Symbolic Arguments PDF

SYMBOLIC ARGUMENTS Mathematics in the Modern World Objectives: After the discussion, the students are able to: a. Write the argument in symbolic form; b. Determine the validity of an argument; and c. Determine a valid conclusion for an argument. ARGUMENTS An argument con...

SYMBOLIC ARGUMENTS Mathematics in the Modern World Objectives: After the discussion, the students are able to: a. Write the argument in symbolic form; b. Determine the validity of an argument; and c. Determine a valid conclusion for an argument. ARGUMENTS An argument consists of two components the initial statements, or hypothesis or premises and the final statements or the conclusion. THE ARGUMENT TYPES An inductive argument uses a collection of specific examples as its premises and uses them to propose a general conclusion (from specific to general). A deductive argument uses a collection of general statements as its premises and uses them to propose a specific situation as the conclusion (from general to specific).  When presented with an argument, a listener or reader may ask, “Does this person have a logical argument? “Does his or her conclusion necessarily follow from the given statements?” An argument is valid if the conclusion of the argument is guaranteed under its given set of hypotheses. Valid Argument An argument is valid if the conclusion is true whenever all the premises are assumed to be true. An argument is invalid if it is not a valid argument. Example 1. Human beings are mortal. Aristotle is human. Therefore, Aristotle is mortal. In the argument above, the two premises and the conclusion is shown below. First Premise: Human beings are mortal. (a generalization) Second Premise: Aristotle is human. (specific example) Conclusion: Therefore, Aristotle is mortal. This is an example of Deductive Argument. Example 2. The premises are:  I ate candy last week and I had toothache.  I ate candy yesterday, and I had toothache.  I ate candy today, and I had toothache. The conclusion is: Therefore, eating candy gives me toothache. This is an example of Inductive Argument.  Classify each argument as deductive or inductive. I ate a hotdog at Julies Resto and I get indigestion. I ate a hotdog at Macdoy’s and I got indigestion. Conclusion: Therefore, eating hotdogs give me indigestion. So, this is inductive argument. All spicy foods upset my stomach. Caldereta is a spicy food. Conclusion: Therefore, eating caldereta will upset my stomach. So, this is deductive argument. ARGUMENTS IN SYMBOLIC FORM Arguments can be written in symbolic form. For instance, If we let “h” represents the statement “Aristotle was human” and “m” represents “Aristotle was mortal” then the argument can be expressed as: h m (first premise) h (second premise) ∴m The three dots ∴ are symbol for “therefore” Example 1: The fish is fresh or I will not order it. The fish is fresh. Therefore, I will order it. Solution: Let f represent the statement “The fish is fresh”. Let O represent the statement “I will order it. The symbolic form of the argument is f ⋁ ∼O f ∴O Example 1: Write the following arguments in symbolic form. “If she doesn’t get on the plane. She will regret it. she doesn’t regret it. Therefore, she got on the plane.” Let p represent “She got on the plane.” Let r represent “She will regret it.” ~p → r ~r ∴p ARGUMENTS AND TRUTH TABLE Truth Table Procedure to Determine the Validity of an Argument 1. Write the argument in symbolic form. 2. Construct a truth table that shows the truth value of each premise and the truth value of the conclusions for all combinations or truth value of the simple statements. 3. If the conclusion is true in every row or the truth table in which all the premises are true, the argument is valid. If the conclusion is false in any row in which all the premises are true, the argument is invalid.  1. Once again, we let h represent the statement “Aristotle was human” and m represent the statement “Aristotle was mortal”. In symbolic form the argument is h m (first premise) h (second premise) ∴m Conclusion 2. Construct a truth table. r g First premise Second premise Conclusion h m h m T T F F T row 1 T F T F F row 2 F T T T T row 3 F F T T F row 4 3. The argument is valid.  