Class Interval

In statistics, class intervals are used to group a large set of data into specific numerical ranges, making it easier to organize, understand, and analyze the data. They are mainly helpful for continuous data, where values can take any value within a range. In mathematical terms, a class interval represents the numerical width of each group in a grouped frequency distribution. 

A class interval is defined by two values - the lower limit and the upper limit. 

• Lower limit: The smallest value that can belong to the class.
• Upper limit: The most significant value that can belong to the class.

The difference between the upper and lower class limits determines the size of the class interval. Class intervals systematically arrange data in a frequency distribution table, with each class mutually exclusive. It means that no data point can belong to more than one class. 

Class Interval Definition

A class interval is a range of values used to group data in a frequency distribution. It is defined by a lower and an upper limit and helps organize continuous data into transparent, manageable groups.

Class Interval Formula

To find the class interval, we subtract the lower class limit from the upper class limit. It gives the class width. And the formula is:
Class interval = Upper limit - Lower limit.

Let us learn more from an example: 

Consider a class given as 10–20. Find the class interval.
Here, 
Lower limit = 10 and upper limit = 20.
By applying the formula: 
Class interval = 20-10 
= 10. 
So, the width of this class interval is 10 units.
 

To find class intervals of datasets accurately, we follow these steps:

Step 1: Find the range. 
To find the range of the dataset, subtract the minimum value from the maximum value. 
Range = maximum value - minimum value.

Step 2: Decide the number of intervals.
Choose how many class intervals you need, depending on the size of the data set and the level of details required. 

Step 3: Calculate the class width.
Determine the class width by dividing the range by the number of intervals selected.
Class width = range ÷ number of intervals.

Step 4: Set the class boundaries.
Start from the minimum value, then repeatedly add the class width to maintain the lower and upper limits of each interval. 

Step 5: Organize the data. 
Place each data point into its appropriate class interval based on the boundaries you have defined.

Let us practice through an example: 

A teacher records the ages of 12 students in a class as 10, 12, 14, 15, 13, 17, 18, 16, 19, 11, 13, 15. Organize the data using class intervals.

Solution: 
 

Step 1: First, we can find the range.
We know that the maximum value = 19, and the minimum value = 10. 
So, range \(= 19 - 10 = 9\).

Step 2: Since, we have 12 data points, we choose 4 intervals.

Step 3: Now, we can calculate the class width. 
We know that, class width = range ÷  number of intervals. 
Then class width \(= 9 ÷ 4 = 2.25\).
We can round it to the nearest whole number, 3. 

Step 4: We can set the class boundaries, starting from the minimum value 10, and by adding the class width 3. 
Hence, the classes will be: 

• 10 -13
• 13 -16
• 16 -19
• 19 -22

Here, we have added one extra interval to ensure that all values are covered. 

Step 5: Now we can organize the data into intervals. 

Class intervals are categorized into different types based on how they are structured and what kind of data we want to represent:

• Exclusive class interval
  
   
• Inclusive class interval

Exclusive class interval: Exclusive class intervals are when the lower bound or the minimum value is included in the interval and the upper bound or maximum value is excluded. This means that any data point equal to the upper bound is assigned to the previous interval. For example, in the interval (10, 20), include from 10 to 19.999, but not 20.

As you can see in this table, in an exclusive class interval, the next class interval’s lower limit is equal to the upper limit of the previous interval.

Inclusive class intervals: In inclusive class intervals, both the lower limit and upper limits are included within the interval. We use inclusive class intervals when we want to ensure that all the data values in the data set are equal to the limits that are part of that interval.

In this case, both lower and upper limits are included in the interval and the upper limit of one class interval is different from the lower limit of the next class interval.
 

We know that in an inclusive class interval, both the lower and upper limits are included. For example, 10–19, 20–29, 30–39 are inclusive. But they needed to be converted to exclusive class intervals for histogram representation. To convert inclusive intervals into exclusive intervals, follow these steps:

Step 1: Identify the gap or overlap. 
See the two consecutive class intervals and check how much they overlap. For example, in classes 10 - 19 and 20 - 29, the gap between 20 and 19 is 1. 

