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.
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
Range of 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
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
How to Find the Range
To understand the method of calculating range for different types of data in statistics, go through the solved examples given below. These examples help in understanding how to find the range for discrete and continuous data.
Solved Examples
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: Age (in years) of 6 boys and 6 girls are recorded as below:
(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
Example 3: Calculate the range for the given frequency distribution.
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 4: Find the range of the following data.
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.
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
&
Win
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