PRESENTIATION OF STATISTICAL DATA

At the end of this sub-topic, you will be able to: Present of ungroup data

Construct the frequency table

Present of group data

FUNDAMENTAL STATISTICS CONCEPTS

Statistics is a scientific method of collecting, analysing, presenting and interpreting data to

assist in making decisions.

TERM EXPLAINATION

Population The collection of all elements if interest.

Sample The collection of a few elements from this population.

Data is a collection of information.

Data variable Discrete data –--- data that obtained by counting.

Continuous data ---- data that is obtained by measuring

Ungrouped data ---- data that is not grouped into classes.

Grouped data---- data that is grouped into classes.

PRESENTATION OF UNGROUP DATA

A. Pictograms

A pictogram represents data by using pictures or symbols to show the frequency of the

information given.

The symbols used may represent one, or more than one unit of given data.

Example: - represent one apple

- represent five persons

Constructing pictograms

Either horizontally or vertically.

Example 1

Number of cars sold by Syarikat ABC shown by table 1 below;

Table 1

Month Number or cars sold

May

June

July

- represent 50 cars

B. Bar charts A bar charts represents data using horizontal or vertical bars.

The width of each bars is the same throughout and the height of each bar is proportional to the

frequency.

Constructing bar charts

There are three types of bar charts:

a. Vertical bar chart

b. Horizontal bar chart

c. Double bar chart (vertical or horizontal)

Example 2

Vertical bar chart Horizontal bar chart

C. Line graphs

A line graph represent data obtained over an interval of time. The points represents the

data are joint by straight lines.

The horizontal axis represents the intervals of time.

Example 3

D. Pie charts

A pie chart represent data in relative quantities by using the areas of sectors in a circle. So, to

construct a pie chart, we relate the quantities of the data to the size of angles of the sectors.

Quantity represented by the sector

Angle of the sectors = X 3600

total quantity

Quantity represented by the sector

% of the sectors = X100 total quantity

To construct a pie chart:

Step 1: Calculate the angle of each sector.

Step 2: Calculate the % of each sector

Step 3: Draw suitable circle.

Step 4: Divide the circle into different sectors according to their angles.

Step 5: Write the title and the labels for the groups of data.

Example 4

Represent the data in table 1 below using pie chart:

Population of pupils in5 primary school

Table 1

School Number of Pupils

A 420

B 400

C 380

D 310

E 290

Solution :

School Number of Pupils Size of angle % of sector

A 420 840 23.3

B 400 800 22.2

C 380 760 21.1

D 310 620 17.2

E 290 580 16.1

1,800

Number of Pupils

Population of pupils in 5 primary school

E

16%

D

17%

A 24%

B

22%

C

21%

SUMMARRY

Advantages and disadvantages of some statistical representations.

Representation of Advantages Disadvantages

data

Pictogram - Data is represent in an attractive - Not accurate manner - Difficult and time

- Easy to understand consuming to draw the

figures.

Bar Chart - Easy to construct - Does not show - Show the exact quantities of comparisons between the

each data category categories of data.

- Two or more type of data can be

displayed simultaneously.

Line graph - Show the rising or a falling - Does not show complete trend in a set of data over a comparisons among the

period of time. data.

- Useful to predict trends.

Pie Chart - Shows clearly the difference in - Long calculations are magnitude between the needed.

categories of data. - Not suitable if too many categories of data

involved.

- Actual quantities are not

displayed.

EXERCISE 1.1 A

1. The monthly incomes for 75 workers in a firm were tabled as table 2 below:

Table 2

Wages (RM) 400-499 500-599 600-699 700-799 800-899 No. of Male 5 9 10 9 12

workers

Female 8 6 15 9 10

Referring to the above, draw a vertical bar chart.

2. The table 3 below shows the number of houses (in thousand units) complete within 5

months in the year 2008.

Table 3

Month Unit of house (in

thousand units)

January 13.5

February 14.6

March 15.3

April 19.0

May 17.3

From the above data, draw a line graph.

