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21 21.2Box-and-whisker Diagrams 21.3Standard Deviation Chapter Summary Case Study Measures of Dispersion 21.1Range and Inter-quartile Range 21.4Applications of Standard Deviation 21.5Effects on the Dispersion with a Change in Data

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P. 2 Although both drivers got the same total mark, it does not mean that both of them had a consistent performance for all ten dives. If we plot a broken line graph for both divers, we find that the marks of diver A fluctuate more than those of diver B. Case Study In the above example, we consider the spread of the data. Both divers get the same total mark. But it seems that A’s performance is less consistent. Is that true? You need to know the meaning of dispersion first. In this chapter, we will learn how to represent this by statistical methods.

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P. 3 Consider two boxes of apples A and B. 21.1 Range and Inter-quartile Range The mean and the median of the weights for both boxes of apples is 104 g and 102 g respectively, and the weights of the apples in Box B are more widely spread than those in Box A. In junior forms, we learnt three measures of central tendency of a set of data, namely mean, median and mode. However, these measures tell us only limited information about the data. The spread or variability of data is called the dispersion of the data. In this chapter, we are going to learn the following measures of dispersion: 1.Range 2.Inter-quartile range 3.Standard deviation The weight (in g) of each apple in each box is given below: Box A: 100, 100, 100, 104, 110, 110 Box B: 85, 95, 100, 104, 116, 124

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P. 4 1.For ungrouped data, the range is the difference between the largest value and the smallest value in the set of data. 21.1 Range and Inter-quartile Range A. Range The range is a simple measure of the dispersion of a set of data. 2.For grouped data, the range is the difference between the highest class boundary and the lowest class boundary Range Largest value – Smallest value Range Highest class boundary – Lowest class boundary

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P. 5 Example 21.1T Solution: 21.1 Range and Inter-quartile Range The weights (in g) of eight pieces of meat are given below: 210, 230, 245, 180, 220, 240, 175, 195 (a) Find the range of the weights. (b) If the meat is sold at $3 per 100 g, find the range of the prices of the meat. (a) Range (245 175) g (b) Range of the prices A. Range

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P. 6 Example 21.2T 21.1 Range and Inter-quartile Range The following table shows the weights of the boys in S6A. (a) Write down the upper class boundary of the class 70 kg – 74 kg. (b) Write down the lower class boundary of the class 50 kg – 54 kg. (c) Hence find the range of the weights. Solution: (a) 74.5 kg (b) 49.5 kg A. Range (b) Range (74.5 49.5) kg Weight (kg)50 – 5455 – 5960 – 6465 – 6970 – 74 Frequency36852

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P. 7 21.1 Range and Inter-quartile Range B. Inter-quartile Range When a set of data is arranged in ascending order of magnitude, the quartiles divide the data into four equal parts. Full set of data arranged in order of magnitude 25% of data Q 1 Q 2 Q 3 Q 1 : lower quartile 25% of data less than it Q 2 : median 50% of data less than it Q 3 : upper quartile 75% of data less than it The inter-quartile range is defined as the difference between the upper quartile and the lower quartile of the set of data. Inter-quartile range Q 3 – Q 1 Q 1, Q 2 and Q 3 are also called the first, the second and the third quartiles respectively.

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P. 8 Example 21.3T The marks of 13 boys in a Chinese test are recorded below: 72 78 80 65 62 62 78 81 84 70 68 69 60 (a) Arrange the marks in ascending order. (b) Find the median mark. (c) Find the range and the inter-quartile range. Solution: (a) Arrange the marks in ascending order: (b) Median (c) Range Inter-quartile range 21.1 Range and Inter-quartile Range B. Inter-quartile Range 60, 62, 62, 65, 68, 69, 70, 72, 78, 78, 80, 81, 84

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P. 9 Example 21.4T Solution: The cumulative frequency polygon shows the lifetimes (in hours) of 80 bulbs. (a) Find the range of the lifetimes. (b) Find the median lifetime. (c) Find the inter-quartile range. (a) Range (b) From the graph, 21.1 Range and Inter-quartile Range B. Inter-quartile Range (780 140) hours Median (c) Q 1 240 hours, Q 3 600 hours Inter-quartile range (600 240) hours

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P. 10 Example 21.5T Solution: Consider the ages of passengers in two mini-buses. Mini-bus A: 18, 24, 25, 19, 12, 10, 34, 39, 45, 23, 34, 40, 24, 28 Mini-bus B: 23, 26, 28, 32, 38, 34, 19, 26, 29, 32, 35, 30, 29, 22 By comparing the ranges and the inter-quartile ranges of the ages, determine which group of passengers has a larger dispersion of ages. For Mini-bus A, 21.1 Range and Inter-quartile Range B. Inter-quartile Range the range 45 10 35 For Mini-bus B, the range 38 19 19 For Mini-bus A, the inter-quartile range 34 19 15 For Mini-bus B, the inter-quartile range 32 26 6 Since the range and the inter-quartile range of the ages of passengers of mini-bus A are larger, the passengers on mini-bus A have a larger dispersion.

