# DATA PROCESSING

**WEEK 2**

**SUBJECT: FURTHER MATHEMATICS**

**CLASS: S.S 1 THIRD TERM**

**TOPIC: Data presentation**

CONTENT: Graphs and charts:

- Pictogram
- frequency distribution
- Bar chart
- Pie chart
- Histogram
- Frequency polygon
- Cumulative frequency.

**Data presentation**

**Statistics is a general word for the presentation, study and interpretation of information (usually numerical information or data). Accurate and well-presented statistics can help decision makers in government, commerce and industry to make sound choices in relation to the distribution of resources, population needs and market trends. Good data presentation is a necessary condition for good statistics. Statistical data when collected are raw materials that needs to be classified and then presented in different ways **

**Pictogram**

**Pictures or caricatures are often used to relay information. For example, if DLHS Lagos campus purchases 50 bags of rice in a term while DLHS Abuja campus purchases 30 bags, then each campus’ purchase of rice can be shown in a diagram by drawing 5 bags for Lagos campus and 3 bags for Abuja campus.**

** (In Tens) Lagos campus **

**(In Tens ) Abuja campus **

Comparison of this nature is often adopted by geographers; and serves as a good example of comparative statistics.

**Frequency Distribution**

The most important form of tabulation is the frequency distribution. For easy access to information, data are normally presented using frequency tables. This table marches each data with the number of times it appeared. The number of times a particular item occurs is called the frequency in any distribution.

**Example 1: **The numbers 2, 2, 1, 1, 2, 3, 3, 4, 4, 3, 5

Item |
Frequency |

1 | 1 |

2 | 3 |

3 | 3 |

4 | 2 |

5 | 1 |

**Example 2**

The scores of 30 students in a mathematics test marked out of 10 are as follows:

3 | 4 | 6 | 3 | 4 | 3 | 5 | 6 | 7 | 6 |

8 | 9 | 5 | 9 | 10 | 7 | 8 | 2 | 6 | 5 |

4 | 10 | 5 | 6 | 7 | 3 | 8 | 9 | 4 | 2 |

Prepare a frequency table for the distribution

Solution

Marks x |
Tally |
Frequency |

2 | 11 | 2 |

3 | 1111 | 4 |

4 | 1111 | 4 |

5 | 1111 | 4 |

6 | 1111 | 5 |

7 | 111 | 3 |

8 | 111 | 3 |

9 | 111 | 3 |

10 | 11 | 2 |

Total |
30 |

**CLASS ACTIVITY**

Prepare a frequency table for each of the following sets of data:

- A die is rolled 50 times and the following numbers are obtained.

2 | 3 | 4 | 4 | 2 | 1 | 3 | 2 | 6 | 5 |

3 | 2 | 1 | 1 | 2 | 5 | 2 | 1 | 4 | 4 |

6 | 5 | 6 | 1 | 6 | 5 | 4 | 5 | 4 | 3 |

6 | 5 | 5 | 3 | 5 | 2 | 1 | 4 | 5 | 2 |

4 | 5 | 4 | 6 | 3 | 1 | 5 | 6 | 6 | 5 |

- The ages of 32 students in class 2 of a Junior Secondary School are:

11 12 11 12 12 14 14 13

15 13 12 13 13 13 13 12

14 14 13 15 14 11 12 14

12 15 14 16 14 14 14 15

**Bar Chart**

The bar chart consists of rectangles with bases on the x-axis and the areas of the rectangles are proportional to the corresponding class frequencies. Each rectangle is separated from each adjacent rectangle by equal intervals. Students choose the scale used but it must be reasonable.[mediator_tech]

**Example 1**

The table below shows the number of babies born to a number of women within a given age range to a number of women within a given age range.

Women ages | 24 | 25 | 26 | 27 | 28 |

No of babies | 1 | 4 | 2 | 3 | 2 |

Draw a bar chart to illustrate the distribution below.

Example 2:

The number of bottles of soft drinks sold in a restaurant one evening is given by the data in the table below.

