 # Discrete Frequency Distribution

In Statistics, data plays a vital role in estimating the different types of parameters. To draw any conclusions from the given data, first, we need to arrange the data in such a way that one can perform suitable statistical experiments. We know that data can be grouped into two ways, namely, Discrete and Continuous frequency distribution. In this article, you will learn about discrete frequency distribution with its definition, representation, and examples in detail.

## Discrete Frequency Distribution Definition

Let the given data consist of n distinct values x1, x2, …, xn occurring with frequencies f1, f2, …, fn, respectively. Then, this data can be represented in the tabular form with variables and the corresponding frequencies as given below and is called discrete frequency distribution:

 x x1 x2 x3 ….. xn f f1 f2 f3 ….. fn

The above table describes the occurrences of data values such as the observation x1 occurs f1 times, x2 occurs f2 times, and so on.

Thus, in a discrete frequency distribution, the values of the variable are determined individually. The number of times each value occurs denotes the frequencies of the particular value or observation. Discrete frequency distribution is also known as ungrouped frequency distribution.

### How to Construct Discrete Frequency Distribution

The discrete frequency distribution table can be made with the help of tally marks for easy understanding and to get the accurate output. This can be understood with the help of the example given below:

Example:

Below are the marks scored for a total of 20 marks by 25 students in a monthly test. Construct a discrete frequency distribution.

15, 13, 16, 16, 15, 16, 17, 14, 15, 16, 16,17, 14, 17, 17, 16, 14, 15, 16, 17, 14, 16, 15, 17, 13

Solution:

Given scores:

15, 13, 16, 16, 15, 16, 17, 14, 15, 16, 16,17, 14, 17, 17, 16, 14, 15, 16, 17, 14, 16, 15, 17, 13 This can be tabulated as:

 x 13 14 15 16 17 f 2 4 5 8 6

Discrete frequency distribution table is constructed when the number of data values are small. If we have more observations, then we have to prepare a grouped frequency table for easy calculation. This way of grouping data will be helpful for solving many statistical problems. Also, we can directly construct a frequency table without actually using tally marks.

## Discrete Frequency Distribution Examples

Example 1:

In a city, 45 families were surveyed for the number of cell phones they used. Prepare a discrete frequency array based on their replies as recorded below.

1 3 2 2 2 2 1 2 1 2 2 3 3 3 3

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

2 4 2 7 4 2 4 3 4 2 0 3 1 4 3

Solution:

For the given data, the discrete frequency distribution table can be constructed as:

Let x = Number of cell phones

f = Number of families

 x 0 1 2 3 4 5 6 7 Total f 1 7 15 12 5 2 2 45 45

Example 2: Prepare a discrete frequency distribution table for the following data.

12, 21, 21, 3, 9, 3, 6, 12, 13, 21, 15, 22, 3, 6, 9, 9, 21, 22, 15, 13, 15, 9, 15, 6, 15, 13, 6, 9, 13, 22

Solution:

Given data:

12, 21, 21, 3, 9, 3, 6, 12, 13, 21, 15, 22, 3, 6, 9, 9, 21, 22, 15, 13, 15, 9, 15, 6, 15, 13, 6, 9, 13, 22

The discrete frequency distribution table is given as:

 x 3 6 9 12 13 15 21 22 f 3 4 5 2 4 5 4 3

### Mean for Discrete Frequency Distribution

We can find the measures of central tendency for the given discrete frequency distribution with suitable formulas. Let’s understand how to find the mean for discrete distribution of data with the help of the solved example below:

Question:

Find the mean for the following data:

 Marks 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Frequency 1 6 6 8 8 2 2 3 0 2 1 0 0 0 1

Solution:

 Marks (x) Frequency (f) fx 2 1 2 3 6 18 4 6 24 5 8 40 6 8 48 7 2 14 8 2 16 9 3 27 10 0 0 11 2 22 12 1 12 13 0 0 14 0 0 15 0 0 16 1 16 Total ∑f = 40 ∑fx = 239

Mean = ∑fx/∑f = 239/40 = 5.975