Range for Ungrouped and Grouped Data in Statistics – Class 11 Notes
Class 11 Mathematics | Written by Neeraj Anand
Published by ANAND TECHNICAL PUBLISHERS
Introduction to Range in Statistics
The range is one of the simplest measures of dispersion in statistics. It represents the difference between the highest and lowest values in a dataset. The range provides a quick measure of how spread out the data values are but does not give information about individual variations within the dataset.
In statistics, the range is used for both ungrouped data (raw data) and grouped data (organized into class intervals).
Table of Contents
Range Definition
Range is a fundamental statistical concept that helps us understand the spread or variability of data within a dataset.
In statistics, the difference between the highest and lowest observations in a given data is called its Range.
Range in statistics is the difference between the highest and lowest values in a dataset.
Range Formula For Ungrouped Data
The formula to find the range of ungrouped data or discrete distribution of data is given as:
Range = Highest value of the data set – Lowest value of the data set
How to Calculate Range For Ungrouped Data?
We can use following steps for range calculation:
- Identify the maximum value (the largest value) in your dataset.
- Identify the minimum value (the smallest value) in your dataset.
- Subtract the minimum value from the maximum value to find the range.
Range = Maximum value − Minimum value
Solved Examples of Range For Ungrouped Data
Example 1: Find the range of the data: 21, 6, 17, 18, 12, 8, 4, 13
Solution:
Given,
21, 6, 17, 18, 12, 8, 4, 13
Highest value = 21
Lowest value = 4
Range = Highest value – Lowest value
= 21 – 4
= 17
Example.2 : Consider the following dataset of exam scores for a class eleventh : 77, 89, 92, 64, 78, 95, 82
Find the Range of the above data
Solution:
Now To Calculate the range
Here, Select The Largest Score as Maximum Value and Smallest score as Minimum Value:
Maximum value = 95
Minimum value = 64
Range = 95 – 64 = 31
So, the range of the exam scores in this dataset is 31.
Example 3: Age (in years) of 6 boys and 6 girls are recorded as below:
| Girls | 6 | 7 | 9 | 8 | 10 | 10 |
| Boys | 7 | 9 | 12 | 14 | 13 | 17 |
(a) Find the range for each group.
(b) Find the range if the two groups are combined together.
Solution:
(a) The range for group of girls = 10 – 6 = 4
The range for group of boys = 17 – 7 = 10
(b) If the ages of the group of boys and girls are combined, then the range will be:
17 – 4 = 13
How to Calculate Range of Grouped Data ?
In the case of continuous frequency distribution or grouped data, the range is defined as the difference between the upper limit of the maximum interval of the grouped data and the lower limit of the minimum interval. It is the simplest measure of dispersion. It gives a comprehensive view of the total spread of the observations. Thus, the formula to calculate the range of a grouped data is given below:
Range = Upper-class boundary of the highest interval – Lower class boundary of the lowest interval.
We can understand it from the example mentioned below:
| Class Interval | Frequency |
|---|---|
| 0-10 | 12 |
| 10-20 | 10 |
| 20-30 | 15 |
| 30-40 | 13 |
| 40-50 | 11 |
Range = Upper Limit of the Last Class Interval – Lower Limit of First Class Interval = 50-0 = 50
Solved Examples of Range For Grouped Data
Example 1: Calculate the range for the given frequency distribution.
| Class Interval | 10 – 20 | 20 – 30 | 30 – 40 | 40 – 50 | 50 – 60 | 60 – 70 | 70 – 80 |
| Frequency | 2 | 3 | 14 | 8 | 3 | 8 | 2 |
Solution:
We know that the range of grouped data is given by the formula:
Range = Upper-class boundary of the highest interval – Lower class boundary of the lowest interval
Here, the Upper-class boundary of the highest interval = 80
Lower class boundary of the lowest interval = 10
Therefore, range = 80 – 10 = 70
Example 2: Find the range of the following data.
| CIass | 16 – 20 | 21 – 25 | 26–30 | 31 – 35 | 36–40 | 41–45 | 46–50 | 51–55 |
| Frequency | 5 | 6 | 12 | 14 | 26 | 12 | 16 | 9 |
Solution:
Given data is not continuous frequency distribution.
Now, we have to convert the given data into continuous frequency distribution by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit of each class interval.
| CIass | 15.5–20.5 | 20.5–25.5 | 25.5 – 30.5 | 30.5 – 35.5 | 35.5 – 40.5 | 40.5 – 45.5 | 45.5 – 50.5 | 50.5 – 55.5 |
| frequency | 5 | 6 | 12 | 14 | 26 | 12 | 16 | 9 |
Here,
Upper-class boundary of the highest interval = 55.5
Lower class boundary of the lowest interval = 15.5
Therefore, range = 55.5 – 15.5 = 40
Advantages and Disadvantages of Ranges in Statistics
The range in statistics has both advantages and disadvantages:
Advantages
- Easy to understand: The concept of range is simple and easy to grasp for people unfamiliar with statistics. It’s essentially the difference between the highest and lowest values in a dataset, making it intuitive.
- Quick to calculate: Computing the range involves only finding the maximum and minimum values in the dataset and subtracting them, making it a fast measure to calculate.
