Chapter 5 - Statistics PDF

MODULE MATHEMATICS IN THE MODERN WORLD CHAPTER 5: STATISTICS Objectives: a. Identify and differentiate Patterns in Nature. b. Understand the Fibonacci Sequence. c. Appreciate the beauty of Math...

MODULE MATHEMATICS IN THE MODERN WORLD CHAPTER 5: STATISTICS Objectives: a. Identify and differentiate Patterns in Nature. b. Understand the Fibonacci Sequence. c. Appreciate the beauty of Mathematics in terms of Patterns and Number in Nature and in the World. Lesson 1: Measures of Central Tendency Many problems in statistics are concerned with averages. The most common measures that attempt to locate the center of a set of data are: average or mean, median and mode. The mean is the part of the distribution around which the values balance. Symbol for mean: X , read as “x bar” The median provides the necessary information about the value of the middle position in the distribution. Symbol for median: X read as “x tilde” The mode is the score with the highest frequency. Symbol for mode: X̂ read as “x hat” Definition: The mean (commonly called the average) of a set of n numbers is the sum of all numbers divided by n. The median is the middle number in a set of data when arranged in decreasing order. When there are even numbers of elements, the median is the mean of two middle number. The mode is the number that occurs most often in a set of data. A set of data can have more than one mode. If all the numbers appear the same number of times, there is no mode for that data set. Page 1  MODULE MATHEMATICS IN THE MODERN WORLD A. The Mean The most widely used average, the arithmetic mean, is defined as the sum of the observations divided by the number of observations. Mean (Ungrouped data) X= X i N Example: A motorist records the time in it takes him to travel to work by car during the peak hour traffic for a 10-day period. The times (to the nearest minute) are as follows: 36,33, 28, 28,32, 29,33,34,32,33 Find the mean time it took him to get to work during the two weeks (ten working days). Solution: X= X i N 36 + 33 + 28 + 28 + 32 + 29 + 33 + 34 + 32 + 33 = 10 318 = 10 = 31.8 minutes Weighted Mean If a set of data is in the form of a frequency distribution in which an observation x occurs with frequency f, we may use the formula for the arithmetic mean as: X=  fX N Where: X = mean, f= frequency, X= score  fX = sum of the products of frequency and score Page 2  MODULE MATHEMATICS IN THE MODERN WORLD N= total frequency Example: Suppose a particular math curse is graded in the following manner: Assignments 10% Project 20% Midterm Exams 30% Final Exams 40% 100 % Jason obtained marks of 85%in assignments, 74% in project, 76% in midterm exams and 87% in his final examples. Find his weighted mean mark for math course. Solution: X=  fX N =  85(10) + 74(20) + 76(30) + 87(40) 100 = 80.9% Mean (Grouped data) X=  fX m N Where: X = mean f = frequency X m = class mark (average of lower interval and upper interval)  fX m = sum of the product of frequencies and class mark Page 3  MODULE MATHEMATICS IN THE MODERN WORLD N= total frequency Example: Shown below in Table are the sources of 50 junior students in an achievement test. Calculate the mean score. Class Interval Frequency (f) 95-99 1 90-94 9 85-89 8 80-84 14 75-79 11 70-74 5 65-69 2 Solution: Add two columns for the class mark ( X m ) and fX m in the given table. Find the summations of f and fX m. Class Interval Frequency Xm fX m (f) 95-99 1 97 97 90-94 9 92 828 85-89 8 87 696 80-84 14 82 1148 75-79 11 77 847 70-74 5 72 360 Page 4  MODULE MATHEMATICS IN THE MODERN WORLD 65-69 2 67 134 f = 50  fX m = 4110 Solve for the mean. X=  fX m = 4110 = 82.2 N 50 B. The Median The median is the value of the middle observation if the data are arranged in the form of an array. Thus, the median is the value of an array which divides it so that there an equal number of observations on either side of it. The median is often used when describing an educational and sociological data, such as ages, income, family size, etc. If there is an odd number (n) of observations, then the median is the value  n +1  of   th observation. If n is an even number, the median is usually defined  2  n  n +1  as the mean of the   th and   th observation. 2  2  Example 1: In a certain clinic, the waiting time (in minutes) for 11 randomly chosen out patients on a particular day are: 12,36,34,15,17,14,8, 40,16,36,17 Find the median waiting time. Solution: Arrange the data in order of magnitude. 