S1 Mathematics in the Modern World.pdf

Mathematics in the Modern World MARINO, WILFREDO | 1st Semester | 24 - 25 ALY & CUI MATHEMATICS numeral system, which is presently the most common...

Mathematics in the Modern World MARINO, WILFREDO | 1st Semester | 24 - 25 ALY & CUI MATHEMATICS numeral system, which is presently the most common numeral system. ★It is a set of tools for problem solving, a language, a ★ The system was invented between the 1st and 4th way of thinking, and a study of patterns, among other centuries by Indian mathematicians. things. EX. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 … ★ Simply the study of numbers. ★ SIPNAYAN - ISIP SANAYAN PATTERN ★ CALCULATION ○ Solving for x ★ Anything that repeatedly occurs ★ APPLICATION ★ A Pattern is an organization that aids observers in ○ Figuring out for y forecasting potential observations or future events. NUMBERS - BILANG (SEQUENCE) The patterns can sometimes be modeled mathematically and they NUMERALS - NUMERO (SYMBOL) include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes. ROMAN NUMERAL SEQUENCE ★ Is an ordered list of numbers, called terms that may have repeated values. The arrangement of these terms is set by a definite rule. I 1 EXAMPLES: Example: Analyze the given sequence for its rule and identify the next three terms. V 5 a. 1,10,100,1000,... b. 2,5,9,14,20,... X 10 Solution: L 50 a. Power by 10. Following the rule, the 5th number= 10000, 6th number= 100000 and the7th number= 1000000. C 100 b. The sequence is obtained by adding the difference of two previous numbers to the last listed next number. Following the D 500 rule, (2 and 5) 3, (5 and 9) 4, (9 and 14) 5, and (14 and 20) M 1000 ARITHMETIC SEQUENCE ★ An arithmetic sequence is an ordered set of numbers that have a common difference between each consecutive term. ★ 3, 9, 15, 21, 27, the common difference is 6. An arithmetic sequence can be known as an arithmetic progression. Formula: 2𝑛𝑑 𝑡𝑒𝑟𝑚 − 1𝑠𝑡 𝑡𝑒𝑟𝑚 = 𝑐𝑜𝑚𝑚𝑜𝑛 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑 EX: -16, -11, -6, -1, 4…..10th? 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑 𝑎10 =− 16 + (10 − 1)5 𝑎10 =− 16 + (9)5 HINDU ARABIC 𝑎10 =− 16 + 45 𝑎10 = 29 ★ Is a positional base ten numeral system for representing integers; its extension to non-integers is the decimal EX: Arithmetic means between 14 and 86. 1 Mathematics in the Modern World MARINO, WILFREDO | 1st Semester | 24 - 25 ALY & CUI 14, _,_,_,_,_, 86 EX: -3,-15-,-75,...5th term? an=a1rn-1 GIVEN: a5=-3(5)5-1 𝑎𝑛 = 86 a5=-3(5)4 𝑎1 = 14 a5=-3(625) a5=-1875 𝑛 = 7 𝑑 =? 4, 16, 64,....7th term? SOLUTION: an=a1rn-1 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑 a7=4(4)7-1 86 = 14 + (7 − 1)𝑑 a7=4(4)6 86 = 14 + (6)𝑑 a7=4(4096) 86 − 14 = 6𝑑 a7=16384 72 6𝑑 = 6 6 1, _,_, 216 12 = 𝑑 GIVEN: an=216 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑 a1=1 𝑎2 = 14 + (2 − 1)12 n= 4 SOLUTION 𝑎2 = 14 + (1)12 an=a1rn-1 𝑎2 = 14 + 12 216=1r4-1 𝑎2 = 26 216=1r3 216=r3 6=r 𝑎3 = 14 + (3 − 1)12 a2=1(6)=6 𝑎3 = 14 + (2)12 a3=6(6)=36 𝑎3 = 38 a4=36(6)=216 If the 3rd term is 12 and the 6th term is 96 in a geometric 𝑎4 = 14 + (4 − 1)12 sequence. Find the common ratio (r) 𝑎4 = 14 + (3)12 𝑎4 = 14 + 36 _,_, 12, _, _, 96 an=a1rn-1 𝑎4 = 50 3−1 a3= 12 = 𝑎1𝑟 6−1 𝑎5 = 14 + (5 − 1)12 a6= 96 = 𝑎1𝑟 𝑎6 𝑎5 = 14 + (4)12 = 12 = 8 𝑎3 96 𝑎5 = 14 + 48 2−5 𝑟 3 8=𝑟 GEOMETRIC SEQUENCE 3 8=2 ★ is a sequence where each new term is obtained by multiplying the preceding term by a constant number. PASCAL’S TRIANGLE ○ The series of numbers 1, 2, 4, 8, 16... is an example of a geometric sequence, named after Blaise Pascal, a famous French Mathematician and Philosopher FORMULA: 2𝑛𝑑 𝑡𝑒𝑟𝑚 ÷ 1𝑠𝑡 𝑡𝑒𝑟𝑚 = 𝑐𝑜𝑚𝑚𝑜𝑛 𝑟𝑎𝑡𝑖𝑜 an=a1rn-1 2 Mathematics