21 21.2box-and-whisker diagrams 21.3standard deviation chapter summary case study measures of...

21

21.2 Box-and-whisker Diagrams

21.3 Standard Deviation

Chapter Summary

Case Study

Measures of Dispersion

21.1 Range and Inter-quartile Range

21.4 Applications of Standard Deviation

21.5 Effects on the Dispersion with a Change in Data

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 StudyCase 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.

P. 3

Consider two boxes of apples A and B.

21.1 21.1 Range and Inter-quartile RangeRange 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

P. 4

1. For ungrouped data, the range is the difference between the largest value and the smallest value in the set of data.


A. A. RangeRange

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

P. 5

Example 21.1T

Solution:


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. A. RangeRange

100

703$

g 70

1.2$

P. 6

Example 21.2T


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. A. RangeRange

(b) Range (74.5 49.5) kgkg 25

Weight (kg)

50 – 54 55 – 59 60 – 64 65 – 69 70 – 74

Frequency 3 6 8 5 2

P. 7


BB. . Inter-quartile RangeInter-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 25% of data 25% of data 25% of data

Q1 Q2 Q3

Q1 : lower quartile 25% of data less than itQ2 : median 50% of data less than itQ3 : 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 Q3 – Q1

Q1, Q2 and Q3 are also called the first, the second and the third quartiles respectively.

P. 8

Example 21.3TThe 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



60, 62, 62, 65, 68, 69, 70, 72, 78, 78, 80, 81, 84

70

6084 24 2

65621

Q

5.632

80783

Q

795.6379

5.15

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,



(780 140) hours hours 640

Median hours 420

(c) Q1 240 hours,

Q3 600 hours

Inter-quartile range (600 240) hours hours 360

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,



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.

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.

21.21.22 Box-and-whisker DiagramsBox-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.

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) Q1 2 and Q3 4


5.2Maximum value 6 and minimum value 1 Range 16

5

Inter-quartile range 24 2

P. 13

Example 21.7TThe 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:


Girls: 22, 23, 24, 25, 25, 25, 26, 26, 26, 27, 28, 28, 29, 30, 32Boys: 25, 26, 27, 28, 28, 28, 28, 29, 29, 30, 31, 32, 32, 32, 34

For girls, median

,inches 62 and inches 521 Q inches 823 QFor boys,

median

,inches 92 and inches 821 Q inches 323 Q

P. 14

Example 21.7T

Solution:(b) For boys, minimum 25 inches

maximum 34 inches


Refer to the figure on the right.

For girls, minimum 22 inchesmaximum 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.

P. 15

Standard deviation describes how the spread out of the data are around the mean. It is usually denoted by .

21.21.33 Standard DeviationStandard Deviation

Consider a set of ungrouped data x1, x2, …, xn.

AA. . Standard Deviation for Ungrouped DataStandard 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.

n

xxxxxx n22

22

1 )(...)()(

n

xxn

ii

1

2)(

x

_(xi – x) is the deviation of the ith data from the mean.

P. 16

Example 21.8TSix 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)

6

52504946455.49

y

55y

6

1

2)(i

i xx 222 )5.4949()5.4946()5.4945(

25.3025.625.025.025.1225.20 5.69

Standard deviation min 6

5.69 min 40.3 (cor. to 3 sig. fig.)



222 )5.4955()5.4952()5.4950(

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 4.715

7868767065

Standard deviation

5

)4.7178()4.7168()4.7176()4.7170()4.7165( 22222

88.4 (cor. to 3 sig. fig.)



Test 1 Test 2 Test 3 Test 4 Test 5

Chinese 65 70 76 68 78English 72 81 85 90 80

P. 18

Example 21.9T

Solution:






For English, mean 6.815

8090858172

Standard deviation

5

)6.8180()6.8190()6.8185()6.8181()6.8172( 22222



Chinese 65 70 76 68 78English 72 81 85 90 80

P. 19

Example 21.9T

Solution:




(b) For Chinese, standard deviation 4.88



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


Chinese 65 70 76 68 78English 72 81 85 90 80

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 5

)15()20()9()6()15(

yyyyy1y

Standard deviation

5

)115()120()19()16()115(

22

222

yyyy

yyyyyy

5

962

9.13(cor. to 3 sig. fig.)

(b) ∵ y 1 34

The five numbers are 20, 41, 44, 15 and 50. ∴ Range

351550

and Median 41



∴ y 35

P. 21

For a set of grouped data, we have to consider the frequency of each group.


BB. . Standard Deviation for Grouped DataStandard Deviation for Grouped Data

Standard deviation, =

= ,

where fi and xi 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.

n

nn

fff

xxfxxfxxf

...

)(...)()(

21

2222

211

n

ii

n

iii

f

xxf

1

1

2)(

x

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)


BB. . Standard Deviation for Grouped DataStandard Deviation for Grouped Data

Class mark

15.5

25.5

35.5

45.5

55.5

Frequency 4 24 12 8 2

50

)5.55(2)5.45(8)5.35(12)5.25(24)5.15(4 5.31

5

1

2)(i

ii xxf 22 )5.315.25(24)5.315.15(4

4800Standard deviation

50

4800 80.9 (cor. to 3 sig. fig.)