8,12,14,15,16,17,17, 26,34,36, 40 Since n = 11 , there are observations. Therefore:  11 + 1  Median waiting time =   th observations  2  Page 5  MODULE MATHEMATICS IN THE MODERN WORLD = 6th observations = 17 minutes Example 2: A sample of 50 students was given an inventory test (based on a possible score of 0-5). The result sore as follows: Score 0 1 2 3 4 5 Frequency 5 9 12 16 6 2 Find the median score. Solution: Construct a cumulative frequency distribution. Score Frequency` Cum. Frequency 0 5 5 1 9 14 2 12 26 3 16 42 4 6 48 5 2 50  f = 50 There are f = 50 observations. The median is the mean of the 25th and 26th observations. Since the data is already in an array form, it can easily be seen that both 25th and 26th observations are 2. Hence, the median score is 2. The first step in the computation of the median of a grouped data is n to determine the class interval which contains the   th score. This can 2 be located under the column  cf of the cumulative frequency distribution. n The class interval that contains the   th score is called the median class 2 of the distribution. To calculate the median, we use the formula: Median (Grouped Data) N   2 − cfb  X = X LB +  i  f m    Where: Page 6  MODULE MATHEMATICS IN THE MODERN WORLD X = median X LB = the lower boundary or true lower limit of the median class N = total frequency cf = cumulative frequency before the median class f m = frequency of the median class i = size of the class interval Example: Calculate the median score of 50 junior students in an achievement test in Math given in the table below. Solution: Achievement test Results in Math of 50 Junior Students Class Interval Frequency (f) 95-99 1 90-94 9 85-89 8 80-84 14 75-79 11 70-74 5 65-69 2 Add the entitles in the column for  cf. Achievement test Results in Math of 50 Junior Students Class Interval Frequency (f)  cf 95-99 1 50 90-94 9 49 85-89 8 40 80-84 14 32 75-79 11 18 70-74 5 7 65-69 2 2 N=50 N  50  th score=   th score 2  2 = 25th score The class interval that contains the 25th score is 80-84. X LB = 79.5 cf b = 18 Page 7  MODULE MATHEMATICS IN THE MODERN WORLD f m = 14 i= 5 N   2 − cfb  X = X LB +  i  f m     50   2 − 18  X = 79.5 +  5  14    X = 79.5 + 2.5 X = 82.2 This means that 50 percent of the students got scores below 82. C. The Mode The mode is defined as the observation which occurs the most often in a set of data This is the observation which has the largest frequency. It is frequently used to determine those products which are in greatest demand. Example: Find the mode of the following scores: 14,17,17,17,18,18,19, 20, 21, 21, 23 Solution: By inspection, the mode is 17 since it occurs 3 times in the distribution. Note: A distribution can have one or more modes. Example: An ice cream parlor sells 6 flavors of ice cream The numbers for each type sold on a particular day are shown below. Flavors of Ice Frequency of Cream sale Cheese 16 Chocolate 12 Vanilla 22 Page 8  MODULE MATHEMATICS IN THE MODERN WORLD Macapuno 26 Strawberry 23 Fruit Salad 18 Determine the most popular flavor of ice cream for that day. Solution: The most popular flavor is that which is most frequently sold. The highest frequency is 26. Hence, the modal (most popular) flavor for that day is macapuno. In the computation of the mode given a frequency distribution, the first step is to get the modal class. The modal class is that class interval with the highest frequency. To compute for the mode, we use the formula: Mode (Grouped data)  d1  Xˆ = X LB +  i  d1 + d 2  Where: X LB = lower boundary of the modal class d1 = difference of the frequency of the modal class and the frequency preceding it. d 2 = difference of the frequency of the modal class and the frequency succeeding it. i = size of the class interval Example: Find the mode for the following grouped frequency distribution. Achievement test Results in Math of 50 Junior Students Class Interval Frequency (f) 95-99 1 90-94 9 85-89 8 80-84 14 75-79 11 70-74 5 65-69 2 Solution: The modals class is the class interval 80-84 since it has the highest frequency. Therefore, X LB = 79.5 d 2 = 14 − 8 = 6 Page 9 MODULE MATHEMATICS IN THE MODERN WORLD d1 = 14 − 11 = 3 i =5  d1  Xˆ = X LB +  i  d1 + d 2   3  Xˆ = 79.5 +  5  3+ 6  Xˆ = 79.5 + 1.67 Xˆ = 81.17 For more knowledge about Measures of Central Tendency, please check the link provided; http://onlinestatbook.com/2/summarizing_distributions/measures.html https://study.com/academy/lesson/central-tendency-measures-definition- examples.html REMEMBER Mean is the most widely used average, the arithmetic mean, is defined as the sum of the observations divided by the number of observations. Median is the value of the middle observation if the data are arranged in the form of an array. Mode is defined as the observation which occurs the most often in a set of data ACTIVITY: Choose 10 of your classmates and ask them if how much money left in their pocket. In your collected data, compute for the mean, median and mode. Page 10  MODULE MATHEMATICS IN THE MODERN WORLD Lesson 2: Measures of Dispersion The measure of central tendency is not in itself sufficient to adequately describe a set of data. In addition, a measure of dispersion (or spread) of data is also required. This measure describes the extent to which individual observations vary above and below the average. The need for a measure of dispersion is just as important as the average. A measure of dispersion gives an indication of the reliability of the average value. The most commonly measure of dispersion are: the