3. Fauzi salary is RM 1200 per month. His monthly expenses is shown in the table 4 below and the

balance is saved in the bank.

Table 4

Expenses Total

House Rental RM 250

Food RM 200

Clothes RM 120

Entertainment RM 130

Show Mr. Fauzi’s expenditure and saving in a Pie Chart.

FREQUENCY TABLE

Frequency is the number of times a value or a piece of information, appears in a given set of

data.

A frequency table indicates the frequency of given data.

Example 5

The marks of a group of students in a mathematics test are stated as below:

40 28 67 30 58 74 49 68 56 71

42 80 50 84 63 44 82 56 34 47 62 71 41 52 25 64 38 74 66 54

61 45 46 55 32 40 62 48 51 75

50 53 69 55 76 54 59 67 87 90

Build a frequency table by using 6 class intervals.

(normally : 5 – 15 classes)

Steps :

1. Find the largest value : 90 2. Find the smallest value : 25 3. Calculate the range of class interval :

largest value – smallest value = 90 – 25

= 65 4. Calculate the size (c) of class interval :

c =

range

no of classes

= 65

= 10.83 => 11 6

5. Frequency table :

Mark Tally Frequency (f)

25 – 35 1111 5

36 – 46 1111 111 8

47 – 57 1111 1111 1111 14

58 – 68 1111 1111 1 11

69 – 79 1111 11 7

80 – 90 1111 5 Frequency table of a group of students in a mathematics test

EXERCISE 1.1 B

1. The data below shows the distances (km) from the school to the homes of 30 teachers in

SK Taman Jaya. Construct a frequency table for the above set of data such beginning

from the first class of 1.1 – 2.0

1.5 2.8 6.5 4.9 4.8 7.1 5.5 2.9 3.2 3.8

1.4 3.7 4.1 2.8 7.3 4.0 3.2 5.2 3.5 8.0

Upper Limit,4.7 Lower4.3 Limit,6.2 Upper1.4 Boundary,2.21Lower.8

1Boundary.36.0 (L),8Midpoint.06.0 (Class Mark) and Cumulative Frequency (F)

2. Based on the data below, built a frequency table by using 7 class intervals.

165 168 165 180 164 179 163 179 182 189

171 169 177 178 173 175 174 175 186 176

173 170 165 176 171 173 162 176 163 179

189 179 168 170 175 169 179 198 165 165

184 171 178 163 178 174 183 169 190 177

Upper Limit, Lower Limit, Upper Boundary, Lower Boundary (L), Midpoint

(Class Mark) and Cumulative Frequency (F)

Lower Boundary (L):

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.

Upper Boundary (U):

Upper Boundary of each class refers to the middle value between the upper limit of the class

and the lower limit of the previous class.

Size of a Class Interval (c) :

The difference between the upper boundary and the lower boundary of the class.

Midpoint / Class Mark (x):

Midpoint / Class Mark of each class refers to the middle value between the lower limit of the

class and the upper limit of the each class.

Example 6

Class lower upper lower upper

Midpoint ( x ) limit limit boundary boundary

25-35 25 35 24.5 35.5 30

36 46 36 46 35.5 46.5 41

47 57 47 57 46.5 57.5 52

58 68 58 68 57.5 68.5 63

69 79 69 79 68.5 79.5 74

80 90 80 90 79.5 90.5 85

Cumulative Frequency (F):

Cumulative frequency for a given class interval is the sum of the frequency in that class

interval and the frequencies of all the class intervals before it.

Example 7

Class f F

14 24 0 0

25 35 5 5

36 46 8 13

47 57 14 27

58 68 11 38

69 79 8 46

EXERCISE 1.1 C

Find the Lower boundary, Upper boundary , Midpoint (Class Mark) and Cumulative Frequency

(F) in the table below.

Mass

Frequency

Lower Upper

Midpoint F (kg) boundary boundary

1-3 2

4-6 4

7-9 6

10-12 8

13-15 10

16-18 12

PRESENTATION OF GROUP DATA

A. Histograms

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.