in the Modern World MARINO, WILFREDO | 1st Semester | 24 - 25 ALY & CUI ★ In algebra, a triangular arrangement of numbers that Leonardo Pisano Bigollo, “filius bonacci” means “the gives the coefficients in the expansion of any binomial son of bonacci”, introduced the series to Western expression, such as (x + y)n. European math around 1202. GOLDEN RATIO Is a number often encountered when taking the ratios of distances in simple geometric figures such as the pentagon, pentagram, decagram and dod ecahedron. ★ The symbol of the golden ratio is the Greek letter "phi" – Ф (uppercase letter) or φ (lowercase letter). ★ It is named after the Greek sculptor Phidias. ★ The Golden Ratio is also known as Divine Ratio or Divine Proportion. You will notice that the bigger Fibonacci numbers you use as a ratio, the closer you get to the approximate value of φ (1.61803398874989484820...) Example: 1. Given (x+y)^5 Pascal’s Triangle row 5: 1, 5, 10, 10, 5, 1 = x5 + x4y + x3y2 + x2y3 + xy4 + y5 Example: 2. Given (x+y)^5 EXAMPLES OF GOLDEN RATIO: Pascal’s Triangle row 5: 1, 5, 10, 10, 5, 1 "Mona Lisa" by Leonardo Da Vinci. = x5 + x4y + x3y2 + x2y3 + xy4 + y5 Parthenon. = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5 Snail shells. Hurricanes. 3. Given (2 - 3x)6, find the 5th term Seed heads. Pascal’s Triangle row 6: 1, 6, 15, 2, 15, 6, 1 Flower petals. 1 (2)6 (-3x)0 = 64 Pinecones. 6 (2)5 (-3x) 1 = 6 (32) (-3x) = -576x "The Last Supper" by Leonardo Da Vinci. 15(2)4 (-3x) 2= 15 (16) (9x2) = 2160x2 Module 2 20(2)3 (-3x) 3= 20(8) (-27x3) = - 43203 15(2)2 (-3x) 4= 15(4) (81x4) = 4860x4 Language is the system of words, signs and symbols 6(2)1 (-3x) 5= 6(2) (-243x5)= -2916x5 which people use to express ideas, thoughts and feelings 1(2) (-3x) 6= (1) (729x6) = 729x6 Mathematical Language is the system used to = 64-576x + 2160x2 - 43203 + 4860x4 -2916x5 + 729x6 communicate mathematical ideas. Four main actions attributed to problem solving and FIBONACCI SEQUENCE reasoning ★ The Fibonacci Sequence is the series of numbers: 1. Modeling and Formulating 2. Transforming and Manipulating 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,... 3. Inferring The next number is found by adding up the two numbers before it. 4. Communicating The first 8 fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34 𝐹9 = 𝐹8 + 𝐹7 Characteristics of Mathematical Language = 21 + 13 = 34 Mathematics is about ideas - relationships, quantities, processes, measurements, reasoning and so on. The Fibonacci sequence's precise origin is uncertain. 3 Mathematics in the Modern World MARINO, WILFREDO | 1st Semester | 24 - 25 ALY & CUI The use of language in mathematics differs from the language ex. 2xy < 3y of ordinary speech in three important ways. (Jamison 2000) A closed sentence, is a mathematical sentence that is known to be either true or false 1. First, mathematical language is non-temporal. There is no past, present and future in mathematics ex. 2(x + y) = 2x + 2y 2. Second, mathematical language is devoid of emotional Conventions in the Mathematical Language content 3. Third, mathematical language is precise. Context refers to the particular topics being studied and it is important to understand the context to understand Advantage of Mathematical Notation mathematical symbols. Convention is a technique used by mathematicians, Symbolic and Graphical is that it is highly compact and focused engineers, scientists in which each particular symbol has particular meaning. Addition Substraction Multiplication Division Four Basic Concepts Set is a well-defined collection of distinct objects. + - x* ÷ Functions are mathematical quantities that give unique outputs to particular inputs. Mathematical