range, the quartile deviation, the mean deviation, the variance and the standard deviation. A. The Range The easiest and the simplest way to determine measure of dispersion is the range. The range is simply defined as the difference of the highest score (H.S) and the lowest score (L.S). It shows the extreme scores of a set of data. When we talk of grouped data, the range can be calculated data by subtracting the lower boundary (L.B) of the lowest class interval from the upper boundary (U.B) of the highest class interval. That is, R = H.S − LS = U.B − LB Example 1: a. The range of the set of scores in 12,14,14,16,16 is 16 − 12 or 4. b. The range of the set of scores in 10,14,14,18, 25 is 25 − 10 or 15. Example 2: Find the range of the frequency distribution below. Class interval Frequency 38-39 1 36-37 3 34-35 3 32-33 3 30-31 6 28-29 6 26-27 8 24-25 6 Page 11  MODULE MATHEMATICS IN THE MODERN WORLD 22-23 10 20-21 14 Solution: Range = U.B − L.B Range = 39.5 − 19.5 Range = 20 B. The Quartile Deviation An extension of the median is the concept of quartiles which divide the data into four equal parts. Quartiles are often used with scores on aptitude test, examinations, and other testing stations. When the data is divided into four equal parts, the points of separation are: 1st quartile ( Q1 ). There are 25% of the observations below Q1 and 75% of the observation above Q1. 2nd quartile ( Q2 ). There are 50% of the observations below Q2 and 50% of the observations above Q2. The second quartile is also the median. 3rd quartile ( Q3 ). There are 75% of the observations below Q3 and 25% of the observations above Q3.  n +1 th If there are n observation in a set of data, then Q1 can be identified as the    4   3 ( n + 1)  th observation, and Q3 as the   observation.  4  Example: Mr. Basanez is interested in the amount of time it takes his bank tellers to service customers. One particular morning, her records the service times for 15 customers. The times (to the nearest minute) are given below. 6,9, 7,5,16,11,9, 7, 4,9, 7,11,10,8, 6 a. Find the median time b. Find Q1 and Q3 of the service times. Solution: Page 12  MODULE MATHEMATICS IN THE MODERN WORLD The number of observations is n=15. Arrange the data in array 4,5, 6, 6 7, 7, 7,8 9,9,9,10 11,11,16    Q1 Q2 Q3 :12th  n +1  th a. The median or Q2 =   observations  2  = 8th observation = 8 minutes  n +1  th b. The first quartile: Q1 =   observation  4  = 4thobservation = 6 minutes 3(n + 1) th c. The third quartile: Q3 = observation 4 = 12th observation = 10 minutes Q1 and Q3 Grouped data N   4 − cfb  Q1 = X LB +  i  f q1    Where: X LB = lower boundary of the Q1 class N = total frequency cf b = cumulative frequency before the Q1 class f q1 = frequency of the Q1 class i = size of the class interval Q1 and Q3 Grouped data  3N   4 − cfb  Q3 = X LB +  i  f q 3    Where: X LB = lower boundary of the Q3 class N = total frequency cf b = cumulative frequency before the Q3 class fq3 = frequency of the Q3 class i = size of the class interval Example: From the given frequency distribution table, compute for Q1 , Q2 and Q3. Page 13  MODULE MATHEMATICS IN THE MODERN WORLD Class interval f 28-32 3 23-27 8 18-22 15 13-17 12 8-12 5 3-7 2 Solution: a. The first step is to add the entitles in the column for  cf. Class interval f  cf 28-32 3 45 23-27 8 42 18-22 15 34 13-17 12 19 8-12 5 7 3-7 2 2 b. Calculate for Q1 : N 45 Q1class = = = 11.25 Class interval 13-17 4 4 X LB = 12.5 cf b = 7 f q1 = 12 i =5 N   4 − cfb  Q1 = X LB +  i  f q1     11.25 − 7  Q1 = 12.5 +  5  12  = 12.5+1.77 = 14.27 C. Calculate for Q2 : 2 N 90 Q2class = = = 22.5 Class interval 18-22 4 4 X LB = 17.5 cf b = 19 f q1 = 15 i =5 Page 14  MODULE MATHEMATICS IN THE MODERN WORLD  22.5 − 19  Q2 = 17.5 +  5  15  = 17.5+1.17 = 18.67 d. Calculate for Q3 : 3N 135 Q3class = = = 33.75 Class interval 18-22 4 4 X LB = 17.5 cf b = 19 f q1 = 15 i =5  33.75 − 19  Q3 = 17.5 +  5  15  = 17.5+4.92 = 22.42 Like the range, the quartile deviation is a measure of dispersion which is determined by the distance between two particular observations. The first step is to calculate the interquartile range. The interquartile range or (I.R) is a more reliable measures of variability. It is the difference of the 75th percentile or Q3 and the 25th percentile or Q1 , hence we can conclude that 50 percent of the distribution will be falling within the interquartile range, 25 percent will be falling below Q1 and 25 percent will be above Q3. Interquartile Deviation I.R = Q3 − Q1 The formula for finding the interquartile range shows the distance between Q3 and Q1. The value obtained half of this distance is called the quartile deviation or (Q.D) and the formula is given by: Quartile Deviation Q3 − Q1 Q.D = 2 Example: A farmer has his corn crop spread over 15 fields each of equal size. The output (in cubic meters) for each of the fields is 226,174,185, 203,193, 216,164, 228, 244, 208, 235, 200, 216,196,188 a. Find the range of the output. b. Find the quartile deviation of the