CHAPTER FOUR
GRAPH
ICAL REPRESENT
ATION OF DATA
4.1Introd
uc
tion
In the
previous cha
pter,
you hav
e learnt how to collect
and tabulate statistical data.
This chapter, will look into h
ow to present frequency distribu
tion data into graph for easy s
eeing, convey information and
also assists in the understa
nding of data.
All graphs m
ust be properly scaled in o
rder to convey correct information. Graphical methods to be discussed include Pie chart, Bar chart, Histogram, and Frequency polygon.
4.2Objectives
By the end of this section, you should be able to:
Construct Pie chart,
Draw bar chart, histogram, and frequency polygon
Draw a cumulative curve
Draw a table of relative frequency distribution (percentage distribution)
Construct a relative frequency histogram
4.3Pie Chart
Pie chart is sometimes known as circle graph. It is a chart that arrange, events, objects or items in a circle form end used for both discrete and continuous data. Each occurrence or item is represented by a sector of the circle, with the angle occupied at the sector’s center corresponding to the frequency of the occurrences.
Example 1:
A Researcher who is interested in studying the performance of students in various department, can present the data using pie chart. For example, the examination Officer of School of Sciences presents the average performance of Students in various Departments in the School as follows: Biology = 55, Chemistry = 30, Computer Science = 40, Mathematics = 25, Integrated Science = 60, Physics = 20.
Solution
The Researcher following the procedures is to present the data in pie chart.
Step 1: Sum up all performance in the six departments, thus:
55 + 30 + 40 + 25 + 60 + 20 = 230.
Step 2: calculate the angle that is subtended at the center by each item.
Biology\\frac{55}{230}\\times 360^{\\circ}= 86.9^{\\circ}
Chemistry\\frac{30}{230} × 360^{\\circ} = 46.96^{\\circ}
Computer Science\\frac{40}{230} × 360^{\\circ} = 62.61^{\\circ}
Mathematics\\frac{25}{230} × 360^{\\circ} = 39.13^{\\circ}
Integrated science\\frac{60}{230} × 360^{\\circ} = 93.91^{\\circ}
Physics\\frac{20}{230} × 360^{\\circ} = 31.30^{\\circ}
Step 3: Use a pair of compasses, draw a circle with any convenient radius and indicate the center of the circle.
Step 4: Mark the angles corresponding to each item using your protractor.
Step 5: Label the six sectors obtained, that is Biology, Chemistry, Computer Science, Mathematics, Integrated Science and Physics.
Fig. 4.1: A pie chart of average performance of departments
Example 2:
The data shows the numbers of lecturers in each department at certain College of Education
Department | Number of Lecturer |
Arabic | 14 |
English | 12 |
French | 8 |
Hausa | 4 |
Nupe | 2 |
Total | 40 |
Solution
To construct the pie chart as earlier outline some steps to be followed. Firstly, sum-up number of lecturers, which gives 40. This numbering 40 lecturers correspond to the number of degrees in circular are 360^{\\circ}
i.e.,
Arabic:\\frac{14}{40}\\times 360=126^{\\circ}
English:\\frac{12}{40}\\times 360=108^{\\circ}
French:\\frac{8}{40}\\times 360=72^{\\circ}
Hausa:\\frac{4}{40}\\times 360=36^{\\circ}
Nupe:\\frac{2}{40}\\times 360=18^{\\circ}
Now use your protractor to obtain required angles for each item after drawing a circle of any continent radius.
Figure 4.7
4.4Bar Chart
Bar chart is known as bar diagram or bar graph, and which contains bar that stand out exclusively from one another. This shows that the measurement scales are not continuous but discrete data. Bar diagrams convey the frequency of cases in each group relative to each other. It has two axes vertical and horizontal base. The vertical axis is known as the ordinate and horizontal axis is called abscissa. The bars can be vertical or horizontal and the line bars or columns are of equal width, but the height varies according to the proportion of the data.
The data in last example can be represented in a bar chart by the following steps:
Step 1: Choose a convenient scale to draw the two axes (vertical and horizontal).
Step 2: Make out the height of each section based on the chosen scale.
Step 3: Draw out the bars of each to represent the height.
