# The Range

## Introduction

• Ram, Arjun, Ravi, Ashok are three friends of the same class and they are awaiting their exam results. All of them got 85% marks which were shocking as their response to the different papers were different, few had written English very well and few had written it bad. So how did this happen?
• Let us have a look at their scorecard of individual subjects which is out of 100:
Subject/Student English Hindi Maths Science Total percentage
Ram 70 96 85 89 85
Arjun 99 45 99 97 85
Ravi 82 88 80 90 85
Ashok 60 82 99 99 85
• It is well known that the total percentage is the average percentage of each subject. Though the total percentage of all the four is the same, the scoring pattern of each of them is very different from each other.
• Hence, Average or Mean gives us detail about the overall picture only and it skips the individual contribution of all the elements.
• In other words, Average gives us information about the size or value of elements of the dataset (total percentage) and not about the spread of the values in elements i.e. how much or how less is the contribution of an element (percentage in each subject).
• The measure of dispersion helps us to overcome the drawback of the Mean observed above, it helps in understanding the contribution of each element in a dataset.
• Dispersion is a measure to find out the extent to which values on element differ from the Mean of dataset i.e. in the above example measure of dispersion will give us an idea that how much score did Ram got in each subject (how much or how less than 85).
• There are various ways to measure the dispersion of a dataset which we will study in upcoming sections.

## Range

• The range is a method used to measure the Dispersion of a dataset, in simple words, it is the difference between two boundaries commonly known as start and endpoint. E.g. Salary in the range of ₹20000 to ₹30000, Shops in the range of 10km, etc.
• In statistics, the definition of a range is a little different, instead of defining it as the upper and lower limit, it is defined as the difference between them. It is the difference between the highest and lowest element of a dataset.

#### Example 1:

• Consider a dataset: 2,5,6,7,8,20, 50,100,200, 500. Calculate its Range.
• Solution: In the above dataset, Range = Value of highest Element – Value of Lowest Element
• Hence, Range = 500-2 = 498

#### Example 2:

• Let us consider our Earlier example of marks of four students and calculate the Range.
Subject/Student English Hindi Maths Science Total percentage
Ram 70 96 85 89 85
Arjun 99 45 99 97 85
Ravi 82 88 80 90 85
Ashok 60 82 99 99 85
• Solution: Rewriting the individual marks in the forms of a dataset and calculating their range.
• Ram: 70,96,85,89 Range = 96-70 = 26
• Arjun: 99, 45, 99, 97 Range = 99-45 = 54
• Ravi: 82,88,80,90 Range = 90-80 = 10
• Ashok: 60,82,99,99 Range = 99-60 = 39
• Sorting the students in ascending order on their Range: Ravi, Ram, Ashok, Arjun. It is clearly established that low range implies more consistency in the score and high range defines variations in the score.
• Ravi performed best in all the subjects as he was consistent in his scores whereas Ashok had few subjects as his strength and few as his weakness due to which he had a large range than others.
• By looking at the average percentage (85%) we could not make out the difference between the individual performance of the four students but by using Range we can easily see the difference in the performance as we have an idea regarding their performance in the individual subjects.

## Important Fact

#### Example 3:

• Let us take a dataset: 2, 5, 7, 9, 10, 21
• Mean = (2+5+7+9+10+21)/6 = 54/6 = 9
• Range = 21 – 2 = 19
• Lowest Element = 2; its distance from Mean = 9 – 2 =7
• Highest Element = 21; its distance from Mean = 21 – 9 = 12
• Adding the above two values: 12+ 7 = 9 + Range
• Hence, it is concluded that Range is the sum of the difference between the smallest and largest elements from the mean.
• Any change in any values except the lowest and highest does not impact the Range of dataset. 