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CHAPTER 7 : STATISTICS 7.1 MEASURES OF CENTRAL TENDENCY (i) Ungrouped data data that is not grouped into classes (ii) Grouped data - data that is grouped in certain classes 7.1.1 Calculating the mean of ungrouped data x x N Where x - mean of the set data. x values in the set of data N total number of data Example 1: Find the mean of 58, 67, 45, 73 and 77 Example : Number 1 3 4 5 Frequency 5 8 4 6 Find the mean of the number. 7.1. Determining mode of ungrouped data Mode is the value which appears the most number of times in a set of data (value that has the highest frequency) Example 1: Determine the mode for the following sets of number. (a), 5, 6,, 6, 7,, 4, 8, (b) 5, 5, 8, 10, 4, 4 (c),, 3, 4, 4, 4, 6, 6 Answer:.. Answer:.. Answer:.. Note: It is possible that a set of data either has more than one mode or has no mode.
Example : Determine the mode of the number of pens a student has. Number of pens 1 3 4 5 Number of students 6 7 5 3 3 Answer:.. 7.1.3 Determining the median of ungrouped data When the values in set of data are arranged in either ascending or descending order, the value that lies in the middle is the median. Example 1: 3, 4, 5, 6, 7, 8, 9 Median Example : 1, 0, 19, 18, 17, 16, 15, 14 4 numbers 4 numbers Median = Activity 1: 1. Calculate the mean, mode and median for the following sets of data. (i) 1, 4, 5, 8, 9, 8, 8, 7, 4 (ii) 5, 8, 1, 10, 5, 3, 7, 5, 0, 10. (i) Find the mean of 6, 8, 4, 9 and 11 (ii) Find the value of x if the mean of 4, 5, 6, 7, 11 and x is 7. 3. Find the mode of each following sets of data. (i) 8, 6, 10, 8, 5 (ii),, 5, 5, 11, 11
4. Find the mean, mode and the median of the following data Time ( hours ) 1 13 14 15 16 Number of cars 3 5 10 6 6 5. The following frequency distribution table shows the score of a group of students in a quiz. Find the mean, mode and median Score 5 6 7 8 9 10 Number of students 5 6 4 3 8 4 7.1.4 Determine modal class of grouped data from the frequency distribution table. Modal class of a set of data is the value of class which occurs most frequently. The value of mode can be obtained by drawing the histogram Example 1: The following table below shows the mark for 50 students in their Additional Mathematics test. Find the modal class for the students. Mark 10-19 0-9 30-39 40-49 50-59 60-69 70-79 Number of students 3 6 13 10 7 7 4 7.1.5 Find mode from a histogram Note: In drawing a histogram, class boundaries are used. HISTOGRAM Plot: frequency against boundary of the class Frequency Boundary of the class
Example 1: The following table below shows the mark for 50 students in their Additional Mathematics test.. Mark 10-19 0-9 30-39 40-49 50-59 60-69 70-79 Number of students 3 6 13 10 7 7 4 Complete the following table. Table 3 Mark Frequency, f Lower class boundary Upper class boundary 10 19 3 From the table, draw a histogram, hence determine the mode from the histogram. Calculate mean of grouped data fx x where f is the frequency for each class f x is the corresponding class midpoint Note : Class mid point = lower limit upper limit Example 1: 1. The table below shows the marks obtained by 30 students in a Mathematics test. Marks 30-39 40-49 50-59 60-69 Number of students 4 8 1 6 Find the mean for the marks.
