Chapter 6

Created By: Mohd Said B Tegoh (Guru Cemerlang Matematik)

CHAPTER 6 : STATISTICS CHAPTER 6 : STATISTICSCLASS INTERVALData obtained by measurement of a quantity can be grouped into several classes. The range of each class is called the class interval

CLASS INTERVAL16 18 33 25 24 20 34 26 34 29 37 27 26 31 35 28 30 30 35 21 40 40 38 23 35 34 39 31 30 36 41 31 26 35 25 38 33 32 25 33

When choosing a suitable class interval, it is suggested that the number of classes are in the range of 5 to 12. The best range of class intervals can be determined as follows.

Lowest Number

Highest Number

16 18 33 25

24 20 34 26

34 29 37 27

26 31 35 28

30 30 35 21

40 40 38 23

35 34 39 31

30 36 41 31

26 35 25 38

33 32 25 33

Range of class intervals = Highest data value Lowest data value Number of classes Range of class intervals = 41 16 4 6 The suitable classes are 15-19, 20-24, 25-29, 30-34, 36-39, 40-44

Lowest Number

Highest Number

28 19 35 44 28

35 27 39 14 38

22 32 34 22 26

40 32 38 39 22

29 37 45 33 17

30 33 21 31 26

32 35 34 28 24

23 40 30 27 20

Range of class intervals = 45 14 4 7 The suitable classes are 11-15, 16-20, 21-25, 26-30, 31-35, 36-40, 41-45

LOWER LIMIT AND UPPER LIMITClass 15-19 15 19 20-24 25-29 30-34 35-39 40-44 Lower Limit 15 20 25 30 35 40 Upper Limit 19 24 29 34 39 44

LOWER B0UNDARY AND UPPER BOUNDARYThe lower boundary of each class refers to the middle value between the lower limit of the class and the upper limit of the previous class The upper boundary of each class refers to the middle value between the upper limit of the class and upper limit of the following class

Class 15-19 20-24 25-29 30-34 35-39 40-44

Lower Boundary 14.5 19.5 24.5 29.5 34.5 39.5

Upper Boundary 19.5 24.5 29.5 34.5 39.5 44.5

15

19

20

24

25

29

Lower Boundary = 19 + 20 2 = 19.5

Upper Boundary = 24 + 25 2 = 24.5

SIZE OF CLASS INTERVAL

The size of class interval is the difference between the upper boundary and the lower boundary of the class

Class 15-19 20-24 25-29 30-34 35-39 40-44

Lower Boundary 14.5 19.5 24.5 29.5 34.5 39.5

Upper Boundary 19.5 24.5 29.5 34.5 39.5 44.5

Size of class intervals = Upper boundary Lower boundary = 19.5 14.5 =5

FREQUENCY TABLE16 18 33 25 24 20 34 26 34 29 37 27 26 31 35 28 30 30 35 21 40 40 38 23 35 34 39 31 30 36 41 31 26 35 25 38 33 32 25 33

Class 15-19 20-24 25-29 30-34 35-39 40-44

Tally Marks

Frequency 2 4 9 13 9 3

28 19 35 44 28 Class 11-15 16-20 21-25 26-30 31-35 36-40 41-45

35 27 39 14 38

22 32 34 22 26

40 32 38 39 22

29 37 45 33 17

30 33 21 31 26

32 35 34 28 24

23 40 30 27 20

Tally Marks

Frequency 1 3 6 10 11 7 2

MODAL CLASSClass 15-19 20-24 25-29 Modal class 30-34 25-29 40-44 Frequency 2 4 9 13 Highest frequency 9 3

Modal class is the class which has the highest frequency Modal class = 30-34

MID POINTClass Frequency 15 19 15-19 2 20-24 25-29 30-34 35-39 40-44 4 9 13 9 3 Mid Point 17 22 27 32 37 4215 + 19 = 17 2

Mid Point of class = (lower limit + upper limit) 2

MEAN OF GROUPED DATA

Class 15-19 20-24 25-29 30-34 35-39 40-44

Frequency 2 4 9 13 9 3

Mid Point 17 22 27 32 37 42

Mean = Sum ( mid point x class frequency) Total frequency Mean = 17x2 + 22x4 + 27x9 + 32x13 + 37x9 + 42x3 40 = 31

HISTOGRAM A histogram with class intervals of equal size represents the frequency of each class interval with rectangles of the same width but different heights that is proportional to its frequency. The difference between a histogram and a bar chart is that there is no space between two adjacent rectangles in the histogram. When the size of the class intervals are different, the area of each rectangle is taken into account because the area of rectangle is proportional to the class frequency.

DRAWING HISTOGRAM Find the lower boundary and the upper boundary of each class interval. On the horizontal axis, mark the class boundaries. On the vertical axis, mark the frequencies. Draw rectangles to represent each class with its width being equal to the size of the class interval and its height proportional to its frequency.