Example 2: Determine the Validity of an Argument  “If it rains, then the game will not be played. It is not raining. Therefore, the game will be played. Solution If we let r represent “it rains” and g represent “the game will be played,” then the symbolic form is r ~g ~r ∴g  Truth table r g First premise Second premise Conclusion r~g ~ r g T T F F T row 1 T F T F F row 2 F T T T T row 3 F F T T F row 4 The conclusion in row 4 is false and the premises are both true, the argument is invalid. Example r f If the stock market rises, then the bond market will fall. The bond market did not fall. ~f ∴ The stock market did not rise. ~r  Truth Table r f r→f ~f ~r Row T T T F F 1 Row T F F T F 2 Row T T F T F 3 Row F F T T T 4 Valid  Example 3: Determine the Validity of an Argument  Determine whether the following argument is valid or invalid. If I am going to run the marathon, then I will buy new shoes. If I buy new shoes, then I will not buy a television. ∴ If I buy a television, I will not run the marathon. SOLUTION: The symbolic form of the Label each simple argument is: statement. m: I am going to run the m→s marathon. s → ~t s: I will buy new shoes. ∴ t → ~m t: I will buy a television. The truth table for this argument follows. First Premise Second Premise Conclusion m s t m →s s → ~t t → ~m T T T T F F Row 1 T T F T T T Row 2 T F T F T F Row 3 T F F F T T Row 4 F T T T F T Row 5 F T F T T T Row 6 F F T T T T Row 7 F F F T T T Row 8 The argument is valid. Determine whether the following argument is valid or invalid. If I arrive before 8 A.M, then I will make the flight. If I make the flight, then I will give the presentation. ∴ If I arrive before 8 A.M, then I will give the presentation. Let a represent “I arrive before 8 A.M.” Solution Let f represent “I will make the flight.” Let p represent “I will give the presentation.” The symbolic form is: a→f f→p ∴a→p The truth table for this argument: First Second Conclusion a f p premise premise a⟶p a⟶f f⟶p T T T T T T Row1 T T F T F F Row 2 T F T F T T Row 3 T F F F T F Row 4 F T T T T T Row 5 F T F T F T Row 6 F F T T T T Row 7 F F F T T T Row 8 The argument is valid. STANDARD FORMS Arguments can be shown to be valid if they have the same symbolic form as an argument that is known to be valid. For instance, we have shown that the argument is valid h→m h ∴m This symbolic form is known as direct reasoning. All arguments that have this symbolic form are valid.  Table 5.15 shows four symbolic form and the name used to identify each form. Any argument that has a symbolic form identical to one of these symbolic forms is a valid argument. TABLE 5.5 Standard Forms of Four Valid Arguments Direct Reasoning Contrapositive Transitive Disjunctive reasoning Reasoning Reasoning p→q p→q p→q p⋁q p⋁q p ~q q→r ~p ~q ∴q ∴ ~p ∴ p→r ∴q ∴p Transitive Reasoning It can be extended to include more then two conditional premises. For instance, if the conditional premises of an arguments as p → q, q → r and r → 3, then a valid conclusion for the argument is p → s. In Example 4 we use standard forms to determine a valid conclusion for an argument.  Example 4: Determine a Valid Conclusion for an Argument Use a standard form from Table 5.15 to determine a valid conclusion for each argument. a. If Kim is a lawyer (p), then she will be able to help us (q) Kim is not able to help us (~q) ∴? b. If they had a good time (g), they will return (r). If they return (r), we will make more money (m). ∴? Solution a. The symbolic form of the premises is: p→q ~q This matches the standard form known as contrapositive reasoning. Thus, a valid conclusion is ~p: “Kim is not a lawyer” b. The symbolic form of the premises is: g→r r→m This matches the standard form known as transitive reasoning. Thus, a valid conclusion is g → m: “If they had a good time, then we will make more money”. Table 5. 16 shows the two symbolic forms associated with invalid arguments. Any argument that has one of these symbolic forms is invalid. TABLE 5.16 Standard form of Two Invalid Arguments Fallacy of the converse Fallacy of the Inverse p→q p→q q ~p ∴p ∴ ~q Other Example: If they eat (f), they will go back. (b) If they go back (b), they will have a party. (p) ∴ If they eat, they will have a party. f→b b→p ∴f→p Example: Use a standard form from Table 5.15 to determine a valid conclusion for each argument. a. If you can dream it (p), you can do it. (q) You can dream it (p). ∴? b. I bought a car (c) or I bought a motorcycle. (m) I did not buy a car. (~c) ∴? Answer: a. If you can dream it (p), you can do it. (q) p→q You can dream it (p). p ∴ You can do it. ∴q b. I bought a car (c) or I bought a motorcycle. (m) c∨m I did not buy a car. (~c) ~c ∴ I bought a motorcycle. ∴m  Example 5: Use a Standard Form to Determine the Validity of an Argument Use a standard form from a Table 5.15 or Table 5.16 to determine the following arguments are valid or invalid. Table 5.15 Direct Contrapositive Transitive Disjunctive Reasoning reasoning Reasoning Reasoning p→q p→q p→q p⋁q p⋁q p ~q q→r ~p ~q ∴q ∴~p ∴p → r ∴q ∴p Table 5.16 Standard forms of Four Valid Arguments Fallacy of the converse Fallacy of the Inverse p→q p→q q ~p ჻p ∴~q Standard forms of Two Invalid Arguments a. The program is interesting, or I will watch the basketball game. The program is not interesting. ∴ I will watch the basketball game. b. If I have a cold, then I find it difficult to sleep. I find it difficult to sleep. ∴ I have a cold.  