Expressions Mathematical expression consists of terms. The term Relations are correspondence between a first set of of a mathematical expression contains a number and a letter variables such that for some elements of the first set separated by at least one of the fundamental operations. variables, there correspond at least two elements of the second set of variables. In algebra, variables or letters are used to represent unknown quantities. In 2x + 5, x is a variable and is also called Binary Operations are rules for combining two values to literal coefficient while 2 is called numerical coefficient. produce a new value. Meanwhile, 5 in the same expression is called constant whose value is irreplaceable. Sets Mathematical expressions may be classified according to Two ways to describe a Set the number of terms as follows: 1. Roster/Tabular Method (ex. E = {a, e, i , o, u}) 1. Monomial contains one term only. 2. Rule/Descriptive Method (ex. E = {x/x is a collection of vowel Examples are 2x; 5y; -3m; 4n; numbers}) 2. Binomial contains two terms. Kinds of Sets Examples are 2x – 3y; 5x + 9y; -3m + 2n; 1. Empty/Null/Void Set is a set that contains no element Ex. C = {x/x is an integer less than 2 but greater than 1} 3. Trinomial contains three terms. Examples are 2a – 3b + 4c; 5x – 3y + 2z 2. Finite Set is a set that contains a countable number of elements. 4. Multinomial contains four or more terms. 3. Unit Set (Singleton) is a set that contains only one Examples are 2a - 3b + 4c - d; 5x -3y +2z + 4 element. Ex. E = {x/x is a whole number greater than 1 but less than 3} Mathematical Sentence Mathematical Sentence is a combination of two mathematical 4. Equal Sets are sets that contain exactly the same expressions using a comparison operator. elements. Ex. A = {0,1,2,3} B = { 2, 0, 1,3} Equation 5. Equivalent Sets are sets that contain the same number ex. 4x + 3 = 19 of elements. Inequality Ex. A = {a, b, c, d} B = { 2, 0, 1,3} ex. 15x - 5 < 3y 6. Joint Sets are sets that have at least one common element. An open sentence in math means that it uses variables, meaning Ex. A = {a, b, c, d} B = {d, e, f, g, h} that it is not known whether or not the mathematical sentence is true or false 7. Disjoint Sets are sets that contain no common element. 4 Mathematics in the Modern World MARINO, WILFREDO | 1st Semester | 24 - 25 ALY & CUI Ex. A = {a, b, c, d} B = { e, f, g, h} Functions Function is a relation in which, for every value of the first component of the ordered pair, there is exactly one value of the second component. Functions have three most important parts: Binary Operations.can be understood as a function f (x, y) that applies to two 1. Input elements of the same set S, such that the result will also be an 2. Relationship element of the set S. Examples of binary operations are the addition of integers, multiplication of whole numbers, etc. 3. Output Example 1. Let set X consists of four students and set Y consists of their favorite subjects, respectively: X = {Alyssa, Elijah, Steph, Shei} Y = {Chemistry, Math, Physics, Statistics} The result is a set of ordered pairs of the form (x, y), written as: {(Alyssa, Chemistry), (Elijah, Math), (Steph, Physics), (Shei, Statistics)} The Composition of Functions f with g is denoted by f ® g and is defined by the equation (f ® g) (x) = f (g(x)). The domain of the composition function f ® g is the set of all x such that: 1. x is in the domain of g and ; 2. g(x) is in the domain of f. Ex 1. Given f(x) = 4x -5 and g(x) = x2 + 4, find (a) (f 0 g) (x) and (b) (g 0 f) (x) Ex 2. Given f(x) = x -1 and g(x) = 2x2 + x - 3, find (a) (f 0 g) (x) and (b) (g 0 f) (x) Ex 3. Given f(x) = x2 + 3 and g(x) = x2 - 1, find (a) (f 