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P. 11 A box-and-whisker diagram is a statistical diagram that provides a graphical summary of the set of data by showing the quartiles and the extreme values of the data. 21.2 Box-and-whisker Diagrams A box-and-whisker diagram shows the greatest value, the least value, the median, the lower quartile and the upper quartile of a set of data. The difference between the two end-points of the line is the range. The length of the box is the inter-quartile range.

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P. 12 Example 21.6T Solution: The following box-and-whisker diagram shows the number of family members of a class of students. (a) Find the median and the range of the number of family members. (b) Find the inter-quartile range. (a)Median (b) Q 1 2 and Q 3 4 21.2 Box-and-whisker Diagrams Maximum value 6 and minimum value 1 Range Inter-quartile range

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P. 13 Example 21.7T The following shows the measurements of the waists (in inches) of the students in a class. Girls:27 26 25 23 26 28 32 24 25 29 30 28 22 25 26 Boys:32 32 30 29 28 25 28 26 29 31 32 34 28 27 28 (a)Find the median, the lower quartile and the upper quartile of the waists for both boys and girls. (b)Draw box-and-whisker diagrams of their waist measurements on the same graph paper. Solution: (a) Arrange the measurements in ascending order: 21.2 Box-and-whisker Diagrams Girls:22, 23, 24, 25, 25, 25, 26, 26, 26, 27, 28, 28, 29, 30, 32 Boys:25, 26, 27, 28, 28, 28, 28, 29, 29, 30, 31, 32, 32, 32, 34 For girls, median For boys, median

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P. 14 Example 21.7T Solution: (b)For boys,minimum 25 inches maximum 34 inches 21.2 Box-and-whisker Diagrams Refer to the figure on the right. For girls,minimum 22 inches maximum 32 inches The following shows the measurements of the waists (in inches) of the students in a class. Girls:27 26 25 23 26 28 32 24 25 29 30 28 22 25 26 Boys:32 32 30 29 28 25 28 26 29 31 32 34 28 27 28 (a)Find the median, the lower quartile and the upper quartile of the waists for both boys and girls. (b)Draw box-and-whisker diagrams of their waist measurements on the same graph paper.

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P. 15 Standard deviation describes how the spread out of the data are around the mean. It is usually denoted by . 21.3 Standard Deviation Consider a set of ungrouped data x 1, x 2, …, x n. A. Standard Deviation for Ungrouped Data Notes: The quantity 2 is called the variance of the data. Standard deviation, , where is the mean and n is the total number of data. _ (x i – x) is the deviation of the ith data from the mean.

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P. 16 Example 21.8T Six students joined the inter-school cross-country race. The times taken (in min) to complete the race are recorded below: 45, 46, 49, 50, 52, y If the mean time is 49.5 min, find (a)the value of y and (b) the standard deviation of the times taken. (Give the answer correct to 3 significant figures.) Solution: (a) (b) Standard deviation (cor. to 3 sig. fig.) 21.3 Standard Deviation A. Standard Deviation for Ungrouped Data

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P. 17 Example 21.9T Solution: The following table shows the marks of Eric in five tests of two subjects. (a)Find the standard deviations of the marks of each subject. (Give the answers correct to 3 significant figures.) (b)In which subject is his performance more consistent? (a)For Chinese, mean Standard deviation (cor. to 3 sig. fig.) 21.3 Standard Deviation A. Standard Deviation for Ungrouped Data Test 1Test 2Test 3Test 4Test 5 Chinese6570766878 English7281859080

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P. 18 Example 21.9T Solution: The following table shows the marks of Eric in five tests of two subjects. (a)Find the standard deviations of the marks of each subject. (Give the answers correct to 3 significant figures.) (b)In which subject is his performance more consistent? 21.3 Standard Deviation A. Standard Deviation for Ungrouped Data For English, mean Standard deviation (cor. to 3 sig. fig.) Test 1Test 2Test 3Test 4Test 5 Chinese6570766878 English7281859080

