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Statistics - Mathematics Form 2

Background
Data refer to collection information obtained through experience or observation.
For example: the marks of a test of a class of 15 students are as follows
25 22 25 30 28
40 30 40 34 28
26 36 42 37 28
When data is not organized, it is referred to as raw data. The data above can be organized in order as follows.
22,25,25,26,28,28,2830,30,34,36,37,40,40,42.

When the data is organized in such a way, it becomes easy to make comparisons and interpretation. It can be used to answer questions such as:

  1. what is the highest and lowest marks?
  2. what mark was scored by most students?
  3. what is the difference between the highest and the lowest marks?

from the data above, some values appear more than once: the number of times a value appears is referred to as its frequency.
To show frequency (f)  the data above can be organized further as follows.


MARKS

TALLY

FREQUENCY

22

/

1

25

//

2

26

/

1

28

///

3

30

//

2

34

/

1

36

/

1

37

/

1

40

//

2

42

/

1

The number of times each value appears is indicated by tarrying in the tarry column. The values in the marks column are written in ascending order. Such a table that shows given values and their frequencies is called a frequency distribution table.

Example
The data below represents the marks obtained in a mathematics test.
61 70 40 43 50
40 30 75 80 75
61 50 30 40 80
61 50 30 40 43
50 61 30 40 30
80 30 50 30 50
Make a frequency distribution table and  answer the following questions .

  1. How many students sat for the test?
  2. What were the lowest and highest marks?

What is the difference between the highest and lowest marks obtained?
Solution
The following is a frequency distribution table for the data


Marks

Tally

F

30

///// //

7

40 

/////

5

43

//

2

50

///// /

6

61

////

4

70

/

1

75

//

2

80

///

3

The total number of students is found by adding the frequencies =40
Highest mark =80
Lowest mark =30
Difference between highest and lowest =80-30 =50
Grouped data
Data can be arranged in groups or classes. The following table shows grouped data of marked obtained by 40 students in a mathematics rest.


Marks

F

10-19    

5

20-29  

10

30-39   

12

40-49    

7

50-59  

6

From the table 5students scored between 10 and 19 marks,10 students scored 20 and 29 marks and so on. The table shows grouped data.
In grouped data

  1. class limits are the end values that specify a class interval.in the grouped data above ,10-19,20-29,50-59 are the classes defund by their respective class limits.
  2. A class boundary is obtained by calculating the average of the upper and the lower limits of corresponding classes. In the grouped data above ,class boundaries are obtained as follows

class limits   

boundaries

Lower

Upper

10-19

?

1

20-29   

2

3

30-39    

4

5

40-49

6

7

50-59    

8

?


(highlight the diagonals whose average is worked out to get the class boundaries)
Therefore, class boundaries
?    -     19.5
19.5     -     29.5
29.5     -     39.5
39.5     -     49.5
49.5     -     ?
To get the missing class boundaries of the first and last classes you assume there us a class above the first class and a class below the last class and then calculate the class boundaries as follows:
9      The class boundary is 10
10 - 19
Similarly for the Last  work out the class boundary as follows:

60 -69

  10 59      11  =59.5

Or these two values can be obtained by observation i.e.

Example
Using the data below;

  1. find the class boundaries of each class
  2. Calculate the class intervals.

Class
9    -  19
20  -  25
26  -  29
30  -  49
Solution
The class boundaries are worked out as follows:
Therefore class boundaries are
8.5    -     19.5
19.5  -     25.5
25.5  -     29.5
29.5  -     49.5
The clas interval is obtained by getting the difference beteen the upprt class boundary and lower class boundary.

Class boundary      

Class interval

8.5    -   19.5         

19.5 8.5 = 11

19.5  -   25.5    

25.5 19.5 = 6

25  -   29.5 

29.5 25.5= 4

29.5  -   49.5

49.5 29.5 = 20

    

LESSON DEVELOPMENT
Introduction
Grouped data can be represented in several ways which include pie chart, pictograms, line graphs, frequency polygon, bar chart abd histogram. In this topic, we are going to work at a bar chart and a histograms.

  1. Bar chart

A bar chart (also refferd to a a bar graph) in a chart with rectangular bars whose lengths are propositional to the values that they represents. For grouped data, frequency is plotted against class limits or vice versa.

