Introduction to Frequency Distribution

Imagine you have collected data on the heights of 100 students in a college. The raw data might look like a long list of numbers, such as 160 cm, 165 cm, 170 cm, and so on. While this raw data contains all the information, it is difficult to understand patterns or trends just by looking at it. This is where frequency distribution comes in.

Frequency distribution is a way to organize raw data into a meaningful format by grouping data into classes or intervals and showing how often each group occurs. This makes large data sets easier to interpret, analyze, and use for decision-making.

In this chapter, we will learn how to collect data, classify it, and create frequency distributions along with different types of frequencies and their graphical representations. These skills are essential for competitive exams and real-world data analysis.

Primary and Secondary Data

Before organizing data, it is important to understand where the data comes from. Data can be broadly classified into two types:

• Primary Data: This is data collected firsthand by the researcher for a specific purpose. For example, measuring the heights of students in your college or conducting a survey on monthly expenses in INR.
• Secondary Data: This is data collected by someone else and made available for use. Examples include census reports, government publications, or data from research articles.

Primary data is usually more reliable for specific studies because it is collected directly. Secondary data is useful when primary data collection is not feasible due to time or cost constraints.

Methods of Data Collection

Collecting accurate data is the first step in any statistical analysis. There are several methods to collect data:

graph TD    A[Data Collection] --> B[Surveys]    A --> C[Experiments]    A --> D[Observation]    A --> E[Existing Records]    B --> B1[Questionnaires]    B --> B2[Interviews]    C --> C1[Controlled Environment]    C --> C2[Field Experiments]    D --> D1[Direct Observation]    D --> D2[Participant Observation]

Each method has its own advantages and challenges. For example, surveys are quick but may have biased responses, while experiments provide controlled data but can be expensive.

Classification and Tabulation

Once data is collected, it needs to be organized for analysis. This involves two important steps:

• Classification: Grouping data into classes or categories based on common characteristics. For example, grouping student heights into intervals like 150-159 cm, 160-169 cm, etc.
• Tabulation: Arranging classified data into tables for clarity and easy reference.

Consider the following raw data of student heights (in cm):

158, 162, 165, 170, 172, 168, 160, 159, 175, 169, 161, 164

We can classify this data into class intervals and count the number of students in each interval:

This table is a simple example of classification and tabulation.

Frequency Distribution

A frequency distribution summarizes data by showing the frequency of each class or category. It helps us understand how data points are spread across different ranges.

There are three important types of frequencies:

• Absolute Frequency (f): The actual count of observations in each class.
• Relative Frequency (rf): The proportion of the absolute frequency to the total number of observations. It is often expressed as a decimal or percentage.
• Cumulative Frequency (CF): The running total of frequencies up to a certain class.

Here is a frequency distribution table showing all three types:

In this example, the total number of students is 12. The relative frequency for the first class is \( \frac{3}{12} = 0.25 \), meaning 25% of students fall in the 150-159 cm range.

Cumulative and Relative Frequency

Cumulative frequency helps us understand how many observations fall below the upper limit of a class. For example, the cumulative frequency of 9 for the second class means 9 students have heights less than or equal to 169 cm.

Relative frequency provides a normalized view of data, useful for comparing classes when total observations differ. It is calculated as:

Similarly, cumulative frequency is calculated by adding frequencies successively:

Graphical Representation

Graphs provide a visual way to understand frequency distributions. The three common graphs are:

• Histogram: A bar graph representing class intervals on the x-axis and frequencies on the y-axis. Bars touch each other to show continuous data.
• Frequency Polygon: A line graph connecting midpoints of class intervals plotted against frequencies.
• Ogive: A curve showing cumulative frequency, useful for finding medians and quartiles.

Here is an example of a histogram based on the frequency distribution of student heights:

Worked Example 1: Constructing a Frequency Distribution Table Easy

Worked Example 2: Calculating Relative and Cumulative Frequencies Medium

Worked Example 3: Drawing a Histogram from Frequency Data Medium

Worked Example 4: Interpreting Frequency Distribution for Decision Making Hard

Worked Example 5: Problem on Grouped Data Frequency Distribution Hard