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P. 19 Example 21.9T Solution: The following table shows the marks of Eric in five tests of two subjects. (a)Find the standard deviations of the marks of each subject. (Give the answers correct to 3 significant figures.) (b)In which subject is his performance more consistent? (b)For Chinese, standard deviation 4.88 21.3 Standard Deviation A. Standard Deviation for Ungrouped Data The standard deviation of Chinese is smaller than that of English, so Eric’s performance in Chinese is more consistent. For English, standard deviation 5.95 Test 1Test 2Test 3Test 4Test 5 Chinese6570766878 English7281859080

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P. 20 Example 21.10T Solution: The numbers of students in five classes are given as: y – 15, y + 6, y + 9, y – 20, y + 15 (a) Find the mean and the standard deviation. (Give the answers correct to 3 significant figures if necessary.) (b) Find the range and the median if the mean is 34. (a) Mean Standard deviation (cor. to 3 sig. fig.) (b) ∵ y 1 34 The five numbers are 20, 41, 44, 15 and 50. ∴ Range and Median 21.3 Standard Deviation A. Standard Deviation for Ungrouped Data ∴ y 35

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P. 21 For a set of grouped data, we have to consider the frequency of each group. 21.3 Standard Deviation B. Standard Deviation for Grouped Data Standard deviation, = =, where f i and x i are the frequency and the class mark of the ith class interval respectively, is the mean and n is the total number of class marks.

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P. 22 Example 21.11T Solution: The following table shows the ages of 50 workers in a company. (a) Find the mean age of the workers. (b) Find the standard deviation of the ages. (Give the answer correct to 3 significant figures.) (a) Mean age (b) 21.3 Standard Deviation B. Standard Deviation for Grouped Data Class mark15.525.535.545.555.5 Frequency4241282 Standard deviation (cor. to 3 sig. fig.) Age11 – 2021 – 3031 – 4041 – 5051 – 60 Number of workers4241282

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P. 23 In actual practice, it is quite difficult to calculate the standard deviation if the amount of data is very large. 21.3 Standard Deviation C. Finding Standard Deviation by a Calculator In such circumstances, a calculator can help us to find the standard deviation. In order to use a calculator, we have to set the function mode of the calculator to standard deviation ‘SD’. We also have to clear all the previous data in the ‘SD’ mode. For both ungrouped and grouped data, we can use a calculator to find the mean and the standard deviation. For grouped data, use the class mark to represent the entire group.

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P. 24 The following table summarizes the advantages and disadvantages of the three different measures of dispersion. 21.3 Standard Deviation Measure of dispersionAdvantageDisadvantage 1. RangeOnly two data are involved, so it is the easiest one to calculate. Only extreme values are considered which may give a misleading impression. 2. Inter-quartile rangeIt only focuses on the middle 50% of data, thus avoiding the influence of extreme values. Cannot show the dispersion of the whole group of data. 3. Standard deviationIt takes all the data into account that can show the dispersion of the whole group of data. Difficult to compute without using a calculator.

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P. 25 21.4 Applications of Standard Deviation Standard score is used to compare data in relation with the mean and the standard deviation . A. Standard Scores Notes: The standard score may be positive, negative or zero. A positive standard score means the given value is z times the standard deviation above the mean while a negative standard score means the given value is z times the standard deviation below the mean. The standard score z of a given value x from a set of data with mean and standard deviation is defined as:

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P. 26 Example 21.12T Ryan sat for a mathematics examination which consisted of two papers. The following table shows his marks as well as the means and the standard deviations of the marks for the whole class in these papers. (a)Find his standard scores in the two papers. (Give the answers correct to 3 significant figures.) (b)In which paper did he perform better? Solution: (a) Paper I: (b) Since 1.34 1.24, Ryan performed better in Paper II than in Paper I. 21.4 Applications of Standard Deviation A. Standard Scores (cor. to 3 sig. fig.) Paper II: (cor. to 3 sig. fig.) Paper IPaper II Marks6671 Mean60.862.4 Standard deviation4.26.4

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P. 27 Example 21.13T Given that the standard scores of Doris’s marks in Art and Music are –2.3 and 1.4 respectively, find (a)her mark in Art if the mean and the standard deviation of the marks are 30 and 2 respectively; (b)the mean mark of Music if Doris got 41.5 marks and the standard deviation of the marks is 3.5. Solution: (a) Doris’s mark in Art (b) The mean mark of Music 21.4 Applications of Standard Deviation A. Standard Scores ( 2.3) 2 + 30 41.5 1.4 3.5