DRAWING HISTOGRAMClass 15-19 20-24 25-29 30-34 35-39 40-44 Frequency 2 4 9 13 9 3 Lower Boundary 14.5 19.5 24.5 29.5 34.5 39.5 Upper Boundary 19.5 24.5 29.5 34.5 39.5 44.5

HISTOGRAM CLASS BOUNDARIESFrequency

14 12 10 8 6 4 214.5 19.5 24.5 29.5 34.5 39.5 44.5 Body masses (kg)

HISTOGRAM MID POINTSFrequency

14 12 10 8 6 4 2 17 22 27 32 37 42Body masses (kg)

HISTOGRAM CLASS INTERVALSFrequency

14 12 10 8 6 4 215-19 20-24 25-29 30-34 35-39 40-44 Body masses (kg)

INTERPRETING HISTOGRAMFrequency

14 12 10 8 6 4 214.5 19.5 24.5 29.5 34.5 39.5

Modal class = 30-34 Number of children who have a body mass of not less 35 kg = 12

9

344.5 Body masses (kg)

FREQUENCY POLYGON

FREQUENCY POLYGON

FREQUENCY POLYGON A frequency polygon is a graph where the consecutive midpoints of the upper base of the rectangles in a histogram are joined using straight line. The frequency polygon is then closed by joining both sides of the graph to the base on x-axis

DRAWING FREQUENCY POLYGON Add classes with zero frequencies, one before the first class and another after the last class. Then find the midpoint of each class. On the horizontal axis, mark the class boundaries. On the vertical axis, mark the frequencies. The midpoint and frequency of each class interval represent a point on the graph. Plot each point and join the points with a straight line.

Class Freq. 10-14 15-19 20-24 25-29 30-34 35-39 40-44 45-49 0 2 4 9 13 9 3 0

Lower Upper Boundary Boundary 9.5 14.5 14.5 19.5 19.5 24.5 24.5 29.5 29.5 34.5 34.5 39.5 39.5 44.5 44.5 49.5

Midpoint 12 17 22 27 32 37 42 47

FREQUENCY POLYGONFrequency

14 12 10 8 6 4 2 x x14.5 19.5 24.5 29.5

x x x

x

x34.5 39.5

x 44.5

Body masses (kg)

FREQUENCY POLYGONFrequency

14 12 10 8 6 4 2 x x 12 17 22 x x

x x

x 27 32 37 42 x 47Body masses (kg)

FREQUENCY POLYGONFrequency

14 12 10 8 6 4 2 x14.5 19.5 24.5 29.5

x x x

x

x34.5 39.5 44.5 Body masses (kg)

CUMULATIVE FREQUENCY The cumulative frequency for a given class interval is the sum of the frequency in that class interval and the frequencies of all class intervals before it. The cumulative frequency table for grouped data is shown in table (a)

CUMULATIVE FREQUENCYWeight 0.0-0.9 1.0-1.9 2.0-2.9 3.0-3.9 4.0-4.9 5.0-5.9 6.0-6.9 7.0-7.9 Frequency Cumulative Frequency 0 4 10 26 24 17 12 3 0+4 4 + 10 14 + 26 40 + 24 64 + 17 81 + 12 93 + 3 0 4 14 40 64 81 93 96

Table (a)

OGIVE

OGIVE

OGIVE

OGIVE An ogive is a graphical representation of a cumulative frequency. To draw an ogive; Add one class with zero frequency before the first class. Determine the upper boundary of each class Choose suitable scales for the x-axis to represent class intervals and the y-axis to represent cumulative frequency Plot the points (upper boundaries, cumulative frequencies) on the graph. Join the points with smooth curve.

CONSTRUCTING A CUMULATIVE TABLE BASED ON THE HISTOGRAM Number of parcels 30 26

24 17

20 10 4

12

10

3 Weights (kg)

1.0 2.0 3.0 4.0 5.0 6.0 7.0 1.9 2.9 3.9 4.9 5.9 6.9 7.9

CONSTRUCTING A CUMULATIVE TABLE BASED ON THE HISTOGRAMWeight 0.0-0.9 1.0-1.9 2.0-2.9 3.0-3.9 4.0-4.9 5.0-5.9 6.0-6.9 7.0-7.9 Frequency 0 4 10 26 24 17 12 3 0+4 4 + 10 14 + 26 40 + 24 64 + 17 81 + 12 93 + 3 Cumulative Frequency 0 4 14 40 64 81 93 96 Upper Boundary 0.95 1.95 2.95 3.95 4.95 5.95 6.95 7.950.9 + 1.0 2 1.9 + 2.0 2

100 90 80 70 60 50 40 30 20 10 0

x x x

x

x

x0.95 1.95 2.95 3.95 4.95 5.95 6.95 7.95

x

x

MEASURES OF DISPERSION A measure of variation which is used to indicate the extent of dispersion of a given set of data is called the measure of dispersion. Two basic types of measures of dispersion are the range and the interquartile range. The range is the measure of dispersion which refers to the difference between the highest value and the lowest value of the data.

