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Additional Mathematics Form 4 Topic: 7 DECISIVE (Version 2012) by NgKL (M.Ed.,B.Sc.Hons.,Dip.Ed.,Dip.Edu.Mgt.&Lship,Cert.NPQH.) edmet-nklpunya.blogspot.com 2 7.1 MEASURES OF CENTRAL TENDENCY IMPORTANT POINTS: Ungrouped Data Ungrouped Data Grouped Data (in a Frequency Table) Data sets which are not grouped into Data sets which are not grouped Data sets which are grouped into classes. into classes but are presented in classes and presented in Frequency Frequency Table. Table. Example: Example: Example: Number of Number of Number of Number of The masses of six pupils in kilogram: Books Read Students Books Read Students 50, 52, 59, 60, 53, 59. 0 5 0–1 11 1 6 2-3 12 2 8 4-5 15 3 4 6-7 8 4 2 8-9 7 Modal Class = The class with Mode = The value which is repeated Mode = The value of data which highest frequency. the most number of times in a set of has the highest frequency. Mode is obtained from the highest data. bar of a histogram with the procedure as shown below. Example: frequency Modal class Mode of the masses of six pupils in kilogram:; 50, 52, 59, 60, 53, 59. Mode = 59 mode Mean, x = _ x _ Mean, x = fx Mean, x = _ fx N f f x = sum of all the values of x = value of data x = class mid-point data. f = frequency f = frequency N = number of values of data. Median, m = the value in the middle Median, m = the value in the position of a set of data after the data middle position of a set of data N are arranged in ascending order. after the data are arranged in F Median, m = Lm + 2 c fm ascending order. Example: The median of the masses of six pupils in kilogram: Lm = lower boundary of the median class. 50, 52, 53, 59, 59, 60. N = sum of frequency. F = cumulative frequency of the 53 59 class before the median class. Median = 56 fm = frequency of the median class 2 c = size of the median class. edmet-nklpunya.blogspot.com 3 EFFECTS OF UNIFORM CHANGES IN A SET OF DATA ON THE MODE, MEAN AND MEDIAN: 1. When a constant number k is added or subtracted to each data in a set, then * the new mode = original mode k * the new mean = original mean k * the new median = original median k 2. When a constant number k is multiplied to each data in a set, then * the new mode = k x original mode. * the new mean = k x original mean. * the new median = k x original median. Exercise 7.1 1. Find the mode, mean and median of the following sets of ungrouped data. (a) 9, 5, 3, 3, 7, 13, 9 (b) 2, 8, 11, 9, 6, 5, 12, 11 (c) 3, 4, 11, 3, 10, 11, 2, 3, 7 (d) −3, −2, 1, 4, 5, 9 2. Find the mode, mean and median of the following sets of ungrouped data. Total Pocket money (RM), x 25 30 35 40 45 50 mode = ∑ (a) Number of Students, f 2 4 4 6 5 5 mean, x = Cumulative frequency, F fx median = (b) Marks, x No. Pupils, f fx F (c) No. Goals, x No. Players, f fx F 0 3 3 12 1 8 4 10 2 6 5 9 3 4 4 3 6 7 5 1 7 5 ∑f ∑fx= mode= mean, x = median= mode= mean, x = median= edmet-nklpunya.blogspot.com 4 No. of Score, x No. pupils, f fx F Marks, x fx F (d) (e) Sudents, f 8 4 13 6 9 8 14 8 12 11 15 12 15 10 16 10 20 5 17 5 21 2 18 3 mode= mean, x = median= mode= mean, x = median= 3. Determine (i) the modal class of each of the following grouped data. Height / cm, No.pupils, (a) LB UB x f (i) Modal class = 141 – 145 7 (ii) On a graph paper and by using a scale of 146 – 150 9 2 cm to 5 unit on x-axis and 2 cm to 2 units on frequency, f axis, draw a histogram of the 151 – 155 16 grouped data. Hence, from the graph, estimate the mode of the data. 156 – 160 6 161 – 165 2 edmet-nklpunya.blogspot.com 5 (a) Marks, x No.pupils, f LB UB (i) Modal class = 20 – 29 2 (ii) On a graph paper and by using a scale of 30 – 39 5 2 cm to 5 unit on x-axis and 2 cm to 2 units