTRODUCTION  In this chapter we will deal with some counting techniques without enumeration of the number of possible outcomes of a particular set. Such techniques are sometimes called combinatorial analysis.
  • 3. INTRODUCTION  Also we will deal with probability theory. These include the topics such as probability distribution, mathematical expectation, binomial distribution, and Poisson distribution.
  • 4. COUNTING RULES  Fundamental Counting Rules Sum Rule Product Rule  Factorial Notation Permutation Permutation with repeated elements. Circular Permutation Combination Combination of different things taken any number at a time.
  • 5. COUNTING RULES  Sum Rule  Suppose that an event can be performed by either two different procedures, with m possible outcomes for the first procedures and n possible outcomes for the second. If the two sets of possible outcomes are disjoint, then the number of possible outcomes for the event is; m + n
  • 6. COUNTING RULES  Example:  A scholarship is available, and the professor to receive this scholarship must be chosen from the Education, Criminology or Information Technology Department. How many different choices are there for this scholarship if there are 15 qualified professors from Education Department, 50 qualified professors from Criminology Dept. and 26 qualified professors form IT Dept.?
  • 7. COUNTING RULES  Example:  First thing, we need to do, is to analyze first what is asked, and then try to list down the following givens. We have: 15 qualified professors from Education 50 qualified professors from Criminology 26 qualified professors from IT. 15 + 50 + 26 = 91 Therefore, there are 91 possible choices towards the scholarship.
  • 8. COUNTING RULES  Product Rule  In a sequence of n events in which the first has n1 possibilities and the second has n2, and the third is n3, and so forth, the total of possibilities of sequence will be; n1(n2)(n3)…(nk)
  • 9. COUNTING RULES  Example:  A student has a choice of 5 sandwiches and 6 juices. In how many ways he can choose 1 sandwich and 1 juice?  Solution: He can choose a sandwich in 5 ways, and with each of these choices there are 6 ways of choosing a juice. Hence, the required number of ways = 5(6) = 30 ways
  • 10. COUNTING RULES  Factorial Notation  Factorial notation n! (which read as n factorial) is the product of the first n consecutive natural numbers. 0! is defined to be 1. n! = n(n-1)(n-2)(n-3)…(3)(2)(1) or 3! = (3)(2)(1) = 6
  • 11. COUNTING RULES  Example:  Evaluate the following: 1. 1! = 2. 5! = 3. 6! – 4! = 4. 2!(3!) = 5. 10!/5! =
  • 12. COUNTING RULES  Example:  Evaluate the following: 1. 1! = 1 2. 5! = (5)(4)(3)(2)(1) = 120 3. 6! – 4! = [(6)(5)(4)(3)(2)(1)] – [(4)(3)(2)(1)] = 720 – 24 = 696 4. 2!(3!) = (2)(1) [(3)(2)(1)] = (2)(6) = 12 5. 10!/5! = (10)(9)(8)…(2)(1)/ (5)(4)(3)(2)(1) = 3, 628, 800/120 = 30, 240
  • 13. COUNTING RULES  Permutation  A permutation is an arrangement of all or part of a number or things (or objects) in a definite order. The number of permutation n of objects taken r at a time is given by; P(n, r) = n Pr = n!/(n-r)!, 0 ≤ r ≤ n.
  • 14. COUNTING RULES  Permutation with repeated Elements  It often happens that objects which are virtually identical get arrange. Our inability to distinguish between these items reduces the number of possible permutation by the number of ways these identical items themselves can be arrange; Pn= n! /n! (n2!) (n3!)…
  • 15. COUNTING RULES  Circular Permutation  When things are arranged in places along a closed curve or a circle, in which any place may regarded as the first or last place, they can for a circular permutation. Thus with n distinguishable objects we have (n-1) arrangement; Pc= (n – 1)!