Example: Draw a bar chart for the information given in
Table 4.3. The table is provided below
Department | Number of Lecturer |
Arabic | 14 |
English | 12 |
French | 8 |
Hausa | 4 |
Nupe | 2 |
Total | 40 |
00Solution
Figure 4.8
A Simple bar chart consist of various components, which each component represents a section and each section corresponds in size to the magnitude of the item it stands for.
4.5Histogram
It is created by graphing the frequencies of the respective class interval against the class borders. It is mostly used to represent continuous data.
The following procedures should be followed in order to create a histogram.
Step 1:create a frequency distribution table with the class interval, frequencies, and class borders changed.
Step 2:Select appropriate scales for both axes and draw the vertical and horizontal axes
Step 3:is to label the axis using the selected scales.
Step 4:On each border, draw rectangular bars with heights according to the frequencies.
Step 5:Draw arrows to indicate what is on the vertical and horizontal axis.
To draw the histogram, you then adopt the listed above procedures 1—5.
To illustrate the histogram scores of 80 students in MAT 224 test at the end of 2nd semester exams in a certain College of Education as follows:
Scores | Number of Students |
50—52 | 5 |
53—55 | 11 |
56—58 | 14 |
59—61 | 10 |
62—64 | 8 |
65—67 | 7 |
68—70 | 6 |
71—73 | 9 |
74—76 | 5 |
77—79 | 5 |
Table 4.1: MAT224 Test Score for 2nd Semester Exams
S/N | Class Interval | Class Boundary | Frequency |
1. | 50—52 | 49.5—51.5 | 5 |
2. | 53—55 | 51.5—55.5 | 11 |
3. | 56—58 | 55.5—58.5 | 14 |
4. | 59—61 | 58.5—61.5 | 10 |
5. | 62—64 | 61.5—64.5 | 8 |
6. | 65—67 | 64.5—67.5 | 7 |
7. | 68—70 | 67.5—70.5 | 6 |
8. | 71—73 | 70.5—73.5 | 9 |
9. | 74—76 | 73.5—76.5 | 5 |
10. | 77—79 | 76.5—79.5 | 5 |
TOTAL | 80 | ||
Fig. 4.3
Example:
The following table indicates the distribution of mark scored by a class of 100 students
Table 4.4: Mark Scores by Students
Marks | Number of students |
11—15 | 20 |
16—20 | 11 |
21—25 | 12 |
26—30 | 28 |
31—35 | 16 |
36—40 | 13 |
Example: Draw a histogram of the distribution
Solution
A new table is prepared showing the class boundaries and class mark
Table 4.5: Adjusted Data of Mark Scores by Students
Marks | Class boundary | Class mark | Frequency |
11—15 | 10.5—15.5 | 13 | 20 |
16—20 | 15.5—20.5 | 18 | 11 |
21—25 | 20.5—25.5 | 23 | 12 |
26—30 | 25.5—30.5 | 28 | 28 |
31—35 | 30.5—35.5 | 33 | 16 |
36—40 | 35.5—40.5 | 38 | 13 |
Total | 100 |
4.6Frequency Polygon
This is a type of graph of a frequency distribution which is obtained by plotting the class frequencies against the class marks. It is polygon because the mid-point of the tops of the rectangles in the histogram are connected.
A frequency polygon is constructed by following the procedures below:
Step 1:Draw both axes (i.e., vertical, and horizontal).
Step 2:Mark out the frequencies along the vertical axis and the mid-points of class intervals on the horizontal axis.
Step 3:Plot the frequency of each class interval at the appropriates height as a point above the mid-point of interval.
Step 4:Join these points with straight lines.
Step 5:Connect the first and last dots with the horizontal axis at the mid-point before the first dot and the one after the last dot.
Using the data in Table 4.1 to present a frequency polygon by adopted step 1—5.
Fig. 4.4: frequency polygon of data in table 4.1
4.7Frequency Curve
Frequency curve is sometimes called smoothed frequency polygon. It is smooth curve that joins the middle of the tops of the histogram. It is similar to the frequency polygon, the only difference is that frequency curve is a smooth curve while, the frequency polygon is a line segment. In drawing the frequency curve is the midpoint of the class intervals against the cumulative frequency. To illustrate this graph using the data in Table 4.2.