outputs. Page 15  MODULE MATHEMATICS IN THE MODERN WORLD Solution: Arrange the data in an array: 164,174,185,188,193,196, 200, 202, 208, 216,, 216, 226, 228, 235, 244 a. The range of the output = 244 − 164 = 80 cubic meters b. The first quartile =4th observations =188 cubic meters The third quartile =12th observations =226 cubic meters The interquartile range = 226 − 1888 =38 cubic meters interquartile range 226 − 188 The quartile deviation = = 2 2 38 = 2 = 19 cubic meters C. The Mean Deviation A measure of dispersion which takes into account each observation is the mean deviation. This is more reliable than the range and the quartile deviation because each makes use of only two values in the distribution, namely: the two most extreme values in the range; and Q3 and Q1 in the quartile deviation. The formulas for the computation of the mean deviation will be shown. Ungrouped Distribution Mean Deviation (M.D) M.D =  X −X N Where: X = represents the scores of the distribution X = is the mean N = is the number of observations The formula tells us that we have to follow the following steps: 1. Calculate the mean of the data. 2. Add a column for X − X. 3. Subtract the mean of each score and record the differences. 4. Write down the absolute values of each of the differences. 5. Get the total of the score under the heading X − X. 6. Divide the sum obtained in Step 3 by N. Example: Find the mean deviation of the following ungrouped distribution. x 5 8 11 Page 16  MODULE MATHEMATICS IN THE MODERN WORLD Solution: a. Calculate the mean. X=  X = 24 = 8 N 8 b. Add the column for X − X. c. X X −X 5 3 8 0 11 3 d.  X −X =6 6 e. M.D = =2 3 Grouped Frequency Distribution: Mean Deviation (M.D) M.D = f X −X or M.D = f Xm − X N N Example: Find the mean deviation of the following distribution. X f 20 5 18 3 16 7 14 15 12 12 Page 17  MODULE MATHEMATICS IN THE MODERN WORLD 10 8 N= 50 Solution: a. Calculate the mean by using the formula X =  fX. This means we N are going to add the entitles in the column for fX. X f fX 20 5 100 18 3 54 16 7 112 14 15 210 12 12 144 10 8 80 N= 50  fX = 700 X=  fX N 700 = 50 = 14 b. Add two columns for X − X and f X − X. Page 18  MODULE MATHEMATICS IN THE MODERN WORLD X f X −X f X −X 20 5 6 30 18 3 4 12 16 7 2 14 14 15 0 0 12 12 2 24 10 8 4 32 N= 50 f X − X = 112 c. Divide f X − X by N. 112 M.D = = 2.24 50 D. The Variance and Standard Deviation The variance and the standard deviation are the most commonly used measures of variation of a set of data. The standard deviation allows us to immediately compare the spread of different sets of score and enables us also to interpret the scores of a given set of data. Like the mean deviation, the variance and standard deviation are based on the deviation of the individual observations about the arithmetic mean. Also, the more widely scattered the observations are about the mean, the larger the value of the standard deviation. The variance is defined as the quotient of the sum of the squared deviations from the mean divided by N-1 while the standard deviation is the square root of the variance. The formulas are given below Variance ( S 2 ) and Standard Deviation ( S ) ( X − X ) ( X − X ) 2 2 S 2 = and S= N −1 N −1 Page 19  MODULE MATHEMATICS IN THE MODERN WORLD These are formulas use the mean deviation method and tell us to follow the following steps: 1. Calculate the mean. 2. Get the difference of each score and the mean, then get the square of this difference. 3. Get the sum of the squared deviations in Step 2. ( X − X ) ( X − X ) 2 2 4. Substitute in the formulas: S 2 = and S = N −1 N −1 Example: Find the variance and standard deviation of the following distribution: X 5 8 11 Solution: a. Calculate the mean: X = 8 b. Add the column for ( X − X ) and sum up the scores. 2 X (X − X ) 2 5 9 8 0 11 9 ( X − X ) 2 = 18 ( X − X ) 2 c. Divide by 2 since N − 1 = 3 − 1 = 2.\ 18 s2 = =9 2 s= 9 =3 Page 20  MODULE MATHEMATICS IN THE MODERN WORLD The following are important points to remember regarding the calculations of the standard deviation. a. The standard deviation cannot be negative. b. The standard deviation of a s et of data is zero if and only if the observations are of equal value. c. As a rough guide, the standard deviation should have a value which is equal to approximately one-third of the range. d. The standard deviation cannot be more than the range of the data. e. If a constant k is added to each observation in a set of data, the standard deviation of the new det of data has the same value as the standard deviation of the original set of data. This method of computing the variance and standard deviation is called the raw score method. The formulas are given below. Variance Standard Deviation For Ungrouped Data N  X 2 − ( X ) N  X 2 − ( X ) 2 2 S = 2 S= N ( N − 1) N ( N − 1) For Ungrouped Frequency Distribution N  fX 2 − (  fX ) N  fX 2 − (  fX ) 2 2 S = 2 S= N ( N − 1) N ( N − 1) For Grouped Frequency Distribution N  fX m 2 − (  fX ) N  fX m 2 − (  fX ) 2 2 S = 2 m S= m N ( N − 1) N ( N − 1) Example: Find the variance and the standard deviation of the following distribution. Page 21  MODULE MATHEMATICS IN THE MODERN WORLD X 5 8 11 Solution: a. Get X. X 5 8 11  X =24 b. Add a column for X 2 , square all the scores and get their sum. X X2 5 25 8 64 11 121  X =24 X 2 = 210 c. Substitute  X = 24 =24 and  X 2 = 210 in the formula. N  X 2 − ( X ) 2 S2 = N ( N − 1) 3(210) − ( 24 ) 2 S = 2 3(3 − 1) 630 − 576 S2 = 6 54 S2 = 6 S2 = 9 Page 22 MODULE MATHEMATICS IN THE MODERN WORLD S= 9 S =3 For more knowledge about Measures of Dispersion, please check the link provided; https://study.com/academy/lesson/measures-of-dispersion-definition- equations-examples.html https://www.slideshare.net/BirinderSinghGulati/measures-of-dispersion- 111028342 REMEMBER The range is simply defined as the difference of the highest score (H.S) and the lowest score (L.S). A measure of dispersion which takes into account each observation is the mean deviation. The standard deviation allows us to immediately compare the spread of different sets of score and enables us also to interpret the scores of a given set of data. ACTIVITY: Calculate the: a. range b. quartile deviation c. mean deviation d. standard deviation e. variance for these data. 1,6,3,5,5,3,4,1,2,7,3,2,4 Page 23  MODULE MATHEMATICS IN THE MODERN WORLD Lesson 3: Measures of Relative Position A measure of position is a method by which the position that a particular data value has within a given data set can be identified. As with other types of measures, there is more than one approach to defining such a measure. Standard Score (z-score) The standard score (often called the z-score) for a given data value x is the number of standard deviations that x is above or below the mean of the data. The following formulas show how to calculate the z-score for a data value x in a population and sample. x− x−x population data z = and sample data z =  s To compute a standard score, only the mean and standard deviation are required. However, since both of those quantities do depend on every value in the data set, a small change in one data value will change every z-score. Example: Scores on a history test have an average of 80 with a standard deviation of 6. What is the z-score for a student who earned a 75 on the test? x− Solution: z=  75 − 80 z= 6 z = −0.833 Example: The weight of chocolate bars from a particular chocolate factory has a mean of 8 ounces with a standard deviation of.1 ounce. What is the z- score corresponding to a weight of 8.17 ounces? x− Solution: z =  8.17 − 8 z=.1 z = 1.7 Page 24  MODULE MATHEMATICS IN THE MODERN WORLD Percentiles A value x is called the pth percentile of a data set provided p% of the data values are less than x. Example: In a recent year, the median annual salary for a physical therapist was Php 74,480. If the 90th percentile for the annual salary of a physical therapist was Php 105,900; find the percent of physical therapists whose annual salary was a. More than Php 74,480 b. Less than Php 105,900 c. Between Php 74,480 and Php 105,900 Solution: a. By definition, the median is the 50th percentile. Therefore, 50% of the physical therapists earned more than Php 74,480 per year. b. Because Php 105,900 is the 90th percentile, 90% of all physical therapists earned less than Php 105,900. c. From parts a and b, 90%-50% = 40% 40% of the physical therapists earned between Php 74,480 and Php 105,900. 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 examination given to 900 students. Benedict’s score of 602 was higher than the scores of 576 of the students who took the examination. What is the percentile for Benedict’s score? Solution: number of data values less than 602 Percentile = 100 total number of data values Page 25  MODULE MATHEMATICS IN THE MODERN WORLD 576 Percentile = 100 900 = 64 Therefore, Benedict’s score of 602 places him at the 64th percentile. Quartile The three numbers Q1, Q2, and Q3 that partitions a ranked data into four equal groups are called quartiles. - 1st quartile ( Q1 ). There are 25% of the observations below Q1 and 75% of the observation above Q1. - 2nd quartile ( Q2 ). There are 50% of the observations below Q2 and 50% of the observations above Q2. The second quartile is also the median. - 3rd quartile ( Q3 ). There are 75% of the observations below Q3 and 25% of the observations above Q3. 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 Q3 is the median of the data values greater than Q2 Example: The following table lists the calories per 100 milliliters of 25 popular sodas. Find the quartiles for the data. Calories per 100 milliliters 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 Solution: Step 1: Rank the data as shown as the following table. Page 26  MODULE MATHEMATICS IN THE MODERN WORLD 1) 26 2) 32 3) 36 4) 36 5) 37 6) 39 7) 39 8) 40 9) 40 10) 41 11) 42 12) 42 13) 43 14) 45 15) 45 16) 48 17) 48 18) 49 19) 50 20) 53 21) 53 22) 56 23) 58 24) 62 25) 73 Step 2: The median of these 25 data values has a rank of 13. Thus, the median is 43. The second quartile Q2 is the median of the data, so Q2 = 43. Step 3: There are 12 data values less than the median and 12 data values greater than the median. The first quartile is the median of the data values less than the median. Thus, Q1 is the mean of the data values with rank of 6 and 7. 