0 g) (x) and (b) (g 0 f) (x) LINEAR FUNCTION Module 5 A linear function f is a constant function if f(x) = mx + b, where m = 0 and b is a real number, thus f(x) = b. While a linear Definition of Statistics function f is an identity function if f(x) = mx + b, where m = 1 or any real number and b = 0, thus f(x) = x. plural sense: numerical facts, e.g. CPI, peso-dollar exchange rate Relations singular sense: scientific discipline consisting of theory and methods for processing numerical information that one can use Relations are correspondence between a first set of variables such that for some elements of the first set variables, when making decisions in the face of uncertainty. there correspond at least two elements of the second set of variables. History of Statistics The term statistics came from the Latin phrase “ratio status” which means study of practical politics or the statesman’s art. In the middle of 18th century, the term statistik (a term due to Achenwall) was used, a German term defined as “the political science of several countries” 5 Mathematics in the Modern World MARINO, WILFREDO | 1st Semester | 24 - 25 ALY & CUI From statistik it became statistics defined as a statement in figures and facts of the present condition of a state. Inferential statistics methods concerned with the analysis of a subset of data Application of Statistics leading to predictions or inferences about the entire set of data. Diverse applications Example of Inferential Statistics “During the 20th Century statistical thinking and methodology A new milk formulation designed to improve the psychomotor have become the scientific framework for literally dozens of fields development of infants was tested on randomly selected infants. including education, agriculture, economics, biology, and medicine, and with increasing influence recently on the hard Based on the results, it was concluded that the new milk sciences such as astronomy, geology, and physics. In other formulation is effective in improving the psychomotor words, we have grown from a small obscure field into a big development of infants. obscure field.” – Brad Efron Comparing the effects of five kinds of fertilizers on the yield of a particular variety of corn Determining the income distribution of Filipino families Comparing the effectiveness of two diet programs Prediction of daily temperatures Evaluation of student performance Two Aims of Statistics Statistics aims to uncover structure in data, to explain variation… Descriptive Statistics includes all the techniques used in organizing, summarizing and presenting the data on hand while Key Definitions A universe is the collection of things or observational Inferential Statistics includes all the techniques used in units under consideration. analyzing the sample data that will lead to generalizations about a A variable is a characteristic observed or measured on population from which the sample was taken every unit of the universe. A population is the set of all possible values of the Areas of Statistics variable. Descriptive statistics Parameters are numerical measures that describe the methods concerned w/ collecting, describing, and population or universe of interest. Usually donated by analyzing a set of data without drawing conclusions (or Greek letters; μ (mu), σ (sigma), ρ (rho), λ (lambda), τ inferences) about a large group. (tau), θ (theta), α (alpha) and β (beta). Statistics are numerical measures of a sample. Example of Descriptive Statistics Parameter is a summary measure describing a specific Present the Philippine population by constructing a graph characteristic of the population while Statistic is a summary indicating the total number of Filipinos counted during the last census by age group and sex measure describing a specific characteristic of the sample Answer the following questions as briefly as possible. 