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P. 28 For a large number of the frequency distributions we meet in our daily life, their frequency curves have the shape of a bell: B. Normal Distribution The bell can have different shapes. This bell-shaped frequency curve is called the normal curve and the corresponding frequency distribution is called the normal distribution. For a normal distribution, the mean, median and the mode of the data lie at the centre of the distribution. In a normal distribution, mean median mode. 21.4 Applications of Standard Deviation Therefore, the normal curve is symmetrical about the mean, i.e., the axis of symmetry for the normal curve is x .

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P. 29 In addition, we can tell the percentage of the data lie within a number of standard deviations from the mean: B. Normal Distribution 1.About 68% of the data lie within one standard deviation from the mean, i.e., – and + . 21.4 Applications of Standard Deviation 2.About 95% of the data lie within two standard deviations from the mean, i.e., – 2 and + 2 . 3.About 99.7% of the data lie within three standard deviations from the mean, i.e., – 3 and + 3 .

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P. 30 Example 21.14T The heights of some soccer players are normally distributed with a mean of 180 cm and a standard deviation of 8 cm. Find the percentage of players (a) whose heights are between 172 cm and 188 cm, (b) whose heights are greater than 188 cm. Solution: Given 180 and 8. (a) 172 180 8 34% of the players’ heights lie between ( ) cm and cm. B. Normal Distribution 21.4 Applications of Standard Deviation 188 180 + 8 34% of the players’ heights lie between cm and ( ) cm. Percentage of players whose heights are between 172 cm and 188 cm 34% + 34% (b) 188 180 + 8 Percentage of players whose heights are greater than 188 cm 50% 34%

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P. 31 81.5% of children have weights between ( ) kg and ( ) kg. Example 21.15T The weights of 2000 children are normally distributed with a mean of 56.4 kg and a standard deviation of 6.2 kg. (a) How many children have weights between 50.2 kg and 68.8 kg? (b) How many children are heavier than 50.2 kg? Solution: B. Normal Distribution 21.4 Applications of Standard Deviation Given 56.4 and 6.2. (a) 50.2 56.4 6.2 68.8 56.4 + 2 6.2 Number of children 2000 81.5% (b) 50.2 56.4 6.2 Percentage of children who are heavier than 50.2 kg 50% + 34% 84% Number of children 2000 84%

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P. 32 21.5 Effects on the Dispersion with a Change in Data Change in Data In junior forms, we learnt that if we remove a datum greater than the mean of the data set, then the mean will decrease. A. Removal of the Largest or Smallest item from the Data from the Data 1.the range will decrease; 2.the inter-quartile range may increase, decrease or remain unchanged; 3.the standard deviation may increase or decrease. If the greatest or the least value (assuming the removed datum is unique) in a data set is removed, then Similarly, if we remove a datum less than the mean of the data set, then the mean will increase. We can deduce that:

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P. 33 21.5 Effects on the Dispersion with a Change in Data Change in Data We have the following conclusion: B. Adding a Common Constant to the Whole Set of Data Set of Data 1.the range, 2.the inter-quartile range and 3.the standard deviation If a constant k is added to each datum, then the mean, median and the mode will also increase by k. If a constant k is added to each datum in a set of data, then the following measures of dispersion will not change:

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P. 34 21.5 Effects on the Dispersion with a Change in Data Change in Data The range, the inter-quartile range and the standard deviation will be k times the original values if each datum in a set of data is multiplied by a constant k. C. Multiplying the Whole Set of Data by a Common Constant Common Constant Notes: If the quartiles are not members of the data set, the conclusion will be the same. We have the following conclusion:

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P. 35 Example 21.16T Consider the nine different numbers: 20, 45, 25, 30, 32, 28, 35, 51, 40 (a) Find the range, the inter-quartile range and the standard deviation. (b) Find the new range, the new inter-quartile range and the new standard deviation of the new set of data if (i)10 is subtracted from each data; (ii)each number is halved; (iii)the datum 45 is removed. (Give the answers correct to 3 significant figures if necessary.) Solution: (a) Range Inter-quartile range Standard deviation 21.5 Effects on the Dispersion with a Change in Data Change in Data C. Multiplying the Whole Set of Data by a Common Constant Common Constant (cor. to 3 sig. fig.)