Solution : Marks Number of students, f Class mark,x fx f fx 7.1.7 Calculating median of grouped data from the cumulative frequency distribution table Median, m = L + 1 N F C f m, where L = Lower boundary of the class in which the median lies. N = total frequency C = Size of class interval F = cumulative frequency before the class in which the median lies. f m = frequency of the class in which the median lies Note: Size of class interval = upper class boundary lower class boundary Activity : The following table shows the mark for 40 students in their Additional Mathematics test. Find the median by using the formula Mark 10-19 0-9 30-39 40-49 50-59 60-69 70-79 Number of students 3 5 7 9 10 4 TABLE 1
Solution: Mark Frequency, f 10 19 3 Cumulative frequency, F 7.1.8 Estimate median of grouped data from an ogive OGIVE Plot: cumulative frequency against upper class boundary Cumulative frequency Upper class boundary
Refer to table 1 Construct a cumulative frequency table and then draw cumulative frequency curve (ogive) From the graph, find the median weight. Mark Frequency, f Upper class Boundary Cumulative frequency, F Activity : 1. The following table shows the marks obtained by 30 students in Mathematics test. Find the mean for the data. Mark 10-19 0-9 30-39 40-49 50-59 60-69 70-79 Number of students 3 6 13 10 7 7 4 Solution : Mark Number of students, f Class mark,x fx
. The data below shows the scores obtained by 30 students in a game. 0 3 3 6 8 9 9 10 10 11 5 0 7 6 9 10 1 17 5 4 5 8 8 10 11 7 8 1 11 Using 3 scores as a size of class interval, construct a frequency distribution table for its data. Then find (a) the mean (b) median (c) modal class Note : Students have to draw the frequency distribution table as below. Number of scores x f fx Cumulative frequency 3. The table below shows the age of factory workers in 1995. Age(year) 0-5 6-31 3-37 38-43 44-49 50-55 56-61 Number of workers 5 4 16 0 13 1 10 (i) (ii) Draw a histogram and hence estimate the mode of the data. Find the median without using an ogive.
7.1.9 Determine the effects on mode, median and mean for a set of data when (a) Every value of the data is change uniformly Measures of central tendency Added by k Subtracted by k Multiplied by k Divided by k New mean Original mean + k Original mean - k k(original mean) New mode Original mode + k Original mode - k K(Original mode) original k original k mean mode New median Original median+ k Original median - k K(Original median) original median k 1. The mean, mode and median of a set data are 7.4, 9 and 8 respectively. Find the new mean, mode and median if every value of the set data is a) divided by b) subtracted by 4 c) multiplied by 3 d) added by 3. Find the mean of 1, 14, 16, 18, 0. By using the result, find the mean of 8,10,1,14,16 7.1 MEASURES OF DISPERSION 7..1 Finding the measures of dispersion of ungrouped data Formulae: (i) Range of ungrouped data = largest value smallest value (ii) Interquartile range = Upper quartile lower quartile = Q 3 Q 1 (iii) Variance, : ( x x) i x N or x, where N (iv) Standard deviation, : = var iance x x N
Example : For the given ungrouped set of data, find (i) range (ii) interquartile range (iii) variance (iv) standard deviation (a) 3, 6, 8, 1, 15, 16 (b) 3, 5, 6, 7, 8, 4, 8, 9, 10, 1, 14
Activity 3: 1. Find range, interquartile range, variance and standard deviation for each set of the following data. (a) 10, 7, 19, 13, 14, 10 (b) 4, 3,, 7, 9, 10, 1, 6, 15 (d) 4, 1, 15, 10, 7, 6, 1. Given that the mean of set data 5, 7, x, 11, 1 is 9. a. Find the value of x. b. Find the variance and the standard deviation of the data. 7.. Finding measures of dispersion of grouped data Example: For the following table, Find (i) range (ii) interquartile range using ogive Age ( year ) Number of workers Upper class boundary Cumulative frequency Solution: 5 9 16 30 34 0 35 39 4 40 44 0 45 49 14 50 54 6
7..3 Determining the variance and standard deviation of grouped data. Formulae : f ( x x) var iance or N fx = x f where fx x mean f var iance f ( x x) standard deviation = N or fx f x
Example 1: From the following table, calculate the variance and standard deviation. Mass ( kg ) Number of students 46 50 3 51 55 7 56 60 10 61 65 5 66 70 6 71-75 9 Solution: Example : The heights of 100 men to the nearest cm are recorded as follows. Calculate the mean ( giving your answer to one decimal place). Calculate the standard deviation for the height of the men. Height, x(cm) Frequency, f 159-160 161 16 14 163 164 8 165 166 6 167 168 19 169 170 9 171-17