1. Differentiate descriptive from inferential statistics. 2. Give specific application of statistics in the following fields: 2.1 Business & Accountancy 2.2 Computer Studies 2.3 Education 2.4 Social Sciences & Humanities 2.5 Agriculture 2.6 Literature & Fine Arts 6 Mathematics in the Modern World MARINO, WILFREDO | 1st Semester | 24 - 25 ALY & CUI 2.7 Technology & Livelihood Records 3. Look for any printed material and identify the statistics mentioned in the material and classify them as to whether it is Methods of Data Presentation descriptive or inferential statistics. Textual Tabular Graphical Identify whether the given situation belongs to the area of descriptive statistics or inferential statistics. 1. Synchronous vs Asynchronous Learning: Their Effects in the Teaching-Learning Process 1. Nominal 2. Average of a student in his 10 subjects Numbers or symbols used to classify 3. Statistics on COVID-19 cases in the world ○ Examples are sex, marital status, occupation, 4. Effect of music in reviewing for the exams nationality, etc 5. One wishes to find out which gives a better salary between 2. Ordinal scale companies in the rural areas or urban areas Accounts for order; no indication of distance between 6. Enrolment rate in tertiary private institutions positions. 7. Percentage of PUIs by municipality in the Province of Rizal ○ Examples are curriculum level, socio-economic 8. Impact of COVID 19 Pandemic in the life of tertiary students status, military ranks, Latin honors, etc 9. Average sales for the first quarter of 2020 3. Interval scale 10.Amount of time spent in studying vs success of passing Equal intervals; no absolute zero. ○ Examples are temperature, test scores, etc Classify the following variables as to qualitative or 4. Ratio scale quantitative. If quantitative, further tell if it is discrete or Has absolute zero. continuous data. Be able to state the scale each is measured. Examples are bank account, cellphone load, etc The ratio level of measurement has all the following properties: 1. breeds of dogs A. the numbers in the system are used to classify a 2. birth order (first, second, etc) person/object into distinct, non-overlapping and 3. monthly income exhaustive categories; 4. cellphone number B. the system arranges the categories according to 5. night differential of cashiers in a convenient store magnitude; 6. spot on a die C. the system has a fixed unit of measurement representing 7. jersey number of a basketball player a set size throughout the scale and 8. IQ test scores D. the system has an absolute zero. 9. Students classification (continuing, irregular, returning) Enumerate five (5) variables that you may think and classify 10.COVID 19 cases in a barangay each as to qualitative or quantitative data. If quantitative, state whether it is discrete or continuous data. State the level each variable is measured. Definition Measurement is the process of determining the value or label of the variable based on what has been observed. For example, we can measure the educational level of a person by using the International Standard Classification of Education designed by UNESCO: 0 pre-primary; 1 primary; 2 lower secondary; 3 upper secondary; 4 post secondary nontertiary; 5 1st stage tertiary; 6 2nd stage tertiary Methods of Data Collection Objective Method Subjective Method Use of Existing 7 Mathematics in the Modern World MARINO, WILFREDO | 1st Semester | 24 - 25 ALY & CUI Descriptive Statistics MEAN A descriptive statistic (in the count noun sense) is a