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P. 36 Example 21.16T (b) (i)If 10 is subtracted from each datum, the range, the inter-quartile range and the standard deviation of the new data remain unchanged. 21.5 Effects on the Dispersion with a Change in Data Change in Data C. Multiplying the Whole Set of Data by a Common Constant Common Constant Inter-quartile range Standard deviation (cor. to 3 sig. fig.) Range Consider the nine different numbers: 20, 45, 25, 30, 32, 28, 35, 51, 40 (a) Find the range, the inter-quartile range and the standard deviation. (b) Find the new range, the new inter-quartile range and the new standard deviation of the new set of data if (i)10 is subtracted from each data; (ii)each number is halved; (iii)the datum 45 is removed. (Give the answers correct to 3 significant figures if necessary.) Solution:

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P. 37 Example 21.16T 21.5 Effects on the Dispersion with a Change in Data Change in Data C. Multiplying the Whole Set of Data by a Common Constant Common Constant (b) (ii)If each datum is halved, the range, the inter-quartile range and the standard deviation of the new data are multiplied by 0.5. Inter-quartile range Standard deviation (cor. to 3 sig. fig.) Range Consider the nine different numbers: 20, 45, 25, 30, 32, 28, 35, 51, 40 (a) Find the range, the inter-quartile range and the standard deviation. (b) Find the new range, the new inter-quartile range and the new standard deviation of the new set of data if (i)10 is subtracted from each data; (ii)each number is halved; (iii)the datum 45 is removed. (Give the answers correct to 3 significant figures if necessary.) Solution:

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P. 38 Example 21.16T 21.5 Effects on the Dispersion with a Change in Data Change in Data C. Multiplying the Whole Set of Data by a Common Constant Common Constant (b) (iii)The remaining data are 20, 25, 28, 30, 32, 35, 40, 51. Inter-quartile range Standard deviation (cor. to 3 sig. fig.) Range Consider the nine different numbers: 20, 45, 25, 30, 32, 28, 35, 51, 40 (a) Find the range, the inter-quartile range and the standard deviation. (b) Find the new range, the new inter-quartile range and the new standard deviation of the new set of data if (i)10 is subtracted from each data; (ii)each number is halved; (iii)the datum 45 is removed. (Give the answers correct to 3 significant figures if necessary.) Solution:

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P. 39 21.5 Effects on the Dispersion with a Change in Data Change in Data We have the following conclusion: D. Insertion of Zero in the Data Set If a zero value is inserted in a non-negative data set, then 1.the range may increase or remain unchanged; 2.the inter-quartile range may increase, decrease or remain unchanged; 3.the standard deviation may increase or decrease.

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P. 40 Example 21.17T The daily income ($) of a hawker during the last two weeks was: 200, 220, 230, 240, 250, 320, 340, 360, 380, 400, 450, 580, 650 (a) Find the inter-quartile range and the standard deviation. (b) There was a thunderstorm last Monday and the income on that day was zero. If the income from last Monday is also considered, find the new inter-quartile range and the new standard deviation. (Give the answers correct to 1 decimal place if necessary.) Solution: (a) 21.5 Effects on the Dispersion with a Change in Data Change in Data D. Insertion of Zero in the Data Set Inter-quartile range The standard deviation (cor. to 1 d. p.) (b) Inter-quartile range The standard deviation (cor. to 1 d. p.)

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P. 41 21.1 Range and Inter-quartile Range 1. The range is the difference between the largest value (highest class boundary) and the smallest value (lowest class boundary) in a set of ungrouped (grouped) data. Chapter Summary 2. The inter-quartile range is the difference between the upper quartile Q 3 and the lower quartile Q 1 of a set of data.

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P. 42 A box-and-whisker diagram illustrates the spread of a set of data. It shows the greatest value, the least value, the median, the lower quartile and the upper quartile of the data. Chapter Summary 21.2 Box-and-whisker Diagrams

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P. 43 Standard deviation is the measure of dispersion that describes how spread out a set of data is around the mean value. Chapter Summary 21.3 Standard Deviation 1. For ungrouped data: 2. For grouped data: Larger values for the range, the inter-quartile range and the standard deviation of the data indicate a larger dispersion and vice versa.

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P. 44 1.The standard score z is the number of standard deviations that a given value is above or below the mean, and is given by Chapter Summary 21.4 Applications of Standard Deviation 2.The curve of a normal distribution is bell-shaped and is called the normal curve. In the normal distribution, different percentages of data lie within different standard deviations from the mean.