  • 16. COUNTING RULES  Example:  Evaluate the following: 1. P(4, 0) 2. P(5, 2) 3. P (7, 7)
  • 17. COUNTING RULES  Example:  Evaluate the following: 1. P(4, 0) = 4!/4! = (4)(3)(2)(1)/(4)(3)(2)(1) = 24/24 = 1 2. P(5, 2) = 5!/(5-2)! = (5)(4)(3)(2)(1)/(3)(2)(1) = 120/6 = 20 3. P (7, 7) = 7!/(7-7)! = 7!/0! = (7)(6)…(2)(1)/1 = 5, 040
  • 18. COUNTING RULES  Example:  How many different ways can a manager and a supervisor can be selected for a company branch in Manila if there are 8 employees available?
  • 19. COUNTING RULES  Example:  How many different ways can a manager and a supervisor can be selected for a company branch in Manila if there are 8 employees available? P(8, 2) = 8!/(8-2)! = 8!/6! = (8)(7)(6)…(2)(1)/(6)(5)…(2)(1) = 40, 320/720 = 56. Hence, there would be 56 ways to select a manager and a supervisor.
  • 20. COUNTING RULES  Example:  In how many ways can 4 students seated at a round table?
  • 21. COUNTING RULES  Example:  In how many ways can 4 students seated at a round table? Solution: Let one of them seated anywhere. Then the 3 remaining can be seated in 3! ways. Thus, there are Pc = (n-1)! = 3! = 6 ways of arranging 4 persons in a circle.
  • 22. COUNTING RULES  Example:  There are 4 copies of Statistics book, 5 copies of Probability book, and 3 copies of Forecasting book. In how many ways can they be arranged on shelf?
  • 23. COUNTING RULES  Example:  Solution: There are 4 + 5 + 3 = 12 books. The arrangement of a book is Pn = n!/n1! n2! n3! = 12!/4!(5!)(3!) = 27, 720 So, the book can be arrange in 27,720 ways in a shelf.
  • 24. COUNTING RULES  Combination  A combination is a grouping or selection of all or part of a number of things (or objects) without reference to the arrangement of the things selected. The number of combinations of n objects taken r at a time is given by ; C(n, r) = n Pr = n!/(n-r)! r!, 0 ≤ r ≤ n.
  • 25. COUNTING RULES  Combination of different things taken any number at a time.  The total number of combinations Cn of n different things taken 1, 2, 3…, n at a time is; Cn = 2n - 1
  • 26. COUNTING RULES  Example:  Evaluate the following: 1. C(5, 0) 2. C(4, 4) 3. C(6, 2)
  • 27. COUNTING RULES  Example:  In how many ways can 4 board members be selected out of 15 board members of a company to represent the body in the stockholders meeting? Solution: Since this is a combination problem, the answer is; C (15, 4) = 15!/(15-4)4! = 15!/11!(4!) = 1,365 Hence, there would be 1, 365 ways to select a committee to represent the body.
  • 28. COUNTING RULES  Example:  In how many ways can 4 board members be selected out of 15 board members of a company to represent the body in the stockholders meeting?
  • 29. WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE  The weighted mean is found by multiplying each value by it’s corresponding weight and dividing by the sum of the weights.
  • 30. WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE = weighted mean = corresponding weight = value of any particular observation or measurements
  • 31. WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE At the mathematics Dept. of PCC there are 18 instructors, 12 assistant professors, 7 associate professors, and 3 professors. Their monthly salary are 30,500, 33,700, 38, 600, and 45, 000. What is weighted mean salary? EXAMPLE
  • 32. WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE = (18)(30,500) + (12)(33,700) + (7)(38,600) + (3)(45,000) /18 + 12 + 7 + 3 = 1, 358, 600/40 = 33, 965 weighted mean salary
  • 33. WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE  The geometric mean of a set of n positive numbers is defined as the nth root of the product of the n numbers.
  • 34. WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE  There are two application of geometric mean: The first one is to average, percents, indexes, and relatives. The second one is to establish the average percent increase in production, sales, or other business transaction or economic series from one period of time to another.
