Statistical graphs - Histogram

Introduction

The purpose of graph in statistics is to convey the data to the viewers into pictorial form. That is after you have organized the data into a frequency distribution, you need to present them in graphical form. The reason is that it is easier for most people to understand the meaning of data presented graphically than data presented numerically in frequency distributions or tables. This is especially true if users have little or no statistical knowledge.

Uses of statistical graphs

1.      Statistical graphs can be used to analyze a data or described it.

2.       They are used to get the audience’s attention in a speaking presentation.

3.      They are also used to discover a pattern in a situation over a period of time.

4.      Statistical graphs are used to summarize a data.

The Histogram:

The histogram is a graph in which the frequency of values is represented by vertical bars of varying height and widths. A histogram may be drawn in one of the two ways.  Thus,

v  Case one [ frequency against class boundaries ]

v  Case two [ frequency against class mid-point ]

NOTE:

In a test, the core maths student is required to determine the mode or modal value. The elective maths student is required to determine the mode as well as the median.

Mode from a histogram:

A data set that has only one value that occurs with the greatest frequency is said to be uni-modal.

If the data set has two values that occur with the same greatest frequency, both values are considered to be the mode and the data set is said to be bimodal.

If the data set has more than two values that occur with the same greatest frequency, each value is considered to be the mode, and the data set is said to be multimodal.

Example one [ungrouped data]

The table below shows the distribution of marks scored by 40 students in a test.

Marks	2	3	4	5	6	7	8	9	10
Frequency	1	3	5	8	9	7	4	2	1

a.       Draw a histogram for the distribution.

b.      From your histogram, estimate the mode.

c.       Calculate the mean of the distribution.

Solution

Steps:

1.      Construct a table with the following columns; marks(x), frequency (f) and fx [NB: the column for fx is added to the table because we it to calculate the mean.]

2.      Draw and label the x and y axes on the graph sheet. The x axis is always the horizontal axis, and the y axis is always the vertical axis.

3.      Represent the frequency on the y axis and the class boundaries or class marks on the x axis.

4.      Using the frequencies as the heights, draw vertical bars for each class. See fig.1 and fig.2.

MARKS(x)	FREQUENCY (f)	f x
2	1	2
3	3	9
4	5	20
5	8	40
6	9	54
7	7	49
8	4	32
9	2	18
10	1	10
	∑ f = 40	∑ fx = 234

                                             Fig.1

 

 

 

ALTERNATIVELY

Using the class boundaries, we have:

Marks (x)	Class Boundaries	Frequency	F (x)
2	1.5 – 2.5	1	2
3	2.5 – 3.5	3	9
4	3.5 – 4.5	5	20
5	4.5 – 5.5	8	40
6	5.5 – 6.5	9	54
7	6.5 – 7.5	7	49
8	7.5 – 8.5	4	32
9	8.5 – 9.5	2	18
10	9.5 – 10.5	1	10
		∑ f = 40	∑ fx = 234

NB: the column for class boundary is prepared because we want to use that to draw a histogram otherwise it wouldn’t be necessary to add that to the table.

 

 

b. Mode = 5.9

c. Mean (x) = 5.85

Example (2) [Grouped data]

The table below shows the distribution of marks obtained by students in an examination.

Marks	11-20	21-30	31-40	41-50	51-60	61-70	71-80	81-90	91-100
Frequency	5	21	14	27	27	12	7	4	3

a.       Draw a histogram for the distribution.

b.      Estimate the mode from the histogram.

Solution

NB: The intention here is to use the class mid-points [NOT THE CLASS BOUNDARIES] to draw the histogram. It will therefore be sheer waste of time to add the class boundaries to the table.

Steps:

1.      Construct a table with the following columns; marks, class-midpoint (x) and frequency (f).

2.      Draw and label the x and y axes on the graph sheet. The x axis is always the horizontal axis, and the y axis is always the vertical axis.

3.      Represent the frequency on the y axis and the class-midpoint on the x axis.

4.      Using the frequencies as the heights, draw vertical bars for each class.

Marks	Class-midpoint (x)	Frequency (f)
11- 20	15.5	5
21-30	25.5	21
31- 40	35.5	14
41-50	45.5	27
51-60	55.5	27
61-70	65.5	12
71-80	75.5	7
81-90	85.5	4
91-100	95.5	3
		∑ f = 120

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