- Provides a basic measure of variability: Despite its simplicity, the range gives a basic indication of the spread or variability of the data. A larger range suggests greater variability, while a smaller range suggests less variability.
Disadvantages
- Sensitivity to outliers: The range is heavily influenced by extreme values (outliers) in the dataset. A single outlier can greatly inflate the range, potentially giving a misleading picture of the variability of the majority of the data.
- Does not consider distribution: The range does not take into account the distribution of values within the dataset. Two datasets with the same range can have very different distributions, leading to different interpretations of variability.
- Limited information: While the range provides a basic measure of variability, it does not provide any information about the distribution’s shape or central tendency. Other measures such as the interquartile range, variance, or standard deviation offer more comprehensive insights into the dataset’s characteristics.
- Sample size dependency: The range does not account for sample size, so datasets with different sample sizes may have similar ranges even if their variability differs significantly. This can lead to misinterpretations, especially when comparing datasets of different sizes.
Practice Questions On Range In Statistics
Q1. Calculate the range for the following dataset: 12, 15, 20, 25, 30, 35, 40, 45?
Q2. A dataset of temperatures in degrees Celsius for a week is given as follows: 18, 22, 20, 25, 19, 28, 17. Find the range?
Q3. You have a dataset of the heights (in inches) of a group of individuals: 62, 67, 71, 68, 70, 75, 61, 66, 69, 70. Determine the range of heights?
Q.4. Consider the following data for the number of books read by students in a year: 4,9,15,7,12,6. Find the range of the dataset.
Q.5. A survey of 10 people recorded their ages as follows: 25,32,28,45,34,50,29,41,33,36. Calculate the range of the ages.
Q.6. Given the following grouped data, calculate the range:
| Class Interval | Frequency |
|---|---|
| 0-20 | 8 |
| 20-40 | 12 |
| 40-60 | 15 |
| 60-80 | 10 |
Q.7. You have the following test scores of 5 students: 85,90,78,92,88. Determine the range of the test scores.
Q.8. A researcher records the temperatures (in °C) of 6 cities on a particular day: 12,15,14,19,21,16. Find the range of temperatures.
Q.9. For the grouped data below, determine the range:
| Class Interval | Frequency |
|---|---|
| 5-15 | 14 |
| 15-25 | 9 |
| 25-35 | 11 |
| 35-45 | 6 |
Q.10. Calculate the range of the following dataset: 17,24,32,28,41,35,30.
Range in Statistics – FAQs
Define Range in Statistics.
The range in statistics refers to the difference between the maximum and minimum values in a dataset. A larger range suggests greater variability, while a smaller range indicates less variation.
What is the formula for range in Statistics?
The formula for range in Statistics = Maximum Value – Minimum Value
How do you find Range in Statistics?
To find range of any dataset, we can use the following steps:
Step 1: Sort the data points in ascending or descending order.
Step 2: Find the difference between the first and last value.
Step 3: The range is the absolute value of the difference obtained in step 2.
What does the Range tell us about data?
The range provides insight into how much the data values vary from the lowest to the highest. It gives a basic sense of the spread of data points but does not provide information about the distribution or central tendency of the data.
When is the Range Useful?
The range is useful when you need a quick and simple measure to understand the spread of data. It’s often used in introductory statistics or when you want a basic overview of data variability.
What are Some Alternatives to the Range for Measuring Data Spread?
Alternatives to the range include measures like the interquartile range (IQR), standard deviation, and variance. These measures provide more comprehensive information about data spread and are less sensitive to outliers.
Can Range be Negative?
No, the range of the dataset can never be negative, as it is the difference between the maximum value and the minimum value. Therefore, the range can be either zero (when maximum and minimum values are same) or positive only.
How can I Interpret the Range?
Interpretation of the range depends on the specific dataset and context. A larger range indicates greater variability in the data, while a smaller range suggests less variability.
What will be the range of the following data: 32, 41, 28, 54, 35, 26, 33, 23, 38, 40?
The range is 54−23=3154 – 23 = 3154−23=31.
How to calculate range of any function?
To find the range of a function, determine all the possible values the function can take for all allowed input values. Analyze the function’s behavior (increasing, decreasing, boundedness) and use calculus if necessary.
How to Find the Range?
Range is calculated by finding the difference between the upper most and lower most value of dataset.
Why Range is called Measure of Dispersion?
Range is called a measure of dispersion because it quantifies the extent to which data points in a dataset are spread out.
Key Properties of Range
✅ Simple to Calculate – Just requires the maximum and minimum values.
✅ Highly Affected by Extreme Values – If the dataset has outliers, the range may not accurately represent variability.
✅ Gives a Quick Overview – Helps in understanding how spread out the data is.
Applications of Range in Real Life
📊 Weather Forecasting: Analyzing the temperature variation in a city.
📈 Stock Market Analysis: Understanding fluctuations in stock prices.
🏫 Education: Measuring the spread of students’ marks in an exam.
🛍 Business & Economics: Checking the price range of products in a market.
Limitations of Range
🔹 Does not consider how data is distributed within the range.
🔹 Can be misleading if there are extreme outliers.
🔹 Not as robust as other measures of dispersion like variance and standard deviation.
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