Type of soft drink |
No of bottles |

Coke | 12 |

Fanta | 10 |

Sprite | 6 |

Lemon | 4 |

Pepsi | 8 |

Draw a bar chart to display this information

Solution

**CLASS ACTIVITY**

- The number of items produced by a company over a five years period is given below:

Year | 1978 | 1979 | 1980 | 1981 | 1982 |

Number produced | 4100 | 2500 | 1500 | 1800 | 9200 |

- The table below shows the weights, to the nearest kilogram of twelve students in a further mathematics class.

Weight in kg | 55 | 57 | 59 | 61 | 63 |

No of students | 2 | 1 | 2 | 4 | 3 |

Draw a bar chart to illustrate the above information

**Pie Chart**

A pie chart is a cyclic diagram in which each sector of the circle represents a given frequency expressed in degrees.

**Example 1:**

In a certain school, the lesson periods for each week are as itemized below: English 10, Mathematics 7, Biology 3, Statistics 4, Ibo 3, others 9.

Draw a pie chart to illustrate this information.

Solution

Total number of periods = 36

Subjects |
periods |
Sector (degrees) |

English | 10 | |

Mathematics | 7 | |

Biology | 3 | |

Statistics | 4 | |

Ibo | 9 |

**Example 2: The student intake at a certain school of Arts and Science for a particular year was distributed among its four departments as follows:**

**Arts and Social Science, 90; Science, 180; Agriculture, 65; Fine Arts and Languages, 25. **

**Illustrate the above data in a pie chart.****What percentage of students was admitted into the department of science?**

**Solution**

Department |
Intake |
Sector (degrees) |

Arts and Social Science | 90 | |

Science | 180 | |

Agriculture | 65 | |

Fine Arts and Languages | 25 |

CLASS ACTIVITY

- The breakdown of the results of school certificate in a certain school in 1980 is as follows:

Distinction 10 candidates

Division I 25 candidates

Division II 40 candidates

Division III 10 candidates

Statement of Result (SR) 5 candidates

Represent the above information on a pie chart.

- The following table shows the proportion in which Ali spends his annual salary:

Food and drinks 30%

Rent 15%

Income Tax 20%

Transport 7.5%

Saving 15%

Miscellaneous 12.5%

- Illustrate the information on a pie chart.
- If Ali’s annual salary is Le 8000, calculate the total amount he spends on rent and transport each year.

**HISTOGRAM**

A histogram is a graphic representation of a frequency distribution. It consists of rectangles which are drawn on a continuous base and the area of each rectangle being proportional to the frequency of the classes they represent.

The rectangles need not have equal widths but their bases should of necessity be proportional to the class intervals. The lower class boundary and the upper class boundary are the extreme values of the base of each rectangle. If the class intervals are not equal in a frequency distribution, the height of each rectangle is determined by the **frequency density** defined as

**Frequency density = **frequency density

Class width

**Example 1: **Drawing histogram using class boundary, draw a histogram for the frequency distribution in the table below. (Use class boundaries to plot against the frequency).

Class | 1 – 5 | 6 – 10 | 11 – 15 | 16 – 20 | 21 – 25 |

Frequency | 2 | 4 | 6 | 5 | 3 |

Class |
Class boundaries |
Frequency |

1 – 5 | 0.5 – 5.5 | 2 |

6 – 10 | 5.5 – 10.5 | 4 |

11 – 15 | 10.5 – 15.5 | 6 |

16 – 20 | 15.5 – 20.5 | 5 |

21 – 25 | 20.5 – 25.5 | 3 |

NB: The use of class boundaries to plot against the frequency is important in using histogram to estimate the mode.

Solution

CLASS ACTIVITY

- The table below shows the marks obtained by forty pupils in a Mathematics test.

Marks | 0- 9 | 10 – 19 | 20 – 29 | 30 – 39 | 40 – 49 | 50 – 59 |

No. of pupils | 2 | 5 | 6 | 12 | 8 | 5 |

Draw a histogram for the mark distribution.

- The following table gives the lengths in mm of a certain tree.

Length (mm) | 20 – 25 | 25 -30 | 30 – 35 | 35 – 40 | 40 – 45 |

Frequency | 8 | 17 | 19 | 12 | 4 |

If the data were represented on a histogram, and the height of the column for 20-25 mm is 4cm, what is the height of the column for 35 – 40mm?