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P. 45 1.If the greatest or the least value (assuming both are unique) in a data set is removed, then the range will decrease. However, the inter- quartile range may increase, decrease or remain unchanged and the standard deviation may increase or decrease. Chapter Summary 21.5 Effects on the Dispersion with a Change in Data 2.If a constant k is added to each datum in a set of data, then the range, the inter-quartile range and the standard deviation will not change. 3.If each item in the data is multiplied by a positive constant k, then the range, the inter-quartile range and the standard deviation will be k times their original values. 4.If a zero value is inserted in a non-negative data set, then the range may increase or remain unchanged, the inter-quartile range may increase, decrease or remain unchanged and the standard deviation may increase or decrease.

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Follow-up 21.1 Solution: The results (in m) of the best eight boys in the long jump are: 5.2, 5.6, 4.8, 4.2, 5.3, 4.5, 5.0, 4.8 Find the range of the results. (5.6 4.2) m Range 21.1 Range and Inter-quartile Range A. Range

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Follow-up 21.2 The following table shows the heights of the students in S6B. (a) What is the upper class boundary of the class 171 cm – 175 cm? (b) What is the lower class boundary of the class 151 cm – 155 cm? (c) Hence find the range of the heights. Solution: (a) 175.5 cm (b) 150.5 cm Height (cm)151 – 155156 – 160161 – 165166 – 170171 – 175 Frequency13624 (c)Range (175.5 150.5) cm 21.1 Range and Inter-quartile Range A. Range

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Follow-up 21.3 The air pollution indices of a city are recorded at noon every day. The following are the indices in the last 15 days. 56 58 62 58 60 64 63 63 59 56 70 69 67 69 65 (a) Arrange the data in ascending order. (b) Find the median. (c) Find the inter-quartile range. Solution: (a) Arrange the marks in ascending order: 56, 56, 58, 58, 59, 60, 62, 63, 63, 64, 65, 67, 69, 69, 70 (b) Median (c) Q 3 67 and Q 1 58 Inter-quartile range 21.1 Range and Inter-quartile Range B. Inter-quartile Range

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Follow-up 21.4 Solution: The figure shows the cumulative frequency polygon of the heights (in cm) of 100 trees. (a) Find the lower quartile and the upper quartile of the heights. (b) Find the inter-quartile range. (a) From the graph, (b) Inter-quartile range 21.1 Range and Inter-quartile Range B. Inter-quartile Range

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Follow-up 21.5 The following table shows the distribution of members’ ages in two sports clubs. (a) How many members are there in each of the clubs? Solution: (a) For Badminton Club, Age121314151617 Badminton Club14108 62 Basketball Club081217130 number of members 14 + 10 + 8 + 10 + 6 For Basketball Club, number of members 0 + 8 + 12 + 17 + 13 + 0 21.1 Range and Inter-quartile Range B. Inter-quartile Range

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Follow-up 21.5 (b) For Badminton Club,range 17 – 12 For Basketball Club, inter-quartile range 15 – 12 range 16 – 13 inter-quartile range 16 – 14 (c)Since the range and the inter-quartile range of the ages of the members in the Badminton Club are larger, Badminton Club has a larger dispersion of ages. 21.1 Range and Inter-quartile Range B. Inter-quartile Range The following table shows the distribution of members’ ages in two sports clubs. (b) Find the range and the inter-quartile range of the distribution of ages of the members in each club. (c) Which club has a larger dispersion of ages? Age121314151617 Badminton Club14108 62 Basketball Club081217130 Solution:

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Follow-up 21.6 Solution: (a) Median (b) Maximum length 19.4 cm and The following box-and-whisker diagram shows the lengths (in cm) of the hands of a class of students. (a) Find the median length of the hands. (b) Find the range of the lengths of the hands. (c) Find the inter-quartile range of the lengths of the hands. minimum length 17.3 cm Range (19.4 17.3) cm (c)(c) Inter-quartile range (19.0 18.2) cm 21.2 Box-and-whisker Diagrams