Solution Table 5.15 a. Label each simple Direct Contrapositive Transitive Disjunctive Reasoning reasoning Reasoning Reasoning statements. i: The program is p→q p→q p→q p⋁q p⋁q interesting. p ~q q→r ~p ~q w: I will watch the ∴q ∴~p ∴p → r ∴q ∴p basketball game. Standard forms of Four Valid Arguments In symbolic form the argument is i⋁w ~i ∴w The symbolic form matches one of the standard forms known as disjunctive reasoning. Thus, the argument is valid. Solution Table 5.16 b. Label each simple Fallacy of the converse Fallacy of the Inverse statements. p→q p→q c: I have a cold. q ~p s: I find it difficult to ჻p ∴~q sleep. Standard forms of Two Invalid Arguments In symbolic form the argument is c→s s ∴c This symbolic form matches the standard form known as the fallacy of the converse. Thus, the argument is invalid. Example: Use a standard form from Table 5.15 or Table 5.16 to determine whether the following arguments are valid or invalid. a. If it is raining, then the grass is wet. It is not raining. ∴ The grass is not wet. b. If you helped solve the crime, then you should be rewarded. You helped solve the crime. ∴ You should be rewarded.  Solution for a: Label the simple statements. g: It is raining. a. If it is raining, then the grass is wet. h: The grass is wet. It is not raining. ∴ The grass is not wet. In symbolic form the argument is g→h ~g ∴ ~h This symbolic form matches one of the standard form known as fallacy of inverse. Thus, the argument is invalid.  Solution for b: Label the simple statements. b. If you helped solve the crime, then you h: You helped solve the crime. should be rewarded. r: You should be rewarded. You helped solve the crime. ∴ You should be rewarded. In symbolic form the argument is h→r h ∴r This symbolic form matches one of the standard forms known as direct reasoning. Thus, the agreement is valid.  Example 6: Determine the Validity of an Argument Determine whether the following argument is valid. If the movie was directed by Steven Spielberg (s), then I want to see it (w). The movie’s production costs must exceed $50 million (c) or I do not want to see it. the movie’s production costs were less than $50 million. Therefore, the movie was not directed by Steven Spielberg.  Solution In symbolic form the argument is s→w Premise 1 c ∨ ~w Premise 2 ~c Premise 3 ∴ ~s Conclusion Premise 2 can be written as ~w ∨ c, which is equivalent to w → c. Applying transitive reasoning to Premise 1, and this equivalent form of Premise 2 produces: s → w Premise 1 w → c Equivalent form of Premise 2 ∴ s → c Transitive Reasoning Combining the conclusion s → c with Premise 3 gives us s→c Conclusion from previous argument ~c Premise 3 ∴ ~s Contrapositive reasoning This sequence of valid arguments has produced the desired conclusion, ~s. thus, the original argument is valid. Example 2: Determine whether the following argument is valid. I start to fall asleep if I read a math book. I drink soda whenever I start to fall asleep. If I drink soda, then I must eat a candy bar. Therefore, I eat a candy bar whenever I read a math book. Hint: p whenever q is equivalent to q → p. Solution Let a represent “I started to fall asleep.” Let m represent “I read a math book.” In symbolic form the argument is Let s represent “I drink soda.” m → a Premise 1 Let c represent “I eat a candy bar.” s → a Premise 2 s → c Premise 3 ∴c → m Conclusion Premise 2 is equivalent to a → s. Applying transitive reasoning to Premise 1 and this equivalent form of Premise 2 produces m → a Premise 1 a → s Equivalent form of Premise 2 ∴m → s Transitive Reasoning Combining the conclusion m → s with Premise 3 gives us m → s Conclusion from previous argument s → c Premise 3 ∴m → c Transitive Reasoning This sequence of valid arguments has produced the desired conclusion m → c. Thus, the original argument is valid.  Example 7 Determine a Valid Conclusion for an Argument Use all of the Premises to determine a valid conclusion for the following argument. We will not go to Japan (~j) or we will go to Hongkong (h). If we visit my uncle (u), then we will go to Singapore (s). If we go to Hongkong, then we will not go to Singapore. Solution In symbolic form of the argument is ~j ∨ h Premise 1 u → s Premise 2 h → ~s Premise 3 ∴? The first premise can be written as j → h. The contrapositive of the second premise is ~s → ~u. Therefore, the argument can be written as j→h ~s → ~u h → ~s ∴? Interchanging the second and third premises yields j→h h → ~s ~s → ~u ∴? An application of transitive reasoning produces j→h h → ~s ~ s → ~u ∴ j → ~u Thus, a valid conclusion for the original argument is “If we go to Japan (j), when we will not visit my uncle (~u).” Example: Use all of premises to determine a valid conclusion for the following argument. 