**Frequency Polygon**

The line graph that is obtained by joining the mid-points of the tops of the rectangle forming a histogram is called a **frequency polygon**. In practice, this line graph is extended to the next lower and higher classes which automatically have zero frequencies.

Frequency density

Variable

If the number of classes becomes very large, we may eventually draw a smooth curve through the mid-points of the tops of the rectangles. Such a curve is called a **frequency curve**

Frequency density

**Example 1:**

Find the median of the distribution

X | 0 – 4 | 5 – 9 | 10 –14 | 15 – 19 | 20 –24 | 25 –29 | 30 –34 | 35 – 39 | 40 – 44 | 45 – 49 |

F | 1 | 4 | 6 | 7 | 10 | 12 | 6 | 2 | 1 | 1 |

Solution

;

From the table 1 + 4 + 6 + 7 = 18

The median class is 20 – 24

The class is from 19.5 to 24.5

Interval = 5 frequency of the class = 10

Median

The result is approximation.

Find median using cumulative frequency

**Example 2:**

Find the median of the distribution table below

X |
F |
Cum. Freq. |

0 – 9 | 5 | 5 |

10 – 19 | 13 | 18 |

20 -29 | 22 | 40 |

30 – 39 | 8 | 48 |

40 – 49 | 2 | 50 |

Median = = 25

**PRACTICE QUESTIONS**

- The table shows the weights, to the nearest kilogram of twelve students in a further mathematics class

Weight in Kg | 55 | 57 | 59 | 61 | 63 |

Number of Students | 2 | 1 | 2 | 4 | 3 |

Draw a bar chart to illustrate the above information.

- The table below shows how a company’s sales manager spent his n1995 annual salary:

Food – 30%

Rent – 18%

Car maintenance – 25%

Savings – 12%

Taxes – 5%

Others – 10%

- Represent this information on a pie chart.
- Find his savings at the end of the year if his annual salary was #60,000.
- The table below shows the distribution of sources of energy for household cooking in a sample of 600 houses in a town.

Coal 150

Gas 72

Electricity 48

Kerosine 200

Wood 130

Using 1 cm to represent 20 houses on the vertical axis, illustrate this information in a bar chart.

- The Pie chart below shows the weekly sales of a motor dealer in Lagos in 1999.
- What fraction of the cars were Toyota?
- What percentages of the Cars were Datsun?
- If the dealer sold 16 Peugeots, How many BMW did he sell in a week?
- Below is the cumulative frequency table of the life-span of 100 rabbits in a controlled environment.

Life Span in Days | 22 | 50 | 75 | 100 | 125 | 150 | 175 | 200 |

Cumulative No. of Rabbits | 5 | 21 | 40 | 60 | 80 | 91 | 98 | 100 |

- Draw a cumulative frequency curve of the distribution.
- Use your curve to estimate
- the semi-interquartile range
- the number of rabbits still alive after 130 days

**ASSIGNMENT**

- The number of items produced by a company over a five years period is given below:

Year | 1978 | 1979 | 1980 | 1981 | 1982 |

Number produced | 4100 | 2500 | 1500 | 1800 | 9200 |

Draw a bar chart from this information.

- In a school, 50 students were entered for a Joint Matriculation Examination and the marks scored out of 100 were:

17 82 48 34 72 55 56 64 31 47

73 41 53 8 44 40 68 50 76 30

13 45 67 54 38 60 80 59 40 93

28 67 55 70 45 62 39 57 81 62

43 79 50 42 78 34 72 57 45 22

- Use the intervals 0 – 9, 10 – 19, 20 – 29, … to construct a frequency table.
- State the class with the highest frequency.
- Illustrate the diagram by a suitable diagram.
- The following is the record of marks of candidates in an examination:

65 84 91 58 43 86 73 33 76 80

57 33 53 29 40 27 72 19 51 67

37 14 18 92 13 45 61 39 23 22

22 41 27 51 63 47 19 35 39 76

Using class-intervals 11 – 20, 21 – 30,…

- Prepare a frequency table;
- Draw a histogram for the distribution.
- The pie chart below shows the allocation of money to the different departments in a secondary school.

If applied Science Department were allocated the sum of N120,000.00. What was the total allocation to Maths Department?