on frequency, f axis, draw a histogram of the 40 – 49 7 grouped data. Hence, from the graph, estimate the mode of the data. 50 – 59 10 60 – 69 6 4. Find the mean of each grouped data of the following. Height / cm, No. pupils, f Mid-point, x fx (a) 141 – 145 7 146 – 150 9 151 – 155 16 156 – 160 6 161 – 165 2 ∑f = ∑fx= edmet-nklpunya.blogspot.com 6 Number of Marks x fx (b) pupils, f 20 -29 2 30 – 39 4 40 – 49 5 50 – 59 10 60 − 69 6 70 − 79 3 ∑f = ∑fx= No. of Mass / kg (c) pupils, f 30 – 39 8 40 – 49 10 50 – 59 7 60 – 69 15 70 − 79 10 (d) The table below shows the duration of telephone calls received in an office on a certain day for 40 calls. Determine the mean of the duration of calls. Duration of No. of Calls / minutes Calls, f 1–3 2 4–6 4 7 – 10 5 11 – 13 10 14 – 17 6 5. For each of the following sets of data, without drawing an ogive, calculate the median of the set of data. Number of (a) Height / cm, pupils, f 141 – 145 7 146 – 150 9 151 – 155 16 156 – 160 6 161 – 165 2 edmet-nklpunya.blogspot.com 7 No. of (b) Marks, x pupils, f 20 – 29 2 30 – 39 4 40 – 49 5 50 – 59 10 60 – 69 6 70 − 799 3 Mass / Number of (c) kg pupils 30 – 39 8 40 – 49 10 50 – 59 8 60 – 69 14 70 − 79 10 (d) The table below shows the duration of telephone calls received in an office on a certain day for 40 calls. Without drawing an ogive, determine the median of the duration of calls. Duration Number of Calls / of Calls min 2–3 9 4–5 12 6–7 10 8–9 7 10 – 11 2 edmet-nklpunya.blogspot.com 8 7.2 OGIVE An ogive is a statistical graph which is drawn of cumulative frequency of a set of grouped data against its frequency class of upper boundary. An ogive can be used to estimate the median, m, first quartile, Q1 and third quartile, Q3 of the grouped data. Cumulative frequency, F 3N N = Sum of frequency 4 N Q1 = First quartile 2 m = Median N Q3 = Third quartile 4 Upper boundary Q1m Q3 To draw an ogive, a Cumulative Frequency & Upper Boundary table has to be built. A class with zero frequency and its upper boundary also need to be created. Example: Frequency, Cumulative frequency, Mass / kg Upper boundary f F 20 – 29 0 0 29.5 30 – 39 8 8 39.5 40 – 49 10 18 49.5 50 – 59 8 26 59.5 60 – 69 14 40 69.5 70 – 79 10 50 79.5 A graph is then plotted with its cumulative frequency against upper boundary to give an ogive. edmet-nklpunya.blogspot.com 9 Exercise 7.2 1. The table below shows marks scored by 30 pupils in a test. Draw an ogive, hence determine the median, m, first quartile, Q1, and third quartile, Q3 of the test. Number of Marks F UB pupils, f 20 – 29 2 30 – 39 4 40 – 49 5 50 – 59 10 60 – 69 6 70 – 79 3 Exercise 7.3 – Effect of Uniform Chances in a Set of Data on the Mode, Mean and Median 1. The mode, mean and median of a set of numbers are 6, 8.5 and 7.8 respectively. Determine the new mode, mean and median if each of the numbers in the set is; (i) added by 3 and then divided by 2. (ii) subtracted by 5 and then multiplied by 4. edmet-nklpunya.blogspot.com 10 2. The mode, mean and median of a set of data are 32.5, 30 and 31.5 respectively. Find the new mode, mean and median if each value in the data is; (i) added by 3 and then multiplied by ½., (ii) subtracted by 1.2. 3. A set of data with 6 numbers has a mean of 21. When a new number is added to the set, the mean becomes 20. Find the value of the number added. 