Range in Statistics

In statistics, the range is used to describe the spread of data in a dataset. The range in math is the difference between the smallest and largest values. This simple calculation shows how much the values differ and gives a basic idea of the data’s overall spread.


However, the definition of range in math also comes with limitations. The range only considers the lower and upper values and does not account for how the other data points are distributed. It also ignores the number of data points in the dataset. Because of this, the range can be misleading when there are outliers, since excessively high or low values can significantly affect the result.


For example,

Consider the dataset as,

12, 18, 25, 30, 37

Highest value = 37

Lowest value = 12

Now,

The range = \(37–12 = 25\)

As discussed, range is the difference between the upper and lower values. So, the formula to calculate the range is \( R = H - L\), where R is the range, H is the maximum value, and L is the minimum value.

Follow these steps to calculate the range - \(\text{Range} = \text{Maximum value} - \text{Minimum value} \)
 

Step 1: Arrange the data set in ascending order


Step 2: Identify the upper and lower limits from the dataset


Step 3: Finding the range using the formula; \(\text{Range} = \text{Maximum value} - \text{Minimum value} \).
 

For instance, find the range of the given dataset: \(5, 12, 8, 20, 15\)
 

Step 1: Arrange the data set in order


That is \(5, 8, 12, 15, 20\)


Step 2: Identify the upper and lower limits from the dataset

The upper limit is 20

The lower limit is 5


Step 3: Find the difference between the minimum and maximum value.

Range = \(20 – 5 = 15\)

According to the rule of thumb, most of the data values fall within four standard deviations, that is, two standard deviations above the mean and two standard deviations below the mean.
The formula for standard deviation (σ) is,

\(\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}} \)

The range is easy to calculate, but it has several drawbacks:
 

• It does not provide information about how many data points are in the dataset.
   
• It cannot be used to determine other measures, like the mean, median, or mode.
   
• It is highly sensitive to outliers, meaning extreme values can significantly change the range.
   
• It is not suitable for open-ended distributions.

The formula used to calculate the range of a dataset is,

\(\text{Range} = \text{Maximum value} - \text{Minimum value} \)

In addition to this basic formula, there are specific methods for finding the range of both grouped and ungrouped data.

For continuous frequency distributions or grouped data, the range is calculated as the difference between the upper boundary of the highest class interval and the lower boundary of the lowest class interval. It is one of the simplest measures of dispersion and provides an overall idea of how spread out the observations are.


The formula to calculate the range for grouped data is:


Range = Upper class boundary of the highest interval - Lower class boundary of the lowest interval.

For continuous frequency distribution or grouped data, the range is the difference between the upper boundary of the highest class interval and the lower boundary of the lowest class interval. It is one of the simplest measures of dispersion and provides a clear picture of the data's overall spread.


The formula for calculating the range of grouped data is,


Range = Upper class boundary of the highest interval - Lower class boundary of the lowest interval.

The range of statistics is a complex mathematical topic; however, with some tips and tricks, it can be better understood. Some valuable tips and tricks are mentioned below.
 

• Make sure to carefully review your dataset to identify the actual highest and lowest values before calculating the range.
   

• Put your numbers from smallest to largest or largest to smallest; it helps you easily find the smallest and largest values.
   

• Big or small odd numbers can change the range, so make sure to check for any unusual values in your data.
   

• When comparing two sets of data, a larger range indicates the values vary widely, while a smaller range suggests the data is steadier and consistent.
   

• Try using the range with real-life examples such as temperature changes, exam marks, or stock prices. It helps you better understand the concept and see how it works in everyday situations.
  
   
• Parents can use familiar situations, such as children’s heights in class, daily temperatures, or pocket money, to explain how a range shows the spread of values.
  
   
• Teachers and parents can show children how to calculate the range for grouped data using class intervals.
  
   
• Children can better understand the concept when it relates to their everyday life.

The concept of range has numerous applications across various fields. Now let’s learn a few applications of range in statistics.   

 

• We use range to compare the prices of similar products; we use range to understand the price spread and to get an understanding of the budget. 
  
   
•  To analyze the performance of the students in an exam, we use range. 
  
   
• To analyze the performance of the players in sports, range is used.
  
   
• Meteorologists use the range to show the difference between the highest and lowest temperatures in a day, week, or month to analyze climate patterns.
  
   
• Teachers use the range of students’ marks to measure the variation in performance within a class or across multiple tests.