39 + 39 Q1 = = 39 2 The third quartile is the median of the data values greater than the median. Thus, Q3 is the mean of the data values with ranks of 19 and 20. 50 + 53 Q3 = = 51.5 2 Box-and-Whisker Plot It is sometimes called as box-plot. It is often used to provide a visual summary of a set of data. A box-and-whisker plot shows the median, the first and the third quartiles, and the minimum and maximum values of a data set. Page 27  MODULE MATHEMATICS IN THE MODERN WORLD Construction of a Box-and-Whisker Plot 1. Draw a horizontal scale that extends from minimum data value the maximum data value. 2. Above the scale, draw a rectangle (a box) with left side at Q1 and its right at Q3. 3. Draw a vertical line segment across the rectangle at the median, Q2. 4. Draw a horizontal line segment, called a whisker, that extends from Q1 to the minimum and another whisker that extends from Q3 to the maximum. Example: Construct a box plot for the following data: 12, 5, 22, 30, 7, 36, 14, 42, 15, 53, 25 Solution: Step 1: Arrange the data in ascending order. Step 2: Find the median, lower quartile and upper quartile Median (middle value) = 22 Lower quartile (middle value of the lower half) = 12 Upper quartile (middle value of the upper half) = 36 (If there is an even number of data items, then we need to get the average of the middle numbers.) Step 3: Draw a number line that will include the smallest and the largest data. Page 28  MODULE MATHEMATICS IN THE MODERN WORLD Step 4: Draw three vertical lines at the lower quartile (12), median (22) and the upper quartile (36), just above the number line. Step 5: Join the lines for the lower quartile and the upper quartile to form a box. Step 6: Draw a line from the smallest value (5) to the left side of the box and draw a line from the right side of the box to the biggest value (53). For more knowledge about Measures of Relative Position, please check the link provided; https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Book%3A_Int roductory_Statistics_(Shafer_and_Zhang)/02%3A_Descriptive_Statistics/2.04%3A_Rel ative_Position_of_Data https://www.slideshare.net/AbdulAleem95/measures-of-relative-position Page 29  MODULE MATHEMATICS IN THE MODERN WORLD REMEMBER The percentile rank and z-score of a measurement indicate its relative position with regard to the other measurements in a data set. The three quartiles divide a data set into fourths. The five-number summary and its associated box plot summarize the location and distribution of the data. ACTIVITY: a. Consider the data set 69 93 70 53 92 75 85 70 68 76 88 70 77 82 85 82 80 100 96 85 1. Find the percentile rank of 82. 2. Find the percentile rank of 68. b. Find the z-score of each measurement in the following sample data set. −5 6 2 −1 0 Page 30  MODULE MATHEMATICS IN THE MODERN WORLD Lesson 4: Normal Distribution The normal distribution forms a bell-shaped curve that is symmetric about a vertical line through the mean of the data. A graph of a normal distribution is shown below: Properties of Normal Distribution Every normal distribution has the following properties. ▪ The graph is symmetric about a vertical line through the mean of the distribution. ▪ The mean, median and mode are equal. ▪ The y-value of each point on the curve I the percent (expressed as a decimal) of the data at the corresponding x-value. ▪ Areas under the curve that are symmetric about the mean are equal. ▪ The total area under the curve is 1. Empirical Rule for a Normal Distribution In a normal distribution, approximately ▪ 68% of the data lie within 1 standard deviation of the mean. ▪ 95% of the data lie within 2 standard deviations of the mean. ▪ 99.7% of the data is within 3 standard deviations of the mean. Example: Page 31  MODULE MATHEMATICS IN THE MODERN WORLD The weights of adorable, fluffy kittens are normally distributed with a mean of 3.6 pounds and a standard deviation of 0.4 pounds. First, draw your Empirical curve with the 4 percentages! (Steps 1-3 are completed below.) What percent of adorable, fluffy kittens weigh between 2.8 and 4.8 pounds? Step 4: We need to shade the region they are asking for. Step 5: We need to add the percent in the shaded areas. 13.5% + 34% + 34% + 13.5% + 2.35% = 97.35% What percent of adorable, fluffy kittens weigh less than 2.4 pounds? Page 32  MODULE MATHEMATICS IN THE MODERN WORLD Step 4: We need to shade the region they are asking for. Step 5: We need to add the percents in the shaded areas. 