summary statistic that quantitatively describes or summarizes features from a collection of information while descriptive statistics (in the mass noun sense) is the process of using and analyzing those statistics. Descriptive statistics is distinguished from inferential (or inductive statistics) by its aim to summarize a sample rather than use the data to learn about the population that the sample of data MEASURES OF LOCATION is thought to represent. This generally means that descriptive statistics, A Measure of Location summarizes a data set by giving a unlike inferential statistics, is not developed on “typical value” within the range of the data values that describes its the basis of probability theory and are frequently location relative to entire data set. non-parametric statistics. Even when a data analysis draws its main conclusions using Common Measures: inferential statistics, descriptive statistics are Minimum, Maximum generally also presented. Central Tendency For example, in papers reporting on human subjects, typically a Percentiles, Deciles, Quartiles table is included giving the overall sample size, sample sizes in important subgroups (e.g., for each treatment or exposure group), Maximum and Minimum and demographic or clinical characteristics such as the average age, the proportion of subjects of each sex, the proportion of Minimum is the smallest value in the data set, denoted as MIN. subjects with related co-morbidities, etc. Maximum is the largest value in the data set, denoted as MAX. Summary Measures Given the following set of data (age of graduate school students) obtained by random sampling 30, 42, 24, 54, 29, 24, 31 Min = 24 Max = 54 Measures of Central Tendency A single value that is used to identify the “center” of the data it is thought of as a typical value of the distribution most representative value of the data Mean, Median, Mode A Measure of Location summarizes a data by giving a Mean “typical value” within the range of the data values that describes its Most common measure of central tendencies location relative to entire data set. Also known as arithmetic average Common Measures: Minimum, MAximum POPULATION MEAN Central Tendency Percentiles, Deciles, Quartiles 𝑁 Σ 𝑖 =1𝑋1 𝑋1+𝑋2+...+𝑋𝑁 Maximum and Minimum µ= 𝑁 = 𝑁 MInimum is the smallest value in the data set denoted as MIN SAMPLE MEAN Maximum is the largest value in the data set, denoted as MAX Σ 𝑛 =1𝑥1 𝑥1+𝑥2+...+𝑥𝑛 𝑖 ○ A single value that is used to identify the ‘center 𝑥= = 𝑛 𝑛 of the data ○ It is thought of as typical value of distribution ○ Precise yet simple ○ Most representative value of the data 8 Mathematics in the Modern World MARINO, WILFREDO | 1st Semester | 24 - 25 ALY & CUI Mean Given the following set of data (age of graduate school students) Properties of a Mode obtained by random sampling can be used for qualitative as well as quantitative data may not be unique 30, 42, 24, 54, 29, 24, 31 not affected by extreme values 𝑛 = 7 can be computed for ungrouped and grouped data Σ 𝑥 = 30 + 42 + 24 + 54 + 29 + 24 + 31 = 234 Σ𝑥 234 𝑥= 𝑛 = 7 = 33. 43 LEVELS OF MEASUREMENT 1. NOMINAL - sex (1 = male, 2 = female) Σ = 𝑠𝑢𝑚𝑚𝑎𝑡𝑖𝑜𝑛 2. ORDINAL - rank 𝑥 = 𝑚𝑒𝑎𝑛 3. INTERVAL - no absolute zero 𝑥͂ = 𝑚𝑒𝑑𝑖𝑎𝑛 4. RATIO - absolute zero 𝑥 = 𝑚𝑜𝑑𝑒 PROPERTIES OF THE MEAN Mean, Median & Mode may not be an actual observation in the data set can be applied in at least interval level Use the mean when: easy to compute sampling stability is desired every observation contributes to the value of the mean other measures are to be computed subgroup means can be combined to come up with a group mean Use the median when: easily affected by extreme values the exact midpoint of the distribution