Solution: Class f x fx x fx Activity 4: 1. The following table shows the number of durians sold in 50 consecutive days. Number of durians Number of days 0 8 10 9 17 15 18 6 0 7-35 5 Calculate the variance and standard deviation
. The table below shows the distribution of marks of 10 students in a Physics test. Marks Number of students 0 9 30 39 14 40 49 35 50 59 50 60 69 17 70-79 Calculate a. mean b. median c. standard deviation for this distribution. 7..4 Determining the effects on measures of dispersion when some values in a set of data are changed. A. If every value of the data is changed uniformly New Measures Added by k Subtracted by k Multiplied by k Divided by k Variance, New Standard Deviation, (original variance) (original standard deviation) (original variance) (original standard deviation) k ( k ( ) ) k original k New Range Original range Original range k(original range) original k range New Interquartile range Original interquartile range Original interquartile range k(original interquartile range) original interquartile k
B. If there are extreme values in the set of data Range Extreme values in a set of data will significantly increase the range of the set of data. Interquartile range Extreme values in a set of data will have little or no effect on the interquartile range. Variance and Standard Deviation Extreme values also significantly increase the value of standard deviation and variance but standard deviation is affected to a smaller degree as compare to variance. C. If certain values are added or removed When a value is added or removed from a set of data, the effect on the measures of dispersion is uncertain. In general, the range and the interquartile range are less affected as compared to the variance and the standard deviation. Variance and standard deviation are more significantly affected when the added or removed value has a greater difference from the mean. Activity: (a) The interquartile range and the standard deviation of a set of data are 5 and.5. Find the new interquartile range and standard deviation if every value of the data is divided by followed by an addition of 10. (b) Given a set of data 4, 5, 8, 1, 14, 18, 0. Determine the standard deviation of the set of data. Explain how the standard deviation will change if a value of 100 is added to the set of data. Activity 5: 1 The following table shows the results in five Additional Mathematics Test obtained by two students A and B. Students A 71 76 80 83 90 Students B 40 67 95 98 100 (a) (b) Find the mean and the standard deviation of the result for each students. If a student with a more consistent performance is selected, which students would be selected? Explain the reason for your selection. Find the mean and standard deviation for the following set of numbers:, 3, 5, 8, 10. Hence, using this result find the mean and standard deviation for (a) five numbers : 7, 8, 10, 13, 15 (b) five numbers : 10, 15, 5, 40, 50 (c ) ten numbers :,, 3, 3, 5, 5, 8, 8, 10, 10
3 The mean for 7, 14, x, x, 16, 9, 10 and is 8. Find the value of (a) x (b) the standard deviation (c ) standard deviation if each of the number is added by. Enrichment SPM 006 1. A set of positive integers consists of, 5 and m. The variance for this set of integers is 14. Find the value of m. [ 3 marks ]. Table 1 below shows the frequency distribution of the scores of a group of pupils in a game. SPM 005 Score Number of pupils 10 19 1 0 9 30 39 8 40 49 1 50 59 k 60 69 1 Table 1 a. It is given that the median score of the distribution is 4. Calculate the value of k. [ 3 marks ] b. Use the graph paper to answer this question Using a scale of cm to 10 scores on the horizontal axis and cm to 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 ] 3. The mean of four numbers is m.the sum of the squares of the numbers is 100 and the standard deviation is 3k. Express m in terms of k. [ 3 marks ] 4. Diagram is a histogram which represents the distribution of the marks obtained by 40 pupils in a test.
frequency 14 1 10 8 6 4 0 0.5 10.5 0.5 30.5 40.5 50.5 Upper class boundary SPM 004 (a) Without using an ogive, calculate the median mark. [ 3 marks ] (b) Calculate the standard deviation of the distribution. [ 4 marks ] 5. A set of data consist of 10 numbers. The sum of the numbers is 150 and the sum of the squares of the numbers is 47. (a) Find the mean and variance of the 10 numbers. (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 003 6. A set of examination marks x 1, x, x 3, x 4, x 5, x 6 has a mean of 5 and a standard deviation of 1.5. (a) (b) Diagram Find (i) The sum of the marks, x, (ii) The sum of the squares of the marks, x [ 3 marks ] Each mark is multiplied by and then 3 is added to it. Find for the new set of marks, (i) The mean, (ii) The variance [ 4 marks ] 7. The mean for 0, 5 + x, 5 + 4x and 5 are 10. Calculate the value of (a) x (b) median
