# Range in Statistics

The range in statistics for a given data set is the difference between the highest and lowest values. For example, if the given data set is {2,5,8,10,3}, then the range will be 10 – 2 = 8.

Thus, the range could also be defined as the difference between the highest observation and lowest observation. The obtained result is called the range of observation. The range in statistics represents the spread of observations.

## Range Formula

The formula of the range in statistics can simply be given by the difference between the highest and lowest values.

 Range = Highest Value – Lowest Value Or Range = Highest observation – Lowest observation Or Range = Maximum value – Minimum Value

## How to Find Range in Statistics?

To find the range in statistics, we need to arrange the given values or set of data or set of observations in ascending order. That means, firstly write the observations from the lowest to the highest value. Now, we need to use the formula to find the range of observations.

### Solved Examples

Example 1: Find the range of given observations: 32, 41, 28, 54, 35, 26, 23, 33, 38, 40.

Solution: Let us first arrange the given values in ascending order.

23, 26, 28, 32, 33, 35, 38, 40, 41, 54

Since 23 is the lowest value and 54 is the highest value, therefore, the range of the observations will be;

Range (X) = Max (X) – Min (X)

= 54 – 23

= 31

Hence, 31 is the required answer.

Example 2: Following are the marks of students in Mathematics: 50, 53, 50, 51, 48, 93, 90, 92, 91, 90. Find the range of the marks.

Solution: Arrange the following marks in ascending order, we get;

48, 50, 50, 51, 53, 90, 90, 91, 92, 93

Thus, the range of marks will be:

Range = Maximum marks – Minimum marks

Range = 93 – 48 = 45

Thus, 45 is the required range.

## Arithmetic Mean and Range in Statistics

In statistics, groups of data are commonly represented by arithmetic mean. Sometimes, the arithmetic mean is also referred to as average or just ‘mean’.

Basically, the mean is the central value of given data. To find the arithmetic mean of the data set, we have to add all the values in the set and then divide the resulting value by the total number of values.

Arithmetic mean = (Sum of all observations)/(Total number of observations)

### Solved Example

Let us find the arithmetic mean of the observations for which we evaluated the range in the above examples.

Example 1: Find the mean of the data set: 32, 41, 28, 54, 35, 26, 23, 33, 38, 40.

Solution: To find the mean, we have to add all the given values first.

Sum of observations = 32 + 41 + 28 + 54 + 35 + 26 + 23 + 33 + 38 + 40 = 350

Total number of observations = 10

Therefore, the mean of observations is:

Mean = (Sum of all observations)/(Total number of observations)

Mean = 350/10 = 35

Hence, 35 is the required arithmetic mean.

Example 2: Following are the marks of students in Mathematics: 50, 53, 50, 51, 48, 93, 90, 92, 91, 90. Find the mean of the marks.

Solution: Given, the marks of the students are:

50, 53, 50, 51, 48, 93, 90, 92, 91, 90

Mean = (Sum of all observations)/(Total number of observations)

Thus,

Sum of observations = 50 + 53 + 50 + 51 + 48 + 93 + 90 + 92 + 91 + 90 = 708

Total observations = 10

Therefore,

Arithmetic mean = 708/10 = 70.8

Hence, 70.8 is the required mean.

## Problems and Solutions

Q.1: If the data set has observations as: 4, 6, 7, 5, 3, 5, 4, 5, 2, 6, 2, 5, 1, 9, 6, 5, 8, 4, 6, 7. Then find:

(a) The maximum value?

(b) The minimum value?

(c) Range of data set

Solution: Let us arrange the given values from lowest to highest (increasing order).

1, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 9.

Clearly from the above arrangement, we can conclude that;

(a) The maximum value is 9.

(b) The minimum value is 1

(c) Range = 9 – 1 = 8

Q.2: What is the arithmetic mean of 4, 6, 7, 5, 3, 5, 4, 5, 2, 6, 2, 5, 1, 9, 6, 5, 8, 4, 6, 7?

Solution: To find the arithmetic mean, we have to use the below formula:

Arithmetic mean = (Sum of all observations)/(Total number of observations)

Sum of all observation = 1 + 2 + 2 + 3 + 4 + 4 + 4 + 5 + 5 + 5 + 5 + 5 + 6 + 6 + 6 + 6 + 7 + 7

+ 8 + 9 = 100

Total Number of Observation = 20

Arithmetic mean = (100/20) = 5

### Practice Questions

1. Find the range of the following data set:
1. 7,9,2,5,9,3,5,4
2. 100,120,113,189,190,201,145
3. 1555, 1670, 1750, 2013, 2540, 2820
4. 135, 150, 139, 128, 151, 132, 146, 149, 143, 141
2. Find the mean of the following data set:
1. 4, 6, 9, 3, 7
2. 12, 34, 45, 50, 24
3. 56, 67, 54, 34, 78, 43, 23
3. If the heights of 6 people are 140 cm, 142 cm, 150 cm, 149 cm, 156 cm, and 153 cm. Find the range and mean of the height.

## Frequently Asked Questions on Range in Statistics

Q1

### What is the range in statistics?

In statistics, the range of observations is the value that is obtained from the difference between the highest and lowest values.

Q2

### How to find the range in Maths?

To find the range of a given set of observations, we need to arrange them in ascending order first. Later, evaluate the difference between maximum and minimum values to get the range.

Q3

### What is the formula for finding range?

The formula for finding the range is given by:
Range (X) = Highest observation – Lowest observation

Q4

### What is the range of 3, 6, 9,12,15, 90?

Here, the highest observation is 90 and the lowest observation is 3, therefore the range is the difference between 90 and 3, i.e. 90 – 3 = 87.

Q5

### What is the difference between mean and range?

Mean represents the average value of a set of data whereas the range denotes the spread of observations. The mean is equal to the ratio of the sum of all observations and the total number of observations but the range is the difference between maximum and minimum values in the observation.