1st premises – I will not go to the mall (m) or I will travel (t) ~m ∨ t 2nd premises – If I will travel (t) then I can’t go to the t → ~d department store (d) e∨g 3rd premises – I will buy envelope (e) or I will eat gulaman (g) e→d 4th premises – If I buy envelope (e) then I will go to the ∴? department store (d) The first premises can be written as m → t. The contrapositive of the second premise is d → ~t. Therefore, the argument can be written as: m→t d → ~t e∨g e→d ∴? Interchanging the second and third premises yields m→t e∨g t → ~d e→d ∴? An application of transitive reasoning produces m→t e∨g t → ~d e→d ∴m → d Thus, a valid conclusion for the original argument is “If I will go to the mall, then I will go to the department store.” interpretation Collection -gathering of information or data. Organization or Presentation -involves summarizing data or information in textual, graphical, or tabular. Analysis -involves describing the data by using statistical methods and procedures. Interpretation -the process of making conclusions based on the analyzed data. Statistics is said to have developed from government records. All cultures with a recorded history also have recorded statistics done mostly by agents of the government for governmental purpose. In the beginning of the 16th century, a large number of statistical handbooks were published. This type of descriptive statistics was referred to as Die Tabellen Statistik. The first scientific analysis of publicly recorded data may be ascribed to Captain John Graunt (1620-1974). The registration of deaths was started by Henry VIII in 1532. Several people who were involved in the development of modern statistics: 1. Abraham De Moivre – who discovered the equation of the normal curve. 2. Karl Pearson 3. Marguis de Laplace 4. Carl Friedrich Gauss In the 12th century, sir Ronald Fisher made significant contributions to the science of statistics. He discovered a unified theory for drawing rigorous conclusions from statistical data. He also contributed to the theory of design od experiments, which provided a technique for collecting primary data in such a way that valid inferences might be drawn from them. APPLICATION OF STATISTICS 1. Business – A business firms collects and gathers data or information from everyday operation. Statistics is used to summarize and describe those data such as the amount of sales, expenditures, and production to enable the management to understand and determine the status of the firm. Data that have been organized and analyzed provide the management baseline data to make wise decisions pertaining to the operation of the business. APPLICATION OF STATISTICS 2. Education – Through statistical tools, a teacher can determine the effectiveness of a particular teaching method by analyzing test scores obtained be their students. Results of this study may be used to improve teaching -learning activities. 3. Psychology – Psychologist are able to interpret meaningful aptitude tests, IQ tests, and other psychological tests using statistical procedures or tools. APPLICATION OF STATISTICS 4. Political and Government– Public opinion and election polls are commonly used to assess the opinions or preferences of the public for issues or candidates of interests. Statistics plays an important role in conducting surveys or interviews for that purpose. 5. Medicine – Statistics is also used in determining the effectiveness of new drug products in treating a particular type of disease. APPLLICATION OF STATISTICS 6. Agriculture– Through statistical tools, an agriculturist can determine the effectiveness of a new fertilizer in the growth of plants or crops. Moreover, crop production and yield can be better analyzed through the use of statistical method. 7. Entertainment – The most favorite actresses and actors can be determined by using surveys. Ratings of the members of the board of judges in a beauty contests are statistically analyzed. APPLLICATION OF STATISTICS 8. Everyday Life – Descriptive and Inferential Statistics The study of statistics is divided into two categories: a. Descriptive b. Inferential  a. Descriptive Statistics A statistical procedure concerned with the describing the characteristics and properties of a group persons, places, or things. b. Inferential Statistics A statistical procedure that is used to draw inferences or information about the properties or characteristics by a large group of people, places, or things on the basis of the information obtained from a small portion of a large group.  End! “If any of you lacks wisdom, let him ask God, who gives generously to all without reproach, and it will be given him.” James 1:5 QUIZ 1 Tell whether the following situations will make use of descriptive statistics or inferential statistics 1. A teacher computes the average grade of her students and then determines the top ten students. 2. A manage of a business firm predicts future sales of the company based on the present sales. 3. A psychologist investigates if there is a significant relationship between mental age and chronological age. 4. A researcher studies the effectiveness of a new fertilizer to increasing food production. 5. A janitor counts the number of various furniture inside the school. 