(c) standard deviation 8. The table shows the mark obtained by a group of students in a competition. Mark 1 3 4 5 Number of students 3 7 5 4 1 9. Find (a) mean, mode and median (b) standard deviation Score 0 1 3 4 5 6 Frequency 3 x 5 x + 4 4 3 (a) (b) (c) The table above shows the score obtained by 30 students in a test. Find the value of x Find score mode If we represent the table by pie chart, calculate the angle sector for the students who get score more than 3. SPM 001 10. (a) Given that 4 positive integers have mean 9. If a number, y, is added, the mean become 10. Find the value of y. [ marks ] (b) Find the standard deviation for the set of number 5, 6, 6, 4, 7 [ 3 marks ] 11. Mark Number of students 6 10 1 11 15 0 16 0 7 1 5 16 6 30 13 31 35 10 36 40 Table Table shows the frequency distribution table for the marks obtained by 100 students. (a) Calculate the variance. [ 3 marks ] (b) Using the graph paper provided to solve this question Construct a cumulative frequency distribution table and draw an ogive. Hence, find the percentage of the number of students who get mark between 6 and 4. [7 marks ]
SPM 000 1. Mark < 10 < 0 < 30 < 40 < 50 < 60 < 70 < 80 Number of students 8 1 4 68 87 98 100 The table above shows the mark obtained by 100 students in a test. (a) Using the table above, copy and complete the following table. frequency Mark 0-9 10-19 0-9 30-39 40-49 50-59 60-69 70-79 (b) [ marks ] Without drawing an ogive, estimate the interquartile range for the data. [ 4 marks ] SPM 1998 13. The mean of set data, k, 3k, 8, 1 and 18 in increasing order is m. If every number in 5m the set data is subtracted by, the median for the new data is. Find 8 (a) the value of m and k [ 4 marks ] (b) Variance for the new data. [ marks ] 14. Set X consist of 50 score, in a certain match, has mean 8 and standard deviation 3. (i) Calculate x and x (ii) SPM 1996 The sum of a number, 180 scores with mean 6 and the sum of the square of a number, 100, are removed from set X. Evaluate the mean and variance for left data in set X. [ 7 marks ] 15. The mean for the list of number x, x + 4, x + 5, x 1, x + 7 and x 3 is 7. Find (a) The value of x [ 1 marks ] (b) Variance [ marks ]
SPM 1995 16. Number of class Number of students 6 35 5 36 4 30 The table shows the number of students in a class. Find (i) the mean (ii) the standard deviation, for the number of students in every class. (b) Age Number of people 1-0 50 1-40 79 41-60 47 61-80 14 81-100 10 (Solution using graph method is not allowed) The table above shows the age of 00 people in a village. Calculate (i) median (ii) Upper quartile, of the age of the people. [ 6 marks ] 17. (a) Given a list of number: 3, 6, 3, 8. Find the standard deviation of the number. [ marks ] (b) Find a set of five possible positive integer with mode 3, median 4 and mean 5. SPM 1994 18. Set A is a set that consist of 10 numbers. The sum of the number is 150 and the sum of the square of the marks are 890. (i) (ii) Find the mean and variance. If one number is added to the set of numbers in Set A, (in case mean is unchanged), find the standard deviation of the 11 numbers. 19. Find the interquartile range for the following data: 10, 4, 11, 18, 8, 6, 19, 13, 17, 16, 5
SPM 1993 P1 0. Mark 1-0 1-40 41-60 61-80 81 100 Number of students 5 8 1 11 4 The table shows the mark obtained by a group of students in a Monthly Test. (a) (b) (c) By using graph paper, draw a histogram and hence determine the mode of the data. Without drawing an ogive, find the median. Calculate mean mark. Answer :Enrichment 1 1 11 (a) k = 4 (b) mode = 43 (c) 48 3 m 5 9k 1. (a) (b) 1.78 4 (a) 4.07 (b) 11.74 10 (a) m = 8, k = (b) 3 5 (a) 15,. (b) (i) k = 6 (ii) 5.494 6 (a) (i) 30 (ii) 163.5 (b) (i) 13 (ii) 9 11 (i) x 400, x 3650 (ii) New variance = 1.5 1 (a) x=4 (b) 19 7 (a) x = 1 (b) 7.5 (c) 5.958 13 (a) (i) 34 (ii).449 (b) (i) 33.16 (ii) 49.44 8. (a).65,,.5 (b) 1.108 14 (a).11 (b) 3,3,4,6,9 or 3,3,4,5,10 or 3,3,4,7,8 9. (a) x= 3 (b) 4 (c) 168 o 16 (a) (i) 15, 64 (ii) 7.68 10. (a) y = 14 (b) 1.00 17 13 11. (a) 61.6 (b) 70% 18 (a) 56.5 (b) 5.17 (c) 51