0.15% Standard Normal Distribution A normal distribution with a mean of 0 (u=0) and a standard deviation of 1 (o= 1). The standard normal distribution (graph below) is a mathematical-or theoretical distribution that is frequently used by researchers to assess whether the distributions of the variables they are studying approximately follow a normal curve. Every score in a normally distributed data set has an equivalent score in the standard normal distribution. This means that the standard normal distribution can be used to calculate the exact percentage of scores between any two points on the normal curve. Page 33  MODULE MATHEMATICS IN THE MODERN WORLD Statisticians have worked out tables for the standard normal curve that give the percentage of scores between any two points. In order to be able to use this table, scores need to be converted into Z scores. For finding the area under the curve, the table is shown below: Page 34 MODULE MATHEMATICS IN THE MODERN WORLD Example: Percent of Population Between 0 and 0.45 Solution: Start at the row for 0.4, and read along until 0.45: there is the value 0.1736 And 0.1736 is 17.36% So 17.36% of the population are between 0 and 0.45 Standard Deviations from the Mean. Because the curve is symmetrical, the same table can be used for values going either direction, so a negative 0.45 also has an area of 0.1736 Example: Percent of Population Z Between -1 and 2 Solution: From −1 to 0 is the same as from 0 to +1: At the row for 1.0, first column 1.00, there is the value 0.3413 From 0 to +2 is: At the row for 2.0, first column 2.00, there is the value 0.4772 Add the two to get the total between -1 and 2: 0.3413 + 0.4772 = 0.8185 And 0.8185 is 81.85% So 81.85% of the population are between -1 and +2 Standard Deviations from the Mean. Page 35  MODULE MATHEMATICS IN THE MODERN WORLD Linear Regression Linear regression finds the line that best fits the data points. There are actually a number of different definitions of "best fit," and therefore a number of different methods of linear regression that fit somewhat different lines. By far the most common is "ordinary least-squares regression"; when someone just says "least-squares regression" or "linear regression" or "regression," they mean ordinary least-squares regression. Naming the Variables. There are many names for a regression’s dependent variable. It may be called an outcome variable, criterion variable, endogenous variable, or regressand. The independent variables can be called exogenous variables, predictor variables, or regressors. Three major uses for regression analysis are: (1) determining the strength of predictors (2) forecasting an effect, and (3) trend forecasting. First, the regression might be used to identify the strength of the effect that the independent variable(s) have on a dependent variable Second, it can be used to forecast effects or impact of changes. That is, the regression analysis helps us to understand how much the dependent variable changes with a change in one or more independent variables. Third, regression analysis predicts trends and future values. If the plot of n pairs of data (x , y) for an experiment appear to indicate a "linear relationship" between y and x, then the method of least squares may be used to write a linear relationship between x and y. The least squares regression line is the line that minimizes the sum of the squares ( d1 + d 2 + d 3 + d 4 ) of the vertical deviation from each data point to the line (see figure below as an example of 4 points). Figure 1. Linear regression where the sum of vertical distances d1 + d2 + d3 + d4 between observed and predicted (line and its equation) values is minimized. Page 36  MODULE MATHEMATICS IN THE MODERN WORLD The least square regression line for the set of n data points is given by the equation of a line in slope intercept form: y = ax + b where a and b are given by Figure 2. Formulas for the constants a and b included in the linear regression Example: Consider the following set of points: {(-2 , -1) , (1 , 1) , (3 , 2)} a) Find the least square regression line for the given data points. b) Plot the given points and the regression line in the same rectangular system of axes. Solution: a) Let us organize the data in a table. x y xy x2 -2 -1 2 4 1 1 1 1 3 2 6 9 x = 2 y = 2 xy = 9 x 2 = 14 Page 37  MODULE MATHEMATICS IN THE MODERN WORLD We now use the above formula to calculate a and b as follows a = ( nx y − xy ) / = ( (3)(9) − (2)(2) ) = 23 nx 2 − ( x ) ((3)(14) − (22 )) 2 38 1  1   23   5 b =   ( y − a x ) =    2 −   (2)  = n  3   38   19 b) We now graph the regression line given by y = a x + b and the given points. Graph of linear regression in problem 1. The Least-Squares Regression Line The Least-Square Regression Line for a set of bivariate data is the line that minimizes the sum of the squares of the vertical deviations from each data point to the line. Page 38  MODULE MATHEMATICS IN THE MODERN WORLD Imagine you have some points, and want to have a line that best fits them like this: We can place the line "by eye": try to have the line as close as possible to all points, and a similar number of points above and below the line. But for better accuracy let's see how to calculate