6. A sports journalist determines the most popular basketball player for this year 7. A school administrator forecast future expansion of a school 8. A market vendor investigates the most popular brand of vinegar 9. An engineer calculates the average height of the building along Megaworld. 10.A dermatologist test the relative effectiveness of a new brand of medicine in curing pimples and other skin disease.  CHAPTER 4.1 Statistics Copyright © Cengage Learning. All rights reserved.  Measures of Central Section 4.1 Tendency Copyright © Cengage Learning. All rights reserved. The Arithmetic Mean 3 The Arithmetic Mean Statistics involves the collection, organization, summarization, presentation, and interpretation of data. The branch of statistics that involves the collection, organization, summarization, and presentation of data is called descriptive statistics. The branch that interprets and draws conclusions from the data is called inferential statistics. 4 The Arithmetic Mean One of the most basic statistical concepts involves finding measures of central tendency of a set of numerical data. We will consider three types of averages, known as the arithmetic mean, the median, and the mode. Each of these averages is a measure of central tendency for the numerical data. 5 The Arithmetic Mean In statistics it is often necessary to find the sum of a set of numbers. The traditional symbol used to indicate a summation is the Greek letter sigma, . Thus the notation x, called summation notation, denotes the sum of all the numbers in a given set. We can define the mean using summation notation. 6 The Arithmetic Mean Statisticians often collect data from small portions of a large group in order to determine information about the group. In such situations the entire group under consideration is known as the population, and any subset of the population is called a sample. It is traditional to denote the mean of a sample by (which is read as “x bar”) and to denote the mean of a population by the Greek letter  (lowercase mu). 7 Example 1 – Find a Mean Six friends in a biology class of 20 students received test grades of 92, 84, 65, 76, 88, and 90 Find the mean of these test scores. Solution: The 6 friends are a sample of the population of 20 students. Use to represent the mean. 8 Example 1 – Solution cont’d The mean of these test scores is 82.5. 9 The Median 10 The Median Another type of average is the median. Essentially, the median is the middle number or the mean of the two middle numbers in a list of numbers that have been arranged in numerical order from smallest to largest or largest to smallest. Any list of numbers that is arranged in numerical order from smallest to largest or largest to smallest is a ranked list. 11 Example 2 – Find a Median Find the median of the data in the following lists. a. 4, 8, 1, 14, 9, 21, 12 b. 46, 23, 92, 89, 77, 108 Solution: a. The list 4, 8, 1, 14, 9, 21, 12 contains 7 numbers. The median of a list with an odd number of entries is found by ranking the numbers and finding the middle number. Ranking the numbers from smallest to largest gives 1, 4, 8, 9, 12, 14, 21 The middle number is 9. Thus 9 is the median. 12 Example 2 – Solution cont’d b. The list 46, 23, 92, 89, 77, 108 contains 6 numbers. The median of a list of data with an even number of entries is found by ranking the numbers and computing the mean of the two middle numbers. Ranking the numbers from smallest to largest gives 23, 46, 77, 89, 92, 108 The two middle numbers are 77 and 89. The mean of 77 and 89 is 83. Thus 83 is the median of the data. 13 The Mode 14 The Mode A third type of average is the mode. 15 Example 3 – Find a Mode Find the mode of the data in the following lists. a. 18, 15, 21, 16, 15, 14, 15, 21 b. 2, 5, 8, 9, 11, 4, 7, 23 Solution: a. In the list 18, 15, 21, 16, 15, 14, 15, 21, the number 15 occurs more often than the other numbers. Thus 15 is the mode. b. Each number in the list 2, 5, 8, 9, 11, 4, 7, 23 occurs only once. Because no number occurs more often than the others, there is no mode. 16 The Weighted Mean 17 The Weighted Mean A value called the weighted mean is often used when some data values are more important than others. For instance, many professors determine a student’s course grade from the student’s tests and the final examination. Consider the situation in which a professor counts the final examination score as 2 test scores. To find the weighted mean of the student’s scores, the professor first assigns a weight to each score. 