Table 4.2: shows the scores in Achievement Test
A Interval | B Exact limit | C Frequency | D Midpoints | E Cumulative frequency |
2—4 | 1.5–4.5 | 2 | 3 | 2 |
5—7 | 4.5–7.5 | 8 | 6 | 10 |
8—10 | 7.5–10.5 | 7 | 9 | 17 |
14—16 | 13.5–16.5 | 17 | 15 | 34 |
17—19 | 16.5–19.5 | 7 | 18 | 41 |
20—22 | 19.5–22.5 | 7 | 21 | 48 |
Fig 4.5: Frequency Curve
4.8Relative Frequency Distribution
The relative frequency is how often a specific type of event occurs within the total numbers of observation. Relative frequency does not make use of raw counts, rather they relate the count for an event to the total number of events using percentage, proportion, or fraction. It is the actual frequency of the class divided by the total frequency of all classes.
4.8.1Computation of Relative Frequency
To calculate for relative frequency, you must know the followings:
The count of events, for a particular category
The total number of observations (events)
The formular is given as:
Relative Frequency (RF) =\\frac{Countofevent}{\\begin{array}{c} Totalnumberofevent\\\\ \\end{array}}
Let us use the Table 4.2 data to calculate the relative frequency.
S/N | Class interval | Frequency | Relative | Percentage |
i. | 2—4 | 2 |
\\frac{2}{98}=0.02 | 2 |
ii. | 5—7 | 8 |
\\frac{8}{98}=0.08 | 8 |
iii. | 8—10 | 7 |
\\frac{7}{98}=0.07 | 7 |
iv. | 14—16 | 17 |
\\frac{17}{98}=0.17 | 17 |
v. | 17—19 | 7 |
\\frac{7}{98}=0.07 | 7 |
vi. | 20—20 | 7 |
\\frac{7}{98}=0.07 | 7 |
When the frequencies in Table 4.2 are replaced by corresponding relative frequency, the result obtained is known as relative frequency distribution.
The relative frequency distribution graph can be draw from the histogram or polygon by using the x-axis for class interval and y-axis for relative frequency.
For example, use the data below to construct
a relative frequency percentage distribution
a relative frequency histogram
a relative frequency polygon.
Solution
Scores of 100 Female Students in Department of Mathematics
Scores X | Frequency F | Relative frequency RF | Percentage 10 |
50—52 | 6 | 0.05 | 5 |
53—55 | 18 | 0.18 | 18 |
56—58 | 42 | 0.42 | 42 |
59—61 | 27 | 0.27 | 27 |
62—64 | 8 | 0.08 | 8 |
TOTAL | 100 | 1 | 100% |
Note that each class frequency is calculated in relation to the total frequency of all classes, which is 1.
FIGURE 4.6: SCORES
Students Activity
Newly admitted Mathematics Students of ABU Zaria spent a total of N60,000.00 with the following details:
Tuition fees = N10,000.00
Game fees = 5,000.00
Clinic fee = N2,000.00
Course materials = N15,000.00
Accommodation = N10,000.00
Stationaries = N5,000.00
Feeding = N10,000.00
Notebooks = N3,000.00
Construct these exercises in pie chart.
Consider the following scores obtained by a Researcher by administering 40 Students in post-test.
30 25 54 50 12 5 18 40
21 25 55 13 40 3 46 3
23 21 34 49 18 8 48 18
39 27 37 30 15 5 23 46
21 21 33 35 5 8 45 38
Assuming a class size of 5, create a frequency distribution table for the data.
Draw a bar chart.
Draw a Histogram.
Draw a frequency polygon.
Explain a relative frequency
Draw the relative frequency histogram for the data below.
Masses of 100 Students at a certain Department of Mathematics
Mass (x) Kg | Number of Students F |
30—32 | 8 |
33—35 | 42 |
36—38 | 27 |
39—41 | 5 |
42—44 | 18 |
Compute the relative frequency and relative percentage of the data given in Table.
Table 4.1: Scores Obtained in Statistics by 90 Students
Mark X | Frequency F |
51—55 | 2 |
56—60 | 12 |
61—65 | 11 |
66—70 | 13 |
71—75 | 22 |
76—80 | 7 |
81—85 | 10 |
86—85 | 5 |
91—95 | 5 |
90 |
References
Awotunde, P. O. & Ugodulunwa (2002). An Introduction to Statistical Methods in Education. Printed and published in Nigeria by Fab Anieh (Nig) Ltd.
National Teachers’ Institute, Kaduna (2000). Nigeria Certificate in Education (NCE/DLS) Course Book on Mathematics.