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Follow-up 21.7 Kelvin and Johnny are comparing their results on 10 mathematics tests. The following are their marks. Kelvin’s marks: 54, 70, 67, 92, 75, 80, 84, 78, 66, 82 Johnny’s marks: 70, 74, 76, 78, 78, 79, 80, 78, 72, 75 (a) Find the median, the lower quartile and the upper quartile of the marks for each student. Solution: (a) Arrange the marks in ascending order: Kelvin’s marks: 54, 66, 67, 70, 75, 78, 80, 82, 84, 92 Median Johnny’s marks: 70, 72, 74, 75, 76, 78, 78, 78, 79, 80 Median 21.2 Box-and-whisker Diagrams

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Follow-up 21.7 Solution: (c)Johnny performs more consistently. 21.2 Box-and-whisker Diagrams Kelvin and Johnny are comparing their results on 10 mathematics tests. The following are their marks. Kelvin’s marks: 54, 70, 67, 92, 75, 80, 84, 78, 66, 82 Johnny’s marks: 70, 74, 76, 78, 78, 79, 80, 78, 72, 75 (b)Draw box-and-whisker diagrams on the same graph to compare the results. (c)Which student performs more consistently in the tests? (b)For Kelvin,minimum 54 marks maximum 92 marks Refer to the figure on the right. For Johnny,minimum 70 marks maximum 80 marks

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Follow-up 21.8 Emily got the following marks in five English tests. 59, 65, 76, 68, 82 (a)Find the mean mark. (b)Find the standard deviation of the marks. (Give the answer correct to 3 significant figures.) Solution: (a) Mean (b) 121 + 25 + 36 + 4 + 144 330 Standard deviation (cor. to 3 sig. fig.) 21.3 Standard Deviation A. Standard Deviation for Ungrouped Data

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Follow-up 21.9 Solution: The following table shows the marks of two students in five tests. (a) Find the standard deviations of the marks of the two students. (Give the answers correct to 3 significant figures.) (b) Which student has a more consistent performance? (a) For Winnie, (b) Since the standard deviation of Cherry’s marks is smaller than Winnie’s, Cherry has a more consistent performance. standard deviation (cor. to 3 sig. fig.) standard deviation (cor. to 3 sig. fig.) For Cherry, 21.3 Standard Deviation A. Standard Deviation for Ungrouped Data Test 1Test 2Test 3Test 4Test 5 Winnie8065826678 Cherry76748072

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Follow-up 21.10 The following are the ages of five people. 66 + x, 65 + 3x, 71 – x, 68 + x, 70 + x (a) Find the mean age in terms of x. (b) If x 4, find the standard deviation. (Give the answer correct to 3 significant figures.) Solution: (a) Mean (b) ∵ x 4 Mean 68 + 4 72 and the five numbers are 70, 77, 67, 72 and 74. Standard deviation (cor. to 3 sig. fig.) 21.3 Standard Deviation A. Standard Deviation for Ungrouped Data

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Follow-up 21.11 Solution: The following table shows the air pollution index recorded daily in a particular district at 5:00 p.m. in June. (a) Find the value of y. (b) Find the mean. (c) Find the standard deviation. (a) 8 + 10 + 7 + y 30 (b) Mean Index51 – 5556 – 6061 – 6566 – 70 Number of days8107y 807.5 30Total 361.2572.25568 85.7512.25763 22.52.251058 33842.25853 fx (c) Standard deviation (cor. to 3 sig. fig.) 21.3 Standard Deviation B. Standard Deviation for Grouped Data

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Follow-up 21.12 The following table shows the means and the standard deviations of the marks for the whole class in two subjects, as well as Helen’s marks. (a) Find the standard scores of Helen’s marks in these subjects. (Give the answers correct to 3 significant figures.) (b) In which subject did she perform better? Solution: (a) Chinese: 21.4 Applications of Standard Deviation A. Standard Scores SubjectChineseHistory Helen’s mark8570 Mean7962 Standard deviation8.24.3 (cor. to 3 sig. fig.) History: (cor. to 3 sig. fig.) (b)Since 1.86 0.732, Helen performed better in History than in Chinese.