18 The Weighted Mean In this case the professor could assign each of the test scores a weight of 1 and the final exam score a weight of 2. A student with test scores of 65, 70, and 75 and a final examination score of 90 has a weighted mean of 19 The Weighted Mean 20 Example 4 – Find a Weighted Mean Table 13.1 shows Dillon’s fall semester course grades. Use the weighted mean formula to find Dillon’s GPA for the fall semester. Dillon’s Grades, Fall Semester Table 13.1 21 Example 4 – Solution The B is worth 3 points, with a weight of 4; the A is worth 4 points with a weight of 3; the D is worth 1 point, with a weight of 3; and the C is worth 2 points, with a weight of 4. The sum of all the weights is 4 + 3 + 3 + 4, or 14. Dillon’s GPA for the fall semester is 2.5. 22 ***End*** 23  CHAPTER 4.2 Statistics Copyright © Cengage Learning. All rights reserved. Section 4.2 Measures of Dispersion Copyright © Cengage Learning. All rights reserved. The Range 3 The Range To measure the spread or dispersion of data, we must introduce statistical values known as the range and the standard deviation. 4 Example 1 – Find a Range Range (R) The difference between the maximum and minimum value in a data set, i.e. R = MAX – MIN Example: Pulse rates of 15 male residents of a certain village 54 58 58 60 62 65 66 71 74 75 77 78 80 82 85 R = 85 - 54 = 31 5 The Standard Deviation 6 The Standard Deviation The range of a set of data is easy to compute, but it can be deceiving. The range is a measure that depends only on the two most extreme values, and as such it is very sensitive. A measure of dispersion that is less sensitive to extreme values is the standard deviation. The standard deviation of a set of numerical data makes use of the amount by which each individual data value deviates from the mean. These deviations, represented by , are positive when the data value x is greater than the mean and are negative when x is less than the mean. The sum of all the deviations is 0 for all sets of data. 7 The Standard Deviation 8 The Standard Deviation Because the sum of all the deviations of the data values from the mean is always 0, we cannot use the sum of the deviations as a measure of dispersion for a set of data. Instead, the standard deviation uses the sum of the squares of the deviations. 9 The Standard Deviation Most statistical applications involve a sample rather than a population, which is the complete set of data values. Sample standard deviations are designated by the lowercase letter s. In those cases in which we do work with a population, we designate the standard deviation of the population by , which is the lowercase Greek letter sigma. 10 The Standard Deviation We can use the following procedure to calculate the standard deviation of n numbers. 11 Example 2 – Find the Standard Deviation The following numbers were obtained by sampling a population. 2, 4, 7, 12, 15 Find the standard deviation of the sample. Solution: Step 1: The mean of the numbers is 12 Example 2 – Solution cont’d Step 2: For each number, calculate the deviation between the number and the mean. 13 Example 2 – Solution cont’d Step 3: Calculate the square of each of the deviations in Step 2, and find the sum of these squared deviations. 14 Example 2 – Solution cont’d Step 4: Because we have a sample of n = 5 values, divide the sum 118 by n – 1, which is 4. Step 5: The standard deviation of the sample is. To the nearest hundredth, the standard deviation is s = 5.43. 15 The Variance 16 The Variance A statistic known as the variance is also used as a measure of dispersion. The variance for a given set of data is the square of the standard deviation of the data. The following chart shows the mathematical notations that are used to denote standard deviations and variances. 17 Example 5 – Find the Variance Find the variance for the sample given earlier in Example 2. Solution: The standard deviation which we found in Example 2 is. The variance is the square of the standard deviation. Thus the variance is 18 Standard Deviation and Variance n n  (x i  x) 2  (x i  x )2 s i 1 s i 1 n 1 n 1 19 Comparing Standard Deviation Data A Mean = 15.5 11 12 13 14 15 16 17 18 19 20 21 s = 3.338 Data B Mean = 15.5 11 12 13 14 15 16 17 18 19 20 21 s =.9258 Data C Mean = 15.5 11 12 13 14 15 16 17 18 19 20 21 s = 4.57 20 Comparing Standard Deviation Example: Team B - Heights of five marathon players in inches Mean = 65” SD = 5.0” 60 “ 60 “ 65 “ 70 “ 70 “ 21 Comparing Standard Deviation Example: Team A - Heights of five marathon players in inches Mean = 65 S =0 65 “ 65 “ 65 “ 65 “ 65 “ 22 Standard Deviation and Variance 23 Standard Deviation and Variance Standard Variance Deviation Standard Variance 1 1 Deviation 6 ? 0 0 ? 49 5 25 8 ? 