7.3 MEASURE OF DISPERSION Ungrouped Data Ungrouped Data Grouped Data (in a Frequency Table) Range = midpoint of the higest Range = largest value – Range = largest value – class – midpoint of smallest value of data. smallest value of data. the lowest class. Inter quartile range Inter quartile range Inter quartile range = Q3 − Q1 = Q3 − Q1 = Q3 − Q1 Variance, Variance, x Variance, 2 fx _ 2 _ 2 = − x2 fx 2 _ = 2 x2 N = x f 2 2 where; f x 2 where; = sum of square of the where; f = frequency. values of data. f = frequency. x = class midpoint. N = number of value of data x = value of data. x = mean x = mean x = mean Standard deviation, Standard deviation, Standard deviation, = x 2 x _ 2 = fx 2 x _ 2 = fx 2 x _ 2 N f f edmet-nklpunya.blogspot.com 11 Effects of uniform changes in a set of data on the range, inter quartile range, variance and standard deviation. 1. When a constant number k is added or subtracted to each data in a set, then * the new range, interquartile range, variance and standard deviation = original range range, interquartile range, variance and standard deviation respectively. 2. When a constant number k is multiplied to each data in a set, then * the new range = k x original range. * the new interquartile range = k x original interquartile range.. * the new variance = k2 x original variance. * the new standard deviation = k x original standard deviation. Exercise 7.3(a) 1. Find the range and inter quartile range of each set of the following data. (a) 46, 35, 41, 40, 32, 38, 44, 40 (b) 17, 4, 6, 10, 12, 12 2. Find the range and inter quarter range of each of the following data. (c ) 22, 20, 25, 19, 24 (b) 3, 12, 8, 4, 10, 6, 7 3. Find the range and inter quartile range of each set of the following data. No. of (a) Score F Pupils, f 1 3 2 6 3 12 4 20 5 18 6 11 edmet-nklpunya.blogspot.com 12 No. of No. of (b) book pupils 0 10 1 14 2 20 3 26 4 18 5 12 (c) Mass / No. of kg pupils 50 2 51 3 52 10 53 20 54 8 55 7 5. The table below shows the number of chicken sold over a period of 60 days. No. of chickens, No. of days, x f 11 – 15 11 16 – 20 16 21 – 25 19 26 – 30 8 31 − 35 6 (a) Find the range of incomes of the workers. (b) Calculate the first quartile, Q1,, the third quartile, Q3 and the inter quartile range. (c) Draw an ogive, hence determine the first quartile, Q1,,third quartile, Q3 and the inter quartile range from the ogive. edmet-nklpunya.blogspot.com 13 Exercise 7.3(b): 1. Find the mean, variance and standard deviation of each set of the following data. (a) 9, 5, 3, 3, 7, 13, 9 (b) 2, 8, 11, 9, 6, 5, 12, 11 (c) 3, 4, 11, 3, 10, 11, 2, 3, 7 2. Find the mean, variance and standard deviation of each of the following data. Score, No. of (a) x pupils, f 1 3 2 6 3 12 4 20 5 18 6 11 No. of No. of (b) book pupils 0 10 1 14 2 20 3 26 4 18 5 12 edmet-nklpunya.blogspot.com 14 (c) Mass / No. of kg pupils 50 2 51 3 52 10 53 20 54 8 55 7 No. No. of (d) of family children 0 1 1 2 2 8 3 2 4 1 5 1 Exercise 7.3(c): 1. The table below shows the duration of telephone calls received in an office on a certain day for 40 calls. Find the mean, variance and standard deviation of the duration of calls. Duration of Number Calls / min of Calls 2–3 9 4–5 12 6–7 10 8–9 7 10– 11 2 edmet-nklpunya.blogspot.com 15 2. The table below shows marks scored by 30 pupils in a test. Find the mean, variance and standard deviation of the test. Number of Marks pupils 20 – 29 2 30 – 39 4 40 – 49 5 50 – 59 10 60 - 69 6 70 - 79 3 4. The table below shows the lengths of 60 mature long beans in a field study. Find the mean, variance and standard deviation of the lengths of the beans. Number Length / cm of Beans 10 – 14 8 15 – 19 15 20 – 24 19 25 – 29 13 30– 34 5 Exercise 7.3(d) – Effect of Uniform Chances in a Set of Data on the Measures of Dispersions 1. The range and the variance of a set of data are 12 and 13 respectively. Each value in the set of data is multiplied by 3 and then subtracted by 5. Find (a) the new range, (b) the new variance 2. A set of data has a range of 30, an inter quartile range of 5 and a standard deviation of 8. Each value