Example 1
1) Represents the data given below using a bar chart.
English     72%
Mathematics  60%
Kiswahili      90%
Geography     80%
Chemistry   65%
Physics     52%
Solution
There are two types of bar charts namely: horizontal and vertical bar charts
Horizontal bar chart
The following is horizontal bar chart representing the given data.
1

(Show animation of the drawing of the bar charts by drawing one bar at a time using different colours).
Note that the marks are indicated on the x-axis while the subjects appear on the y-axis.

Vertical bar chart
In this case the marks are plotted on the y-axis. The marks represents frequency.
2
Note that:

  1. The bars are of equal width
  2. The distance between the bars is the same.
  3. the length of the bars represents the frequency

Example 2
The table below shows marks obtained in an English examination


Classes

frequency(f)

10-19

5

20-29

11

30-39

15

40-49

15

50-59

10

60-69

18

70-79

8

  1. draw a bar chart to represent the data
  2. How many pass mark was 40% how many students failed the exam?
  3.  

Solution
The vertical bar chart is obtained by plotting the frequency on the vertical axis and the classes on the horizontal axis as follows.
3

  1. the number of students  who scored 50marks and above is worked out as follows

50   -    59  10 students
60   -    69    8   
70   -    79   8 
Therefore total of students =10 + 18 + 8 = 36

  1. the students who failed the examinations scored 39% and below The number of students is worked out as follows:

10-19 5students
20-29 11  
30-39 15   
Therefore total number of students =5 + 11 + 5 = 31


LESSON DEVELOPMENT
Introduction
Grouped data can be represented in several ways which include pie chart, pictograms, line graphs, frequency polygon, bar chart abd histogram. In this topic, we are going to work at a bar chart and a histograms.

  1. Bar chart

A bar chart (also refferd to a a bar graph) in a chart with rectangular bars whose lengths are propositional to the values that they represents. For grouped data, frequency is plotted against class limits or vice versa.

Example 1
1) Represents the data given below using a bar chart.
English     72%
Mathematics  60%
Kiswahili      90%
Geography     80%
Chemistry   65%
Physics     52%
Solution
There are two types of bar charts namely: horizontal and vertical bar charts
Horizontal bar chart
The following is horizontal bar chart representing the given data.
1

(Show animation of the drawing of the bar charts by drawing one bar at a time using different colours).
Note that the marks are indicated on the x-axis while the subjects appear on the y-axis.

Vertical bar chart
In this case the marks are plotted on the y-axis. The marks represents frequency.
2
Note that:

  1. The bars are of equal width
  2. The distance between the bars is the same.
  3. the length of the bars represents the frequency

Example 2
The table below shows marks obtained in an English examination


Classes

frequency(f)

10-19

5

20-29

11

30-39

15

40-49

15

50-59

10

60-69

18

70-79

8

  1. draw a bar chart to represent the data
  2. How many pass mark was 40% how many students failed the exam?
  3.  

Solution
The vertical bar chart is obtained by plotting the frequency on the vertical axis and the classes on the horizontal axis as follows.
3

  1. the number of students  who scored 50marks and above is worked out as follows

50   -    59  10 students
60   -    69    8   
70   -    79   8 
Therefore total of students =10 + 18 + 8 = 36

  1. the students who failed the examinations scored 39% and below The number of students is worked out as follows:

10-19 5students
20-29 11  
30-39 15   
Therefore total number of students =5 + 11 + 5 = 31


   
Objective:
by the end of the lesson you should be able to represent data in form of bar chart and histogram.

Statistics is a branch of mathematics that deals with collection, organization, presentation and interpretation of data. It is important in decision making.

   
Objective:
by the end of the lesson you should be able to represent data in form of bar chart and histogram.

Division of numbers using logarithms


Example
Evaluate 48.12 6.43 using logarithm tables.

Solution
Arrange the walking as

Number

Standard form

log

48.12
6.43


4.812 x 101
6.43 x 10o


1.6823   -
0.88082

0.7741

To divide the two numbers we subtract their logarithms
Then find the antilogarithm of 0.7741 which is 5.944.
Note that the characteristic is 0. Therefore the answer becomes
      = 5.944 x 10o = 5.944 x 1
= 5.944

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