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Follow-up 21.13 Given that the standard scores of Kelvin’s marks in the Chinese reading and written tests are 1 and –1 respectively, find (a)his mark in the reading test if the mean and the standard deviation of the marks for his whole class are 6.9 and 1.1 respectively; (b)the mean mark of the written test if his mark in the test and the standard deviation of the marks are 5.5 and 2 respectively. Solution: (a) Kelvin’s mark in the reading test (b) The mean mark of the written test 21.4 Applications of Standard Deviation A. Standard Scores

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Follow-up 21.14 The foot sizes of a group of children in a child care centre are normally distributed with a mean of 20 cm and a standard deviation of 2.5 cm. (a)Find the percentage of children having foot sizes between 15 cm and 22.5 cm. (b)Find the percentage of children having foot sizes less than 12.5 cm. Solution: (a) Given 20 and 2.5. (b) B. Normal Distribution 21.4 Applications of Standard Deviation 15 20 2 2.5 47.5% of the children’s foot sizes lie between ( ) cm and cm. 22.5 20 + 2.5 34% of the children’s foot sizes lie between cm and ( ) cm. Percentage of children having foot sizes between 15 cm and 22.5 cm 47.5% + 34% 12.5 20 3 2.5 Percentage of children having foot sizes less than 12.5 cm 50% – 49.85%

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95% of bulbs have a lifetime between ( ) hours and ( ) hours. Follow-up 21.15 The lifetimes of a pack of 1500 bulbs is normally distributed with a mean of 1200 hours and a standard deviation of 50 hours. (a) How many bulbs have a lifetime between 1100 hours and 1300 hours? (b) How many bulbs have a lifetime less than 1150 hours? Solution: B. Normal Distribution 21.4 Applications of Standard Deviation Since 1200 and 50, (a) 1100 1200 2 50 1300 1200 + 2 50 Number of bulbs 1500 95% (b) 1150 1200 50 Percentage of bulbs having a lifetime less than 1150 hours 50% – 34% 16% Number of bulbs 1500 16%

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Follow-up 21.16 Consider the following ten numbers: 0 2 3 5 6 8 8 9 10 12 (a) Find the range, the inter-quartile range and the standard deviation. (b) Find the change in the range, the inter-quartile range and the standard deviation after each of the following changes: (i)1 is added to each datum(ii)each datum is multiplied by 5 (c) Find the range, the inter-quartile range and the standard deviation of the positive numbers. (Give the answers correct to 3 significant figures if necessary.) Solution: 21.5 Effects on the Dispersion with a Change in Data Change in Data C. Multiplying the Whole Set of Data by a Common Constant Common Constant (a) Range Inter-quartile range Standard deviation (cor. to 3 sig. fig.)

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Follow-up 21.16 21.5 Effects on the Dispersion with a Change in Data Change in Data C. Multiplying the Whole Set of Data by a Common Constant Common Constant (b)(i)If 1 is added to each datum, the range, the inter-quartile range and the standard deviation of the new data remain unchanged. (ii)If each datum is multiplied by 5, the range, the inter-quartile range and the standard deviation of the new data are 5 times the original values. Solution: Consider the following ten numbers: 0 2 3 5 6 8 8 9 10 12 (a) Find the range, the inter-quartile range and the standard deviation. (b) Find the change in the range, the inter-quartile range and the standard deviation after each of the following changes: (i)1 is added to each datum(ii)each datum is multiplied by 5 (c) Find the range, the inter-quartile range and the standard deviation of the positive numbers. (Give the answers correct to 3 significant figures if necessary.)

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Follow-up 21.16 21.5 Effects on the Dispersion with a Change in Data Change in Data C. Multiplying the Whole Set of Data by a Common Constant Common Constant Inter-quartile range Standard deviation (cor. to 3 sig. fig.) (c) Range Solution: Consider the following ten numbers: 0 2 3 5 6 8 8 9 10 12 (a) Find the range, the inter-quartile range and the standard deviation. (b) Find the change in the range, the inter-quartile range and the standard deviation after each of the following changes: (i)1 is added to each datum(ii)each datum is multiplied by 5 (c) Find the range, the inter-quartile range and the standard deviation of the positive numbers. (Give the answers correct to 3 significant figures if necessary.)

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Follow-up 21.17 The figure shows the stem-and-leaf diagram of the marks of the girls who took the same English test. (a)Find the inter-quartile range and the standard deviation of the marks. (b)Fiona was also absent, so she got a zero mark. If Fiona’s mark is also considered, find the new inter-quartile range and the new standard deviation of the marks. (Give the answers correct to 3 significant figures if necessary.) Solution: 21.5 Effects on the Dispersion with a Change in Data Change in Data D. Insertion of Zero in the Data Set (a)The inter-quartile range The standard deviation (cor. to 3 sig. fig.) (b)The total number of girls is 14. The new standard deviation (cor. to 3 sig. fig.) The new inter-quartile range Stem (Tens digit)Leaf (Units digit) 00 0 0 0 2 6 7 8 11 22 4 31

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