Age 11 – 20

21 – 30

31 – 40

41 – 50

51 – 60

Number of workers

4 24 12 8 2

222 )5.315.55(2)5.315.45(8)5.315.35(12

P. 23

In actual practice, it is quite difficult to calculate the standard deviation if the amount of data is very large.


CC. . Finding Standard Deviation by a CalculatorFinding 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.

P. 24

The following table summarizes the advantages and disadvantages of the three different measures of dispersion.


Measure of dispersion Advantage Disadvantage

1. Range Only 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 range It 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 deviation It takes all the data into account that can show the dispersion of the whole group of data.

Difficult to compute without using a calculator.

P. 25

21.21.4 Applications of Standard 4 Applications of Standard DeviationDeviation

Standard score is used to compare data in relation with the mean and the standard deviation .

AA. . Standard ScoresStandard 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:

xx

z

x

P. 26

Example 21.12TRyan 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.



2.4

8.6066 z


Paper II: 4.6

4.6271z


Paper I

Paper II

Marks 66 71Mean 60.8 62.4

Standard deviation

4.2 6.4

P. 27

Example 21.13TGiven 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



(2.3) 2 + 30 4.25

41.5 1.4 3.5 6.36

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:

BB. . Normal DistributionNormal 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.


Therefore, the normal curve is symmetrical about the mean, i.e., the axis of symmetry for the normal curve is x .x

P. 29

In addition, we can tell the percentage of the data lie within a number of standard deviations from the mean:


1. About 68% of the data lie within one standard deviation from the mean, i.e., – and + .


xx

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.

xx

xx

P. 30

Example 21.14TThe 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.



xx

x x188 180 + 8 x 34% of the players’ heights lie between cm and ( ) cm. xx

Percentage of players whose heights are between 172 cm and 188 cm 34% + 34% %68

(b) 188 180 + 8 x Percentage of players whose heights are greater than 188 cm

50% 34% %16

P. 31

81.5% of children have weights between ( ) kg and ( ) kg.

Example 21.15TThe 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:



Given 56.4 and 6.2.(a) 50.2 56.4 6.2

xx

x68.8 56.4 + 2 6.2 2x

2x Number of children 2000 81.5%

1630

(b) 50.2 56.4 6.2 xPercentage of children who are heavier than 50.2 kg 50% + 34%

84% Number of children 2000 84%

1680

P. 32

21.21.55 Effects on the Dispersion with a Effects on the Dispersion with a Change in DataChange 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.

AA. . Removal of the Largest or Smallest item Removal of the Largest or Smallest item from the Datafrom 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:

P. 33


We have the following conclusion:

BB. . Adding a Common Constant to the Whole Adding a Common Constant to the Whole Set of DataSet 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:

P. 34


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.

CC. . Multiplying the Whole Set of Data by a Multiplying the Whole Set of Data by a Common ConstantCommon Constant

Notes:If the quartiles are not members of the data set, the conclusion will be the same.


P. 35

Example 21.16TConsider 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



312051

5.422

4540 and 5.26

2

282531 QQ

165.265.42 31.9 (cor. to 3 sig. fig.)

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.



Inter-quartile range Standard deviation

31


Range 16

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:

P. 37

Example 21.16T



(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 65.45.03095.9 (cor. to 3 sig. fig.)

Range 85.016



Solution:

5.155.031

P. 38

Example 21.16T



2

2825

2

4035

(b) (iii)The remaining data are 20, 25, 28, 30, 32, 35, 40, 51.

Inter-quartile range

Standard deviation 97.8 (cor. to 3 sig. fig.)

Range



Solution:

312051

11

P. 39



DD. . Insertion of Zero in the Data SetInsertion 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.

P. 40

Example 21.17TThe 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)


DD. . Insertion of Zero in the Data SetInsertion of Zero in the Data Set

235$2

240230$1

Q

425$2

450400$3 Q

Inter-quartile range )235425$(

The standard deviation 9.133$ (cor. to 1 d. p.)

190$

(b) 023$1 Q004$3 Q

Inter-quartile range )023004$(

The standard deviation 58.21$ (cor. to 1 d. p.)

071$

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 Chapter SummarySummary

2. The inter-quartile range is the difference between the upper quartile Q3 and the lower quartile Q1 of a set of data.

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 Chapter SummarySummary21.2 Box-and-whisker Diagrams

P. 43

Standard deviation is the measure of dispersion that describes how spread out a set of data is around the mean value.

Chapter Chapter SummarySummary21.3 Standard Deviation

1. For ungrouped data:n

xxn

ii

1

2)(

2. For grouped data:

n

ii

n

iii

f

xxf

1

1

2)(

Larger values for the range, the inter-quartile range and the standard deviation of the data indicate a larger dispersion and vice versa.

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 Chapter SummarySummary21.4 Applications of Standard Deviation

.

xxz

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.

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 Chapter SummarySummary21.5 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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