Collected, unorganized data is called raw data. Data should be arranged and presented concisely for easy interpretation. Arranging and presenting data in a table is called tabulation. The resulting table is a frequency distribution table. In this table, data is organized using tallies, and the number of entries in a class is called frequency (f).

Data can be presented in two forms:

1. Ungrouped data: Data arranged as single entities.
2. Grouped data: Data arranged into classes of specified interval sizes.

Tallying procedures

1. List the data values in ascending order without repetition.
2. Mark a tally against each value as it occurs in the data.
3. Repeat step 2 until all data is tallied.

Example 6.1

Make a frequency distribution table for the following data:

5, 10, 7, 19, 25, 12, 15, 7, 6, 8, 17, 17, 22, 21, 7, 7, 24, 5, 6, 5

a) Without grouping:

Data value	Tallies	Frequencies
5	III	3
6	II	2
7	IIII	4
8	I	1
10	I	1
12	I	1
15	I	1
17	II	2
19	I	1
21	I	1
22	I	1
24	I	1
25	I	1
N=		20

b) By grouping into five classes:

Class width = (Highest value − Lowest value) / Number of classes = (25 − 5) / 5 = 4

Class intervals	Tallies	Frequencies
5 – 9	IIII I	6
10 – 14	II	2
15 – 19	IIII	4
20 – 24	III	3
25 – 29	I	1
N=		20

Cumulative frequency is the sum of frequencies up to a defined level. The table containing data values, frequencies, and cumulative frequencies is a cumulative frequency distribution table.

Example 6.2

Data shows the number of books possessed by students.

Books	Frequencies
1	1
2	3
3	2
4	4
5	8
6	2

Books	Frequencies	Cumulative Frequencies
1	1	1
2	3	4 (1+3)
3	2	6 (4+2)
4	4	10 (6+4)
5	8	18 (10+8)
6	2	20 (18+2)
N =		20

1. Class Limits: The smallest (lower limit) and largest (upper limit) values in a class interval.
2. Class Mark: (Lower limit + Upper limit) / 2
3. Class Boundary: Numbers separating classes. Lower boundary: Lower limit − (Gap/2). Upper boundary: Upper limit + (Gap/2).
4. Class Size (Width): The difference between consecutive lower/upper limits or class boundaries.

Example 6.3

Class Intervals	Frequency
6 – 10	4
11 – 15	5
16 – 20	9
21 – 25	2
26 – 30	6

Class Intervals	Frequencies	Class Marks	Lower Limits	Upper Limits	Lower Boundaries	Upper Boundaries
6 – 10	4	8	6	10	5.5	10.5
11 – 15	5	13	11	15	10.5	15.5
16 – 20	9	18	16	20	15.5	20.5
21 – 25	2	23	21	25	20.5	25.5
26 – 30	6	28	26	30	25.5	30.5

Class size = 11 − 6 = 5

Graphical representations visually display data and statistical results. They are often more effective for interpretation than tables.

a) Frequency polygon

A line graph connecting plotted points of frequencies at the midpoints (class marks) of classes. The frequencies are represented by the heights of the points.

Example 6.4

Marks	Frequencies
21 – 30	3
31 – 40	4
41 – 50	6
51 – 60	12
61 – 70	15
71 – 80	9
81 – 90	7
91 – 100	4

Marks	Class Marks	Frequencies
11 – 20	15.5	0
21 – 30	25.5	3
31 – 40	35.5	4
41 – 50	45.5	6
51 – 60	55.5	12
61 – 70	65.5	15
71 – 80	75.5	9
81 – 90	85.5	7
91 – 100	95.5	4
101 – 110	105.5	0

(Graph needed: Frequency polygon with Class Marks on x-axis and Frequencies on y-axis)

b) Cumulative frequency curve (ogive)

A smooth line graph of cumulative frequency plotted against the upper class boundary.

Example 6.5

Length (mm)	Frequencies (f)
11 – 15	2
16 – 20	4
21 – 25	8
26 – 30	14
31 – 35	6
36 – 40	4
41 – 45	2

Length (mm)	Frequencies (f)	Upper Boundaries	Cumulative Frequencies
6 – 10	0	10.5	0
11 – 15	2	15.5	2
16 – 20	4	20.5	6
21 – 25	8	25.5	14
26 – 30	14	30.5	28
31 – 35	6	35.5	34
36 – 40	4	40.5	38
41 – 45	2	45.5	40

(Graph needed: Ogive with Upper Boundaries on x-axis and Cumulative Frequencies on y-axis)

c) Histogram

The histogram is the graph of frequency (vertical axis) against the class marks (horizontal axis) or data values presented by rectangles (blocks). There are no gaps between the blocks as histograms represent continuous data.

Example 6.6

Marks	Frequencies	Class Marks
21 – 30	3	25.5
31 – 40	4	35.5
41 – 50	6	45.5
51 – 60	12	55.5
61 – 70	15	65.5
71 – 80	9	75.5
81 – 90	7	85.5
91 – 100	4	95.5

(Graph needed: Histogram with Class Marks on x-axis and Frequencies on y-axis)

d) Bar graph

(Graph needed: Bar Graph with Class Marks on x-axis and Frequencies on y-axis. Note the difference: There should be gaps between the bars in a bar graph.)

e) Pictogram

Pictograms represent data frequencies using pictures or symbols.

Example 6.7

Time	Number of cars
06:00 – 08:00	150
08:00 – 10:00	200
10:00 – 12:00	150
12:00 – 14:00	100
14:00 – 16:00	150
16:00 – 18:00	200

(Pictogram needed: Each symbol represents 50 cars. Draw 3 symbols for 150, 4 symbols for 200, and 2 symbols for 100.)