10 100 24 Measures of Central Tendency/ sja 10/3/2024 Measures of Central Tendency/ sja 10/3/2024 Measures of Central Tendency/ sja 10/3/2024 Measures of Central Tendency/ sja 10/3/2024 Measures of Central Tendency/ sja 10/3/2024 Measures of Central Tendency/ sja 10/3/2024 Measures of Central Tendency/ sja 10/3/2024  92 + 84 + 65 + 76 + 88 + 90 = 6 495 = 6 = 82.5 The mean of the test scores is 82.5 Measures of Central Tendency/ sja 10/3/2024 Measures of Central Tendency/ sja 10/3/2024 Measures of Central Tendency/ sja 10/3/2024 Measures of Central Tendency/ sja 10/3/2024 Measures of Central Tendency/ sja 10/3/2024 Measures of Central Tendency/ sja 10/3/2024 Measures of Central Tendency/ sja 10/3/2024 Measures of Central Tendency/ sja 10/3/2024 Measures of Central Tendency/ sja 10/3/2024 Measures of Central Tendency/ sja 10/3/2024 Measures of Central Tendency/ sja 10/3/2024  Dillon’s Grades, Fall Semester Table 13.1 Measures of Central Tendency/ sja 10/3/2024 Measures of Central Tendency/ sja 10/3/2024  - is a table that lists observed events and the frequency of occurrence of each observed event, is often used to organize raw data. Measures of Central Tendency/ sja 10/3/2024  Number of No. of Computers Households (x) (f) Number of Laptop Computers per 0 5 Household 1 12 2 0 3 1 2 1 0 4 2 1 1 7 2 0 1 1 2 14 0 2 2 1 3 2 2 1 3 3 1 4 2 5 2 3 1 2 4 2 2 1 2 2 5 0 2 5 5 3 6 0 Find the mean of the data. 7 1 σ(𝑥 ∙ 𝑓) N=40 𝑀𝑒𝑎𝑛 = σ𝑓 0 ∙ 5 + 1 ∙ 15 + 2 14 + 3 ∙ 3 + 4 ∙ 2 + 5 ∙ 3 + 6 ∙ 0 + (7 ∙ 1) = Measures of Central Tendency/ sja 40 10/3/2024  Thank You and God bless! Proverbs 3:5 Trust in the Lord with all your heart, and lean not to your own understanding. Measures of Central Tendency/ sja 10/3/2024 STATISTICS Chapter 4.2 Measures of Dispersion Range To measure the spread or dispersion of data, we must introduce statistical values known as the range and the standard deviation. The difference between the maximum and minimum value in a data set, i.e. R = MAX – MIN Range Example: Pulse rates of 15 male residents of a certain village 54 58 58 60 62 65 66 71 74 75 77 78 80 82 85 R = 85 - 54 = 31 The Standard Deviation The range of a set of data is easy to compute, but it can be deceiving. The range is a measure that depends only on the two most extreme values, and as such it is very sensitive. A measure of dispersion that is less sensitive to extreme values is the standard deviation. The standard deviation of a set of numerical data makes use of the amount by which each individual data value deviates from the mean. These deviations, represented by (𝑥 − 𝑥), ҧ are positive when the data value x is greater than the mean 𝑥ҧ and are negative when x is less than the mean 𝑥ҧ.The sum of all the deviations (𝑥 − 𝑥)ҧ is 0 for all sets of data. The Standard Deviation The Standard Deviation Because the sum of all the deviations of the data values from the mean is always 0, we cannot use the sum of the deviations as a measure of dispersion for a set of data. Instead, the standard deviation uses the sum of the squares of the deviations. The Standard Deviation Most statistical applications involve a sample rather than a population, which is the complete set of data values. Sample standard deviations are designated by the lowercase letter s. In those cases in which we do work with a population, we designate the standard deviation of the population by , which is the lowercase Greek letter sigma. The Standard Deviation We can use the following procedure to calculate the standard deviation of n numbers. The Standard Deviation The following numbers were obtained by sampling a population. 2, 4, 7, 12, 15 Find the standard deviation of the sample. Solution: Step 1: The mean of the numbers is The Standard Deviation Step 2: For each number, calculate the deviation between the number and the mean.  The Standard Deviation Step 3: Calculate the square of each of the deviations in Step 2, and find the sum of these squared deviations. The Standard Deviation Step 4: Because we have a sample of n = 5 values, divide the sum 118 by n – 1, which is 4. 118 = 29.5 4 Step 5: The standard deviation of the sample is 𝑠 = 29.5 To the nearest hundredth, the standard deviation is s = 5.43.  The Standard Deviation Example: A consumer group has tested a sample of 8 size-D batteries from each of 3 companies. The results of the tests are shown in the table below. According to these tests, which company produces batteries for which the values representing hours of constant use have smallest standard deviations? Company Hours of constant use per battery EverSoBright 6.2, 6.4, 7.1, 5.9, 8.3, 5.3, 7.5, 9.3 Dependable 6.8, 6.2, 7.2, 5.9, 7.0, 7.4, 7.3, 8.2 Beacon 6.1, 6.6, 7.3, 5.7, 7.1, 7.6, 7.1, 8.5  Variance A statistic known as the variance is also used as a measure of dispersion. The variance for a given set of data is the square of the standard deviation of the data. The following chart shows the mathematical notations that are used to denote standard deviations and variances.  Variance Find the variance for the sample given earlier in Example. Solution: The standard deviation which we found in Example is 𝑠 = 29.5 The variance is the square of the standard deviation. Thus the variance is 𝑠 2 = ( 29.5)2 = 29.5 Standard Deviation and Variance Comparing Standard Deviation Data A Mean = 15.5 11 12 13 14 15 16 17 18 19 20 21 s = 3.338 Data B Mean = 15.5 11 12 13 14 15 16 17 18 19 20 21 s =.9258 Data C Mean = 15.5 11 12 13 14 15 16 17 18 19 20 21 s = 4.57 Comparing Standard Deviation Example: Team B - Heights of five marathon players in inches Mean = 65” SD = 5.0” 60 “ 60 “ 65 “ 70 “ 70 “ Comparing Standard Deviation Example: Team A - Heights of five marathon players in inches Mean = 65 S =0 65 “ 65 “ 65 “ 65 “ 65 “ Standard Deviation and Variance Standard Deviation and Variance Standard Variance Standard Variance Deviation Deviation 1 1 6 ? 