  • 35. WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE
  • 36. COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Suppose the profits earned by the MSS Construction Company on five projects were 5, 6, 4, 8, and 10 percent, respectively. What is geometric mean profit? EXAMPLE GEOMETRIC MEAN WEIGHTED MEAN
  • 37. COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE EXAMPLE GEOMETRIC MEAN WEIGHTED MEAN
  • 38. COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Badminton as a sport grew rapidly in 2008. From January and December 2008 the number of badminton clubs in Manila increased from 20 to 115. Compute the mean monthly percent increase in the number of badminton clubs. EXAMPLE GEOMETRIC MEAN WEIGHTED MEAN
  • 39. COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE EXAMPLE GEOMETRIC MEAN WEIGHTED MEAN
  • 40. WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE  Note: The geometric cannot be computed if one of the numbers is zero or negative.
  • 41. WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE  The combined mean is the grand mean of all the values in all groups when two or more groups are combined.
  • 42. WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE = Combined mean = Sample mean = sample size
  • 43. COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE A study comparing the typical household incomes for 3 districts in the City of Manila was initiated to see where differences in household incomes lie across districts. The mean household incomes for a sample of 45 different families in three districts of Manila are shown in the following table. Calculate a combined mean to obtain the average household income for all 45 families in the Manila sample EXAMPLE WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN
  • 44. COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE EXAMPLE WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN District 1 District 2 District 3 = 30,400 = 27, 300 = 42, 000 = 12 = 18 = 15 = 30, 400 (12) + 27, 300 ( 18) + 42, 500 (15) / 12 + 18 + 15 = 1, 493, 700/45 = 33, 193.33 combined mean in 3 districts of Manila
  • 45. COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE The median is the midpoint of the data array. When the data set is ordered whether ascending or descending, it is called a data array. WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN
  • 46. COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Median is an appropriate measure of central tendency for data that are ordinal or above, but is more valuable in an ordinal type of data. WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN
  • 47. COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Properties of Median: 1. The median is unique, there is only one median for a set of data. 2. The median is found by arranging the set of data from lowest to highest or vice versa and getting the value of the middle observation. 3. Median is not affected by the extreme small or large values. WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN
  • 48. COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Properties of Median: 4. Median can be computed for an open-ended frequency distribution. 5. Median can be applied for ordinal, interval, and ratio data. WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN
  • 49. COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Median for Ungrouped Data: To determine the value of median for ungrouped data we need to consider two rules. 1. If n is odd, the median is the middle ranked. 2. If n is even, then the median is the average of two middle ranked values. Median (Rank Value) = n + 1/2 n = the population/sample size. WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN
  • 50. COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Find the median of ages of 9 middle-management employees of a certain company. The ages are 53, 45, 59, 48, 54, 46, 51, 58, and 55. WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN EXAMPLE
  • 51. COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE The daily rates of a sample of eight employees at GMS Inc. are 550, 420, 560, 500, 700, 670, 860, 480. Find the median daily rate of employees. WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN EXAMPLE
  • 52. COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Median for Grouped Data: Take note that the median is located in the middle value of frequency distribution. It is the value that separates the upper half of the distribution from lower half. WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN
  • 53. COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Median for Grouped Data: It is also obvious to note that it is a measure of central tendency because it is the exact center of the scores in a distribution. Median (Ranked Value) = N/2 WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN
  • 54. COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Median for Grouped Data: WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN
  • 55. COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Determine the median of the frequency distribution on the ages 50 people taking travel tours. Given the table WEIGHTED MEAN COMBINED MEAN GEOMETRIC MEAN EXAMPLE Class Limits f 18-26 3 27-35 5 36-44 9 45-53 14 54-62 11 63-71 6 72-80 2
  • 56. WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE The mode is the value in a data set that appears most frequently.
  • 57. WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Properties of Mode: 1. The mode is found by locating the most frequently value. 2. The mode is the easiest average to compute.
  • 58. WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE Properties of Mode: 3. There can be more than one mode or even no mode in any given data set. 4. Mode is not affected by the extreme small or large values. 5. Mode can be applied for nominal, interval, and ratio data.
  • 59. WEIGHTED MEAN GEOMETRIC MEAN COMBINED MEAN MEDIAN TYPES OF DISTRIBUTION EFFECTS OF CHANGING THE UNITS ON MEAN AND MEDIAN MIDRANGE MODE EXAMPLE