the line using Least Squares Regression. The Line Our aim is to calculate the values m (slope) and b (y-intercept) in the equation of a line : y = mx + b Where: y = how far up x = how far along m = Slope or Gradient (how steep the line is) b = the Y Intercept (where the line crosses the Y axis) Steps To find the line of best fit for N points: Step 1: For each ( x, y ) point calculate x 2 and xy Step 2: Sum all x, y, x 2 and xy , which gives us x, y, x 2 and xy Step 3: Calculate Slope m : m = N  ( xy ) − x yN  ( x 2 ) − ( x ) 2 (N is the number of points.) Step 4: Calculate Intercept b : Page 39  MODULE MATHEMATICS IN THE MODERN WORLD b =y − m xN Step 5: Assemble the equation of a line y = mx + b Example: Sam found how many hours of sunshine vs how many ice creams were sold at the shop from Monday to Friday: "x" "y" Hours of Ice Creams Sunshine Sold 2 4 3 5 5 7 7 10 9 15 Let us find the best m (slope) and b (y-intercept) that suits that data y = mx + b Step 1: For each ( x, y ) , calculate x 2 and xy x y x2 xy 2 4 4 8 3 5 9 15 5 7 25 35 7 10 49 70 9 15 81 135 Page 40  MODULE MATHEMATICS IN THE MODERN WORLD Step 2: Sum all x, y, x 2 and xy , which gives us x, y, x 2 and xy x y x2 Xy 2 4 4 8 3 5 9 15 5 7 25 35 7 10 49 70 9 15 81 135 Σx: 26 Σy: 41 Σx2: 168 Σxy: 263 Also N (number of data values) = 5 Step 3: Calculate Slope m: m = N  ( xy ) − x yN  ( x 2 ) − ( x ) 2 m = 5 x 263 − 26 x 415 x 168 − 262 m =1315 − 1066840 − 676 m = 249164 m = 1.5183... Step 4: Calculate Intercept b: b = y − m xN b = 41 − 1.5183 x 265 b = 0.3049... Step 5: Assemble the equation of a line: y = mx + b y = 1.518 x + 0.305 Page 41  MODULE MATHEMATICS IN THE MODERN WORLD Let's see how it works out: x y y = 1.518x + 0.305 error 2 4 3.34 −0.66 3 5 4.86 −0.14 5 7 7.89 0.89 7 10 10.93 0.93 9 15 13.97 −1.03 Here are the (x,y) points and the line y = 1.518x + 0.305 on a graph: Sam hears the weather forecast which says "we expect 8 hours of sun tomorrow", so he uses the above equation to estimate that he will sell y = 1.518 x 8 + 0.305 = 12.45 Ice Creams Sam makes fresh waffle cone mixture for 14 ice creams just in case. Yum. Linear Correlation To determine the strength of a linear relationship between two variables, statisticians use a statistics called the linear correlation coefficient, which is denoted by the variable r and is defined as follow: Linear Correlation Coefficient For the n ordered pairs ( x1, y1 ),( x2, y2 ),( x3, y3 ),...,( xn, yn ) , the linear correlation coefficient r is given by n( xy) − ( x)( y) r= n(  x ) − (  x ) 2  n(  y 2 ) − (  y ) 2 2 Page 42  MODULE MATHEMATICS IN THE MODERN WORLD Properties of Linear Correlation Coefficient 1. The value of r lies between −1 and 1, inclusive. 2. The sign of r indicates the direction of the linear relationship between x and y: 1. If r r  0 then y tends to decrease as x is increased. 2. If r  0 then y tends to increase as x is increased. 3. The size of |r| indicates the strength of the linear relationship between x and y: 1. If |r| is near 1 (that is, if r is near either 1 or −1) then the linear relationship between x and y is strong. 2. If |r| is near 0 (that is, if r is near 0 and of either sign) then the linear relationship between x and y is weak. Example: Calculate the linear correlation coefficient for the following data. Page 43  MODULE MATHEMATICS IN THE MODERN WORLD X = 4, 8 ,12, 16 and Y = 5, 10, 15, 20. Solution: Given variables are, X = 4, 8 ,12, 16 and Y = 5, 10, 15, 20. For finding the linear coefficient of these data, we need to first construct a table for the required values. X y x2 y2 XY 4 5 16 25 20 8 10 64 100 80 12 15 144 225 180 16 20 256 400 320 Σx= Σ y =50 480 750 600 40 According to the formula of linear correlation we have, 400 r ( xy ) = 320  500 400 r ( xy ) = 17.89  22.36 400 r ( xy ) = 400 r ( xy ) = 1 Therefore, r ( xy) = 1 For more knowledge about Normal Distribution please check the link provided; https://statisticsbyjim.com/basics/normal-distribution/ https://study.com/academy/lesson/standard-normal-distribution-definition- example.html Page 44  MODULE MATHEMATICS IN THE MODERN WORLD REMEMBER The linear correlation coefficient measures the strength and direction of the linear relationship between two variables x and y. The sign of the linear correlation coefficient indicates the direction of the linear relationship between x and y. When r is near 1 or −1 the linear relationship is strong; when it is near 0 the linear relationship is weak. ACTIVITY: Solve the following problem. 1. X is a normally distributed variable with mean μ = 30 and standard deviation σ = 4. Find a) P(x < 40) b) P(x > 21) c) P(30 < x < 35) https://www.abs.gov.au/websitedbs/a3121120.nsf/home/statistical+language+- +measures+of+central+tendency https://www.toppr.com/guides/business-mathematics-and-statistics/measures-of-central- tendency-and-dispersion/measure-of-dispersion/ https://stattrek.com/descriptive-statistics/measures-of-position.aspx https://statisticsbyjim.com/basics/normal-distribution/ Page 45