Session 1: Presentation of Data – Frequency Distribution (Steps and Examples)

The word data is plural, the singular being ‘datum’. The dictionary meaning of the word datum is ‘fact’; therefore, in the plural, the word data signifies more than one fact. In the broader sense, the term data denotes evidence or facts describing a group or a situation. 

In the modern world, vast amounts of data are collected every week by many different people.  

To collect data, we need a suitable method of data collection. Three simple methods are:

• Conducting surveys, e.g. in India, the census is conducted every ten years to review the country’s progress in the past decade, monitor the ongoing Schemes of the Government and most importantly, plan for the future.
• Conducting experiments, e.g. experiments are carried out to find out the average lifespan of worms.
• Observations, e.g. scientists conduct experiments to study behavioural patterns of different species.

Presentation of Data

After collecting data from the sample group, the investigator organises the data in tabular form to study their salient features.

Raw data can be arranged in the following ways:

• Alphabetical order
• Ascending order
• Descending order

Arranging the data in ascending or descending order is called an array. Arrayed data is much easier to understand and interpret.

If the data collected is extensive and repeated several times, arranging data in ascending and descending order is tough and tedious. For this, we need to organise it in tabular form. 

A systematical arrangement of the data in a tabular form is called tabulation or presentation of the data. This grouping results in a table called the frequency table, which indicates the number of observations within each group. Many conclusions about the characteristics of the data, the behaviour of variables, etc., can be drawn from this table. 

Frequency Distribution

Frequency distribution is a method to present raw data in a table to easily understand and interpret the information contained in the raw data. The process of preparing this type of distribution is very simple.

In constructing a frequency distribution from the given raw data, we write the variates (variables) in the first column. We take each observation from the given raw data, one at a time, and mark a stroke  called tally mark in the second column opposite to the variate (variable). To facilitate counting, we write tally marks in bunches of five, the fifth one crossing the four diagonally . The number of tally marks opposite to a variate (variable) is its frequency and is written in the third column opposite to tally marks of the variate (variable).

Below is the data showing the outcomes when a die is rolled 25 times.  

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

The data may be put in the form of a discrete frequency table as below:

Number on the die	Tally Marks	Number of times (Frequency)
1		5
2		5
3	||||	4
4	|||	3
5	||||	4
6	||||	4
Total		25

Steps to Construct a Frequency Distribution

In summary, we can use the following steps to construct a frequency distribution from the given raw data:

Step 1: Prepare a table. Its first column consists of the variate (or variable) under study, the second column their respective tally marks and the third column represents the corresponding frequency. 

Step 2: Place all the variates (or variables) in the first column in ascending or descending order. 

Step 3: Take each observation from the given raw data and place a bar in the second column next to it. For convenience during counting, record tallies in groups of five () with the fifth tally crossing the first four tallies. 

Step 4: Count the number of tallies corresponding to each variate, which gives us the frequency. 

Step 5: Check that the total of all frequencies is the same as the total number of observations. 

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Presentation of Data – Examples

Example 1

Here are the shoe sizes of 30 adults:

7	8	9	6	5
6	7	10	9	8
9	8	5	6	10
6	7	6	5	9
10	7	8	9	10
8	8	9	9	9

Show the data in a frequency table. 

The frequency distribution of shoe sizes of 30 adults is as follows: 

Shoe Size	Tally Marks	Frequency
5	|||	3
6		5
7	||||	4
8	|	6
9	|||	8
10	||||	4
Total		30

Example 2

In a factory making mobile phones, a sample of 50 mobile phones is selected at random each day to test for defects. The number of defects found on 40 consecutive days is listed below. 

3	8	7	3	6
2	0	8	4	7
3	6	4	7	3
8	0	5	0	7
1	6	5	8	7
3	6	2	3	0
8	8	7	3	7
1	0	4	5	3

Show the data in a frequency table. On how many days were there more than seven mobile phones with defects?

The frequency distribution is as follows. 

Number of mobile phones with defects	Tally Marks	Frequency
0		5
1	||	2
2	||	2
3	|||	8
4	|||	3
5	|||	3
6	||||	4
7	||	7
8	|	6
Total		40

There are six days when more than seven mobile phones have defects.

 Remember this!

• Raw or Ungrouped Data is the data obtained in original form. 
• A systematic arrangement of the data in tabular form is called tabulation or presentation of the data. 
• Frequency distribution is a method to present raw data in the form of a table from which one can easily understand and interpret the information contained in the raw data.