When a set of data is arranged in order of magnitude The value of the first quartile is such that of the total number of data have values less than this value. The value of the third quartile is such that of the total number of data have values less than this value. The interquartile range refers to the difference between the third quartile and the first quartile.

Cumulative Frequancy

100 90 Third Quartile 80 (3/4 x 100 = 75) 70 60 Median (1/2 x 100 = 50) 50 40 First Quartile 30 (1/4 x 100 = 25) 20 10 0x 45.5

x x x

x

First Quartile = 57.5 Median = 61.5 Third Quartile = 65.0

x

x x50.5 55.5 60.5 65.5 70.5 75.5 80.5 Mass(g)

100 90 80 72 70 60 50 48 40 30 24 20 10 0

x x x

x

The third quartile = 5.35 The first quartile = 3.35 Median = 4.15

x

x0.95 1.95 2.95 3.95 4.95 5.95 6.95 7.95

x

x

3.35

4.15 5.35

100 90 83 80 72 70 60 50 40 30 24 20 10 0

x x x

x

The inter quartile range = 5.35 - 3.35 = 2.00 The number of parcels that weigh more than 6.05 kg = 96 - 83 = 13

x

x6.05 0.95 1.95 2.95 3.95 4.95 5.95 6.95 7.95

x

x

The data in Diagram 1 shows the donations, in RM, of 40 people to a charity fund. 12 45 22 38 48 35 42 23 27 56 38 47 43 16 21 35 33 24 47 40 40 49 32 28 18 32 46 35 29 31 17 43 41 53 37 58 30 13 44 26

DIAGRAM 1 a) Based on the data in Diagram 1 and using a class interval of RM10, complete Table 1 in the answer space. b) By using a scale of 2 cm to RM10 on the x-axis and 2 cm to 5 persons on the y-axis, draw an ogive based on the data. c) From your ogive in b), i) find the third quartile, ii) explain briefly the meaning of the third quartile.

12 45 22 38 Donation (RM) 0-9 10 - 19 20 - 29 30 - 39 40 - 49 50 - 59

48 35 42 23

27 56 38 47

43 16 21 35

33 24 47 40

40 49 32 28

18 32 46 35

29 31 17 43

41 53 37 58

30 13 44 26 Upper Boundary 9.5 19.5 29.5 39.5 49.5 59.5

Tally Marks

Freq. 0 5 8 11 13 3

Cumulative frequency 0 5 13 24 37 40

Cumulative Frequancy

40 35 30 25 20 15 10 5 0 9.5

x x

x

x x x19.5 29.5 39.5 49.5 59.5 Donation (RM)

Cumulative Frequancy

Third quartile = 43.5040 35 30 25 20 15 10 5 0 9.5

x xThird Quartile (3/4 x 40 = 30)

x

There are 10 persons donated RM 43.50 or more There are 30 persons donated less than RM 43.50Donation (RM)

x x x19.5 29.5 39.5 49.5 59.5

The data below shows the payment, in RM, of 40 drivers at a toll booth. 38 32 21 24 30 34 25 42 18 23 41 32 36 26 37 25 27 30 35 48 19 42 33 47 39 32 23 37 22 34 28 18 43 38 41 42 46 25 33 27

(a) Based on the data, complete Table 14 in the answer space, (b) Based on Table 14 (a), calculate the estimated mean of the toll paid by a driver. (c) By using the scale of 2 cm to RM 5, on the horizontal axis and 2 cm to 1 pupil on the vertical axis, draw a frequency polygon for the data. (d) Based on the frequency polygon in 14 (c), state the number of drivers who paid more than RM 34 for the toll.

Answer (a) Class Interval 15-19

Midpoint 17

Frequency

Table 14 (b) (c) (d)

38 32 21 24 30 (a) Class Interval 15-19 20-24 25-29 30-34 35-39 40-44 45-49

34 25 42 18 23

41 32 36 26 37

25 27 30 35 48

19 42 33 47 39

32 23 37 22 34

28 18 43 38 41

42 46 25 33 27

solution

Midpoint 17 22 27 32 37 42 47

Frequency

Tally Marks

3 5 7 9 7 6 3

P1 P1 P2

Class Interval 15-19 20-24 25-29 30-34 35-39 40-44 45-49

Midpoint 17 22 27 32 37 42 47

Frequency (b) Estimated mean of the toll paid by a driver?

3 5 7 9 7 6 3

17x3 22x5 27x7 32x9 37x7 42x6 47x3 40 ! 32.25

K2 N1

(c)

P19 8 7 6 5 4 3 2 1 0x

x x x x x x xK2 N1

12 17 22 27 32 3

42 47 52

x

9 8 7 6 5 4 3 2 1 0x

x x x x x x x

(d) Number of drivers who paid more than RM 34 for the toll?

7 6 3 ! 16

K1

12 17 22 27 32 37 42 47 52

x