0 0 ? 49 5 25 8 ? 10 100 1. The fuel efficiency, in miles per gallon, of 10 small utility truck was measured. The results are recorded in the table below. Fuel efficiency (mpg) 22 25 23 27 15 24 24 32 23 22 25 22 Find the mean and the sample standard deviation of these data. Round to the nearest hundredth. 2. A customer at a specialty coffee shop observed the amount of time, in minutes, that each of 20 customers spent waiting to receive an order. The results are recorded in the table below. Time (min) to receive order 3.2 4.0 3.8 2.4 4.7 5.1 4.6 3.5 3.5 6.2 3.5 4.9 4.5 5.0 2.8 3.5 2.2 3.9 5.3 2.9 Find the mean and the sample standard deviation of these data. Round to the nearest hundredth. STATISTICS Measures of Relative Position Z-score The z-score for a given data value x is the number of standard deviations that is above or below the mean of the data. 𝑥−𝜇 𝑥−𝑥ҧ Population: 𝑧𝑥 = sample: 𝑧𝑥 = 𝜎 𝑠 Z-score 1. Raul has taken two tests in Chemistry class. He scored 72 on the first test, for which the mean of all scores was 65 and the standard deviation was 8. He received a 60 on the second test, for which the mean of all the scores was 45 and the standard deviation was 12. In comparison to the other student, did Raul do better on the 1st test or the second test? Solution: Find the z-score for each test. 72 − 65 60 − 45 𝑧72 = = 0.875 𝑧60 = = 1.25 8 12 Z-score 2. A consumer group tested a sample of 100 light bulbs. If found that the mean life expectancy of the bulbs was 842 h, with standard deviation of 90. One particular light bulb from the DuraBright Company had a z-score of 1.2. What was the life span of the light bulb? 𝑥 − 𝑥ҧ 108 = 𝑥 − 842 𝑧𝑥 = 𝑠 𝑥 = 950 𝑥 − 842 1.2 = 90 𝑇ℎ𝑒 𝑙𝑖𝑔ℎ𝑡𝑏𝑢𝑙𝑏 ℎ𝑎𝑠 𝑎 𝑙𝑖𝑓𝑒 𝑠𝑝𝑎𝑛 𝑜𝑓 950 ℎ Percentiles pth Percentile A value x is called the pth percentile of data set provided p% of the data values are less than x. Example: In a recent year, the median annual salary for physical therapist was $74,480. If the 90th percentile for the annual salary for a physical therapist was $105, 900, find the percent of physical therapists whose annual salary was a. more than $ 74,480 b. less than $105, 900 c. between $74,480 and $105,900. Percentiles Percentile for a Given Data Value Given a set of data and a data value x, 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 𝑥 𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒 𝑜𝑓 𝑠𝑐𝑜𝑟𝑒 𝑥 = ∙ 100 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠 Example: On a reading examinations given to 900 students, Elaine’s score of 602 was higher than the scores of 576 of the students who took the examinations. What is the percentile for Elaine’s score? 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 𝑥 𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒 𝑜𝑓 𝑠𝑐𝑜𝑟𝑒 𝑥 = ∙ 100 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 602 𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒 𝑜𝑓 𝑠𝑐𝑜𝑟𝑒 𝑥 = ∙ 100 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠 576 = ∙ 100 900 = 64 𝐸𝑙𝑎𝑖𝑛𝑒 ′ 𝑠 𝑠𝑐𝑜𝑟𝑒 𝑜𝑓 602 𝑝𝑙𝑎𝑐𝑒𝑠 ℎ𝑒𝑟 𝑎𝑡 𝑡ℎ𝑒 64𝑡ℎ 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒. Quartiles The three numbers 𝑄1 , 𝑄2 𝑎𝑛𝑑𝑄3 that partition a ranked data set into four (approximately) equal groups are called the quartiles The Median Procedure for Finding Quartiles 1. Rank the data. 2. Find the median of the data. This is the second quartile, Q2. 3. The first quartile, Q1 ,is the median of the data values less than Q2. The third quartile, Q2 , is the median of the data values greater that Q2. 2, 5, 5, 8, 11, 12, 19, 22, 23, 29, 31, 45, 83, 91, 104, 159, 181, 312, 354 𝑄1 𝑄2 𝑄3 Quartiles Example: The following table lists the calories per 100 milliliter of 25 popular sodas. Find the quartiles for the data. Calories, per 100mL, of selected sodas 43 37 42 40 53 62 36 32 50 49 26 53 73 48 45 39 45 48 40 56 41 36 58 42 39  Assignment 1. Roland received a score of 70 on a test for which the mean score is 65.5. He learned that the z- score of the test is 0.6 What is the standard deviation for the set of test scores? 2. The median annual salary for the police dispatcher in a large city was $44,528. If the 25th percentile for the annual salary of a police dispatcher was $32,761, find the percent of police dispatcher whose annual salaried were a. less than $44,528 b. more than $32761 c. between $32,761 and $44,528 3. The following table lists the weights, in ounces, of 15 avocados in random sample. Find the quartiles for the data. 12.4, 10.8, 14.2, 7.5, 10.2, 11.4, 12.6, 12.8, 13.1, 15.6, 9.8, 11.4, 12.2, 16.4, 14.5