Range in Statistics

Range is the simplest way to see how spread out a set of data is. It is found by taking the difference between the largest and smallest values. By calculating the range, we can quickly understand how much variation exists within the data.

In this lesson, you will learn what range means, the formula for calculating it, and how to find the range for both ungrouped and grouped data. You will also cover solved examples, practice questions, and compare the range with other measures such as mean, median, and mode. At the end of it, you will not only know how to calculate range but also why it is significant in statistics and everyday situations.

 

Table of Contents

• Range Meaning
• Range Formula
• Range of Ungrouped Data
• Range of Grouped Data
• How to Find the Range?
• Solved Examples on Range
• Practice Problems
• Conclusion
• FAQs On Range

 

Range Meaning  

In statistics, we often want to know not just the average of numbers but also how far apart the values are. This is where the range comes in. The range is the simplest way to measure how spread out a set of data is. It is found by subtracting the smallest value from the largest value. Unlike the mean, median, mode, it does not indicate where most of the data lies, but it gives a quick sense of how much the values differ.  

For instance, if the lowest score in a class is 10 and the highest score is 90, then the range is:  

• Range = 90 - 10 = 80  

This tells us that the scores are spread out over 80 marks. The range is quick and easy to calculate, and it helps us understand the variability of data in both small and large datasets.

 

Range Formula  

The range is a way of measuring how spread out the values in a dataset are. It tells us the distance between the smallest and the largest number in the data.

The general formula is:

Range = Maximum value - Minimum value  

This Range formula applies to both grouped and ungrouped data, though the calculation method differs slightly based on the data type.  

 

Range of Ungrouped Data  

When we collect numbers or observations, we often want to know not just the average but also how spread out the data is. One simple way to measure this spread is by finding the range.

The Range of ungrouped data is the difference between the largest value (maximum) and the smallest value (minimum) in the dataset. Ungrouped data consists of raw numerical data that has not been sorted into groups or intervals.  

To calculate the range of ungrouped data, follow these steps:  

1. Arrange the data in ascending order (if needed).

2. Identify the highest value and the lowest value in the dataset.

3. Apply the range formula:
   
   Range = Highest value - Lowest value

Example:  

Data: 4, 7, 15, 20, 10, 12  

Highest value = 20  

Lowest value = 4  

Range = 20 - 4 = 16  

The range of ungrouped data helps analyze small sets of data without frequency distribution.  

 

Range of Grouped Data  

When the data is very large, it is usually arranged into class intervals (like 0-10, 10-20, etc.). This is called grouped data. In grouped data, we don’t have the exact values of every number, but we can still find the range using the boundaries of the classes.

To calculate the range of grouped data, use these steps:  

1. Identify the lowest class and the highest class from the frequency table.

2. Find the lower boundary of the lowest class and the upper boundary of the highest class.

3. Apply the range formula:
   
   Range = Upper boundary of highest class - Lower boundary of lowest class

Example:

Upper boundary of the highest class = 40

Lower boundary of the lowest class = 10

Range = 40 - 10 = 30

The range of grouped data helps in summarizing large data sets and understanding their spread.

 

How to Find the Range? 

To find the range in statistics, the first step is to arrange the given data in ascending order. This means we write all the observations starting from the smallest value to the largest value. Arranging the data makes it easier to clearly see which value is the lowest and which is the highest.

Once the data is arranged, we apply the range formula:

Range = Largest Value - Smallest Value

By subtracting the smallest observation from the largest observation, we get the range of the dataset. This result tells us how much the data is spread out.

Example:
The ages of 6 children are: 11, 8, 13, 10, 9, 12

Step 1: Arrange in ascending order → 8, 9, 10, 11, 12, 13
Step 2: Largest value = 13, Smallest value = 8
Step 3: Apply the formula → Range = 13 – 8 = 5

So, the range of ages is 5 years.

 

Knowing how to find the range is important for analyzing data, especially when comparing the spread of different datasets. 

 

Solved Examples on Range

Example 1: Find the range of given observations: 8, 12, 15, 19, 23.

Solution: Arrange the data in ascending order.

8, 12, 15, 19, 23

Here, the lowest value = 8 and the highest value = 23.

Range (X) = Max (X) - Min (X)
= 23 - 8
= 15

Hence, the required range is 15.

 

Example 2: Find the range of the following grouped data.

 

Solution:

The lowest class interval is 0 – 10 → Lower class limit = 0

The highest class interval is 20 – 30 → Upper class limit = 30

Range (X) = Highest limit - Lowest limit
= 30 - 0
= 30

Hence, the required range is 30.

 

Example 3: Find the mean, median, mode, and range of the given observations: 5, 7, 8, 9, 10, 30.

Solution: Arrange the data in ascending order.

5, 7, 8, 9, 10, 30

Step 1: Find the range.

Lowest value = 5

Highest value = 30

Range (X) = 30 - 5 = 25

Step 2: Find the mean.

Mean = (5 + 7 + 8 + 9 + 10 + 30) ÷ 6
= 69 ÷ 6
= 11.5

Step 3: Find the median.
There are 6 values (even number), so the median is the average of the 3rd and 4th terms.

Median = (8 + 9) ÷ 2 = 8.5

Step 4: Find the mode.
All values appear only once, so there is no mode.

Hence, the results are:

Range = 25

Mean = 11.5

Median = 8.5

Mode = None

 

Example 4: Find the range of the following grouped data.

Solution:

• The lowest class = 30 - 40 → Lower class limit = 30
  
  

• The highest class = 70 - 80 → Upper class limit = 80
  
  

Range (X) = Highest limit - Lowest limit
= 80 - 30
= 50

Hence, the required range is 50.

This example shows how the range helps in measuring data spread, while mean, median, and mode provide information about central values.

 

Practice Problems

1. Find the range for the following ungrouped data:
   3, 7, 10, 2, 9, 12

2. The class intervals and frequencies are given below. Find the range of grouped data:

3. A data set has a minimum value of 11 and a maximum value of 59. What is the range?
   
   

4. Given the data: 10, 15, 20, 25, 30
   What are the mean, median, mode, and range?
   
   

5. Calculate the range for the grouped data below:
   
   

This example shows how the range measures data spread, while the mean, median, and mode provide insight into central values.  

 

Conclusion  

The range in statistics is a basic yet powerful tool for measuring how data is spread. It is easy to calculate and helps to understand the variability of a dataset. By knowing the range of ungrouped and grouped data, and comparing them with the mean, median, and mode, one can get a complete view of how the data behaves. Learn how to find the range correctly and practice regularly with solved examples and practice problems to master this fundamental concept.

 

Frequently Asked Questions on Range

1. What will be the range of the following data: 32, 41, 28, 54, 35, 26, 33, 23, 38, 40?

Answer: To calculate range:

Step 1: Identify the highest value = 54

Step 2: Identify the lowest value = 23

Step 3: Subtract lowest from highest:

Range = 54 - 23 = 31

 

2. How to calculate the range?

Answer: The range of a data set is calculated as:

• Range = Highest value - Lowest value

It shows how spread out the values in a data set are.

 

3. What is an example of a range?

Answer:

Example:

In the data set: 5, 10, 15, 20

• Highest value = 20

• Lowest value = 5

• Range = 20 - 5 = 15

 

4. What is called range?

Answer: Range is a measure of spread in statistics. It is the difference between the largest and smallest values in a data set. It tells us how widely the numbers are spread apart.

 

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