Step 2: Find the correction factor.
Divide the overlap by 2 for that. For example, \(\frac{1}{2} = 0.5\).

Step 3: Adjust the class limits.
Subtract the factor from each lower limit and add the correction factor to each upper limit.
For instance, 10 - 19 is the inclusive and when converted to exclusive class interval, it will be
Lower limit = \(10 - 0.5 = 9.5\)
Upper class = \(19 + 0.5 = 19.5\).

Step 4: Do the same for all intervals.
Apply the correction factor to every interval in the table.

Class intervals can be represented graphically using graphs such as histograms or frequency polygons. Class intervals show how data is distributed across different ranges. These graphs help in visualizing trends and frequencies in the dataset.

To represent a class interval graphically:

Step 1: Prepare a frequency table by organizing the data into class intervals, and then we count the number of values in each interval.

Step 2: Label the axes, the x-axis will represent the class intervals and the y-axis will represent the frequency. 

Step 3: Plot the data by taking the frequency for each interval which is marked as a bar, point, or curve depending on the graph used. In most cases, intervals are continuous, meaning the bars or points are connected without any gaps.

This is a general approach that gets applied to different types of graphs like histograms or frequency polygons, etc.

Class Interval Histogram
 

A class interval histogram is a graphical tool for displaying data grouped into class intervals. It is primarily applicable when showing the distribution of continuous data. In a histogram: 

• The x-axis represents the class intervals.
• The y-axis represents the frequency of each interval. 
   

Exclusive class intervals like 10-20, 20-30, etc. can be plotted directly on a histogram because there are no overlapping values. And inclusive class intervals like 10 - 19, 20 - 29, etc. must first be converted to exclusive to ensure that classes do not overlap. 

Once the class intervals and frequencies are ready, each interval is represented by a bar whose height corresponds to its frequency. This visual representation makes it easier to observe patterns such as skewness, data concentration, and spread.
 

Example: A teacher records the test scores of a group of students and organizes them into the following frequency distribution:

The class interval histogram will look like: 

 

To master class intervals, focus on grouping data into meaningful ranges, creating accurate frequency tables, and using visual tools like histograms.

• Learn to group data into meaningful and equal ranges for clarity.
   
• Practice creating accurate frequency tables using different data sets.
   
• Make sure class intervals are mutually exclusive and cover all data sets.
   
• Use histograms and bar charts to visualize class intervals effectively.
   
• Apply class intervals to real-life examples like test scores, income, or age groups.
   
• Teachers and parents can teach students using familiar examples such as age groups, height ranges, or daily temperatures, to help children understand how data is grouped in real life. 
   
• Give students small datasets like colored beads, survey results or fancy items and let them physically sort items into boxes or containers representing intervals. 
   
• Guide students to draw number lines, bar graphs, and histograms so they can clearly see how intervals represents continuous ranges.
   
• Remind learners that intervals must be of the same width for accuracy. Show how unequal intervals can mislead or confuse interpretations.
   
• Encourage students to use online class interval worksheets or graphing apps to create frequency tables and histograms for better engagement. 
   

Class intervals are widely used in various fields in the real world. Here are a few real-world applications of class intervals:

 

• Education: Schools and colleges use class intervals to group students' scores into ranges. This helps teachers analyze the students’ performance.
  
   
• Market research: Class intervals are used in market research to categorize survey responses or customer feedback into different ranges.
  
   
• Financial Analysis: In finance, we use class intervals to group income levels or investment returns into ranges to identify the trends or plan budgeting.
  
   
• Weather analysis: Meteorologists use class intervals to group temperature or rainfall data into ranges for easier analysis of climate patterns.
  
   
• Healthcare: Hospitals and researchers use class intervals to categorize patients age, blood pressure, or cholesterol levels to study trends and risks.