Frequency

12

8

4

9.5 14.5 19.5 24.5 29.5 34.5 39.5

*** No space between two adjacent rectangles.

Steps to drawing a histogram:

Time (hours)

1. Find the lower boundary and the upper boundary of each class interval.

2. On the horizontal axis, mark the class boundaries.

3. On the vertical axis, mark the frequencies.

4. 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.

Example 8

From the given frequency table, draw a histogram

Class f

25 35 5

36 46 8

47 57 14

58 68 11

69 79 7

80 90 5

Frequency table of a group of students in a mathematics test

Solution :

Class f Lb Up

25 35 5 24.5 35.5

36 46 8 35.5 46.5

47 57 14 46.5 57.5

58 68 11 57.5 68.5

69 79 7 68.5 79.5

80 90 5 79.5 90.5

Histogram of a group of students in a mathematics test

f

14

12

10

8

6

4

2

``

24.5 35.5 46.5 57.5 68.5 79.5 90.5 boundaries

In the above histogram, the x-axis is marked with the class boundaries. Other methods used

to mark the x-axis are by using class intervals or midpoints.

B. Frequency Polygons

i. Drawing from Histograms

A frequency polygon is a graph where the consecutive midpoints of the upper base of

the rectangles in a histogram are joined using straight lines.

The frequency polygon is closed by joining both side of the graph to the base on the x-axis.

Example 9

ii. Drawing Frequency Polygons from Frequency Tables

Example 10

From the given frequency table, draw a frequency polygon

Class f

30 39 10

40 49 30

50 59 20

60 69 10

70 79 35

80 89 15

Solution :

Step 1:

Add classes with zero frequencies, one before the first class and another after the last class.

Then find the midpoint of each class interval.

Class f x

20 29 0 24.5

30 39 10 34.5

40 49 30 44.5

50 59 20 54.5

60 69 10 64.5

70 79 35 74.5

80 89 15 84.5

90 99 0 94.5

Step 2:

i. On the horizontal axis, mark the class boundaries or midpoint.

ii. On vertical axis, mark the frequencies.

iii. The midpoint and frequency of each class interval represent a point on a graph.

Plot each point and join the points with a straight line.

Frequency polygon

f

35

30

25

20

15

10

5

24.5 34.5 44.5 54.5

64.5 74.5 84.5

94.5 boundaries

c. An Ogive

An ogive is a graphical representation of a cumulative frequency.

Two type of ogive or cumulative frequency curves: “less than” or “more than”.

Cumulative frequency for a given class interval is the sum of the frequency in that class

interval and the frequencies of all the class intervals before it.

Less than

ogive

Example 11

From the given frequency table, draw a less than an ogive

Class f

5 9 4

10 14 18

15 19 28

20 24 16

25 29 8

30 34 4

35 39 2

Frequency table of the height of 80 students in a class

Solution :

Drawing “less than” an ogive :

a. Add one class with zero frequency before the first class.

b. Determine the upper boundary of each class.

c. Determine the cumulative frequency of each class.

Class f Upper F

boundaries

0 4.5 0

5 9 4 9.5 4

10 14 18 14.5 22

15 19 28 19.5 50

20 24 16 24.5 66

25 29 8 29.5 74

30 34 4 34.5 78

35 39 2 39.5 80

d. Choose suitable scales for the x-axis to represent class interval and the y-axis to

represent cumulative frequency. e. Plot the point (upper boundaries) on graph. Then joint the points with a smooth

curve.

Ogive of the height of 80 students in a class

EXERCISE 1.1 D

1. Data below show the weight of 40 students :

No. of

Weight students

60 64 6

65 69 10

70 74 14

75 79 12

80 84 8

a. Draw a histogram b. From the histogram draw a polygon frequency

2. Data below show, draw a less than ogive:

No. of

Distance students

35 39 1 40 44 4 45 49 15 50 54 11 55 59 6 60 64 2 65 69 1