is desired there are extreme observations Median Divides the observation into two equal parts Use the mode when: ○ If the number of observations is odd, the median when the “typical” value is desired is the middle number. when the dataset is measured on a nominal scale ○ If the number of observations is even, the median is the average of the 2 middle numbers. Sample median denoted as 𝑥͂ ORDINAL RANKING Population median is denoted as μ (curly bar sa taas μ) Given the following set of data Maria Liza Lexie 8, 14, 10, 15, 26, 28 Determine the median. A 99 (1) 90 (2) 80 (3) Arrange the data from lowest to highest 8, 10 , 14, 15, 26, 28 B 90 (2) 93 (1) 85 (3) Since, the number of observation is even, then 14 + 15 C 88 (3) 92 (2) 98 (1) Properties of a Median MEAN 92 91 98 may not be an actual observation in the data set can be applied in at least ordinal level ORDINAL 6 5 (WINNER) 7 a positional measure; not affected by extreme values MODE occurs most frequently Percentiles nominal average Numerical measures that give the relative position of a may or may not exist data value relative to the entire data set. Given the following set of data (age of graduate school students) Divide an array (raw data arranged in increasing or obtained by random sampling decreasing order of magnitude) into 100 equal parts. 30, 30, 42, 24, 54, 29, 24, 31 The jth percentile, denoted as Pj, is the data value in the 24 and 30 are the most frequent data set that separates the bottom j% of the data from the top (100-j)%. 𝑥 = 24 𝑎𝑛𝑑 30 EXAMPLE 9 Mathematics in the Modern World MARINO, WILFREDO | 1st Semester | 24 - 25 ALY & CUI Position Value = 1.4 Suppose Vinh was told that relative to the other scores on a Decimal = 0.4 certain test, his score was the 95th percentile. LV = 8 UV = 10 This means that 95% of those who took the test had scores less than or equal to Vinh’s score, while 5% had scores higher than Step 3: Use interpolation method Vinh’s. 𝐿𝑗 = 𝐿𝑜𝑤𝑒𝑟 𝑉𝑎𝑙𝑢𝑒 + (𝐷𝑒𝑐𝑖𝑚𝑎𝑙) (𝑈𝑝𝑝𝑒𝑟 𝑉𝑎𝑙𝑢𝑒 − 𝐿𝑜𝑤𝑒𝑟 𝑉𝑎𝑙𝑢𝑒) Deciles Divide an array into ten equal parts, each part having ten 𝐿𝑗 = 𝐿𝑉 + (𝐷) (𝑈𝑉 − 𝐿𝑉) percent of the distribution of the data values, denoted by Dj. 𝐿20 = 8 + (0. 4) (10 − 8) The 1st decile is the 10th percentile; the 2nd decile is the 𝐿20 = 8 + (0. 4) (2) 20th percentile….. 𝐿20 = 8 + 0. 8 Quartiles 𝐿20 = 8. 8 Divide an array into four equal parts, each part having 25% of the distribution of the data values, denoted by Qj. Example: DECILE The 1st quartile is the 25th percentile; the 2nd quartile is Given the following set of data the 50th percentile, also the median and the 3rd quartile 6, 8, 14, 10, 15, 26, 29, 30 is the 75th percentile. Find 𝐷7 Steps in computing Percentile, Decile and Quartile N=8 Step 1: Arrange the data from lowest to highest 𝑗(𝑛+1) Step 2: Determine the position of the lower and upper value 𝐷𝑗 = 10 7(8+1) 𝑗(𝑛+1) 𝑗(𝑛+1) 𝑗(𝑛+1) 𝐷7 = 10 𝑃𝑗 = 100 , 𝐷𝑗 = 10 , 𝑄𝑗 = 4 7(9) 𝐷7 = 10 Step 3: Solve using interpolation 63 𝐷7 = 10 𝐿𝑗 = 𝐿𝑜𝑤𝑒𝑟 𝑉𝑎𝑙𝑢𝑒 + (𝐷𝑒𝑐𝑖𝑚𝑎𝑙) (𝑈𝑝𝑝𝑒𝑟 𝑉𝑎𝑙𝑢𝑒 − 𝐿𝑜𝑤𝑒𝑟 𝑉𝑎𝑙𝑢𝑒) 𝐷7 = 6. 3 (is between 6 and 7) Lower Value = 6th = 26 Example: PERCENTILE Upper Value = 7th = 29 Given the following set of data 8, 14, 10, 15, 26, 28 Position Value = 6.3 Find 𝑃20 Decimal = 0.3 LV = 26 Step 1: Arrange the data from lowest to highest UV = 29 8, 10, 14, 15, 26, 28 𝐿𝑗 = 𝐿𝑉 + (𝐷) (𝑈𝑉 − 𝐿𝑉) N=6 𝐿7 = 26 + (0. 3) (29 − 26) Step 2: Determine the position of the lower and 𝐿20 = 26 + (0. 3) (3) upper value of 𝑃20 𝐿20 = 26 + 0. 9 𝑗(𝑛+1) 𝐿20 = 26. 9 𝑃𝑗 = 100 20(6+1) 𝑃20 = 100 Percentile 100 𝑃1 𝑃21 … 𝑃99 20(6+1) Decile 10 𝐷1 𝐷 2 … 𝐷9 𝑃20 = 100 Quartile 4 𝑄1 𝑄2 𝑄3 20(7) 140 𝑃20 = 100 = 𝑃20 = 100 = 𝑃20 = 1. 4 (is between 1 and 2) Lower Value = 1st = 8 Upper Value = 2nd = 10 10