in the set of the data is divided by 4 and then added by 3. Find (a) the new range, (b) the new inter quartile range, (c) the new standard deviation. edmet-nklpunya.blogspot.com 16 Exercise 7.4: Problem Solving I 1. Given the mode and the mean of the following set of data, 9, p, 14, q, 33, q are 33 and 20 respectively. Determine the values of p and q. 2. The median of the set data 4, 5, 6, 8, k, 9, is 7. Determine the value of k. 3. A set of data has seven numbers. Its mean is 9. If a number p is added to the set, the new mean is 12. What is the possible value of p? 4. A set of data x1, x2, x3, x4, x5 has a mean of 10 and a variance of 4. A value of x 6 is added to the set of data, the mean remains unchanged. Determine (a) the value of x6, (b) the variance of the new set of data. 5. A set of data consists of 6 numbers. The sum of the numbers is 39 and the sum of the squares is 271. (a) Find the mean and variance of the set of data. (b) If a number 5 is taken out from the set of data, find the new mean and standard deviation of the new data. edmet-nklpunya.blogspot.com 17 Past SPM Papers 1. The diagram below is a histogram which represents the distribution of the marks obtained by 40 pupils in a test. Number of Pupils 14 12 10 8 6 4 2 0 Marks 0.5 10.5 20.5 30.5 40.5 50.5 (a) Without using an ogive, calculate the median mark. [3 marks] (b) Calculate the standard deviation of the distribution. [4 marks] (SPM 2005/SectionA/Paper 2) edmet-nklpunya.blogspot.com 18 2. A set of data consists of 10 numbers. The sum of the numbers is 150 and the sum of the squares of the numbers is 2472. (a) Find the mean and variance of the 10 numbers. [3 marks] (b) Another number is added to the set of data and the mean is increased by 1. Find (i) the value of this number, (ii) the standard deviation of the set of 11 numbers. [4 marks] (SPM 2004/SectionA/Paper 2) 3. A set of examination marks x1, x2, x3, x4, x5, x6 has a mean of 5 and a standard deviation of 1.5. (a) Find (i) the sum of the marks, x, (ii) the sum of the squares of the marks, x2. [3 marks] (b) Each mark is multiplied by 2 then 3 is added to it. Find, for the new set of marks, (i) the mean, (ii) the variance. [4 marks] (SPM 2003/Section A/Paper 2) edmet-nklpunya.blogspot.com 19 4. The positive integers consists of 2, 5 and m. The variance for this set of integers is 14. Find the value of m. [4 marks] (SPM 2006/Paper 1) 5. A set of data consists of five numbers. The sum of the numbers is 60 and the sum of the squares of the numbers is 800. Find, for the five numbers (a) the mean, (b) the standard deviation. [3 marks] (SPM2007/Paper 1) 6. Table 1 shows the cumulative frequency distribution for the scores of 32 students in a competition. Score < 10 < 20 < 30 < 40 < 50 Number of students 4 10 20 28 32 Table 1 (a) Based on Table 1, copy and complete Table 2. Score 0–9 10 – 19 20 – 29 30 – 39 40 – 49 Number of students Table 2 (b) Without drawing an ogive, find the interquartile range of the distribution. [5 marks] (SPM2007/Section A/Paper 2) edmet-nklpunya.blogspot.com 20 7. Table 1 shows the frequency distribution of the scores of a group of pupils in a game. Score Number of pupils 10 – 19 1 20 – 29 2 30 – 39 8 40 – 49 12 50 – 59 k 60 – 69 1 (a) It is given that the median score of the distribution is 42. Calculate the value of k. [3 marks] (b) Use the graph paper provided to answer this question. Using a scale of 2 cm to 10 cm scores on the horizontal axis and 2 cm to 2 pupils on the vertical axis, draw a histogram to represent the frequency distribution of the scores. Find the mode score. [4 marks] (c) What is the mode score if the score of each pupil is increased by 5? [1 mark] (SPM2006/Section A/Paper 2)