Calculation of Range | Quartile Deviation | Coefficient - Measures of Dispersion | Absolute | Relative

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In this article, we will discuss the Calculation of Range in Statistics, coefficient of range, Quartile, Interquartile Deviation, and Coefficient of Quartile Deviation, Absolute and relative measures – Measures of Dispersion are used to represent the scattering of data. These are the numbers that show the various aspects of the data spread across various parameters.

Measures of Dispersion in Statistics

Let’s learn about the measure of dispersion in statistics, its types, formulas, and examples in detail.

Dispersion in Statistics

Dispersion in statistics is a way to describe how spread out or scattered the data is around an average value. It helps to understand if the data points are close together or far apart.

Dispersion shows the variability or consistency in a set of data. There are different measures of dispersion like range, variance, and standard deviation.

Measure of Dispersion in Statistics

Measures of Dispersion measure the scattering of the data. It tells us how the values are distributed in the data set. In statistics, we define the measure of dispersion as various parameters that are used to define the various attributes of the data.

These measures of dispersion capture variation between different values of the data.

Types of Measures of Dispersion

Measures of dispersion can be classified into the following two types :

Absolute Measure of Dispersion
Relative Measure of Dispersion

These measures of dispersion can be further divided into various categories. They have various parameters and these parameters have the same unit.

Measures of Dispersion Types

Let’s learn about them in detail.

Absolute Measure of Dispersion

The measures of dispersion that are measured and expressed in the units of data themselves are called Absolute Measure of Dispersion. For example – Meters, Dollars, Kg, etc.

Some absolute measures of dispersion are:

Range: It is defined as the difference between the largest and the smallest value in the distribution.

Mean Deviation: It is the arithmetic mean of the difference between the values and their mean.

Standard Deviation: It is the square root of the arithmetic average of the squares of the deviations measured from the mean.

Variance: It is defined as the average of the square deviation from the mean of the given data set.

Quartile Deviation: It is defined as half of the difference between the third quartile and the first quartile in a given data set.

Interquartile Range: The difference between the upper(Q₃ ) and lower(Q₁) quartile is called the interquartile range. Its formula is given as Q₃ – Q_1.

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Relative Measure of Dispersion

We use relative measures of dispersion to measure the two quantities that have different units to get a better idea about the scattering of the data.

Here are some of the relative measures of dispersion:

Coefficient of Range: It is defined as the ratio of the difference between the highest and lowest value in a data set to the sum of the highest and lowest value.

Coefficient of Variation: It is defined as the ratio of the standard deviation to the mean of the data set. We use percentages to express the coefficient of variation.

Coefficient of Mean Deviation: It is defined as the ratio of the mean deviation to the value of the central point of the data set.

Coefficient of Quartile Deviation: It is defined as the ratio of the difference between the third quartile and the first quartile to the sum of the third and first quartiles.

Calculation of Range in Statistics

Calculation of Range in Statistics is a fundamental statistical concept that helps us understand the spread or variability of data within a dataset. Range in Statistics provides valuable insights into the extent of variation among the values in a dataset. Range quantifies the difference between the highest and lowest values in the dataset.

In this article, we will learn the definition, application, calculation, and constraints of range in statistics. Let’s start learning about range with a clear definition of range in statistics and range applications.

Table of Content

What is Range?
Range Formula
Range in Dataset
Range Applications

What is Range?

Calculation of Range in statistics is the difference between the highest and lowest values in a dataset. The range offers a straightforward measurement of the data’s spread or variability. The range statistic is simple to calculate, but it has limitations because it only takes into consideration the maximum and minimum values and ignores the distribution of values across the dataset.

Range Formula

Calculation of Range in Statistics

Range = Maximum Value – Minimum Value

Here’s a step-by-step explanation of how to calculate the range:

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.

Here Is An Solved Example To Find Range

Example: Consider the following dataset of exam scores for a class tenth:

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.

Range in Dataset

calculation of range in statistics – The range of a dataset is quite simple to understand. It is the difference between the highest (maximum) and lowest (minimum) values in that dataset. Mathematically, the formula for calculating the range is as follows:

Range = Maximum Value – Minimum Value

This simple formula provides a quick way to quantify the spread of data.

Calculation of Range in Grouped Data

In Grouped data where the datasets are arranged in Class Intervals, the Range is find by subtracting the lower limit of the first class interval and the upper limit of the last class 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

Range Applications

The applications of the calculation of range in statistics are mentioned below:

The range has its application in various fields, such as mathematics, science, economics, and social sciences.
Range is used to analyze the variation and dispersion of a dataset.
The range is used in educational assessments to understand the variation in scores of Students
In clinical trials and medical research, the range of outcomes for a particular treatment or medication is studied to determine its effectiveness and potential side effects.
In sports, range can be applied to analyze a player’s performance.

Solved Examples on Range

Example 1: You are given a dataset of the ages of students in a classroom:

18, 19, 20, 21, 22, 35, 18, 23

Solution:

Maximum Value = 35

Minimum Value = 18

Range = 35 – 18 = 17

The range of ages among the students is 17 years.

Example 2: Consider a dataset of exam scores for a class:

Scores: 85, 92, 78, 96, 64, 89, 75, find the range?

Solution:

Maximum Value = 96

Minimum Value = 64

Range = 96 – 64 = 32

So, the range of the exam scores is 32.

Example 3: Imagine a dataset of monthly rainfall (in millimeters) for a city for the past year:

Rainfall: 50, 48, 52, 58, 45, 70, 65, 80, 40, 42, 75, 90, find the range of monthly rainfall for the city?

Solution:

Maximum Value = 90

Minimum Value = 40

Range = 90 – 40 = 50

The range of monthly rainfall for the city is 50 mm

Coefficient of Range

The ratio of the difference between two extreme items (the largest and smallest) of the distribution to their sum is known as the Coefficient of Range. The coefficient of the range is a relative measure of dispersion. Symbolically, range can be expressed as:

$Coefficient~of~Range=\frac{Largest~Item~(L)-Smallest~Item~(S)}{Largest~Item~(L)+Smallest~Item~(S)}$

Merits of Range

1. It is easy to understand and calculate.
2. It provides a quick measure of variability.
3. Range provides an overview of the data at once.

Demerits of Range

1. Range is not based on all of the observations. The range of distribution remains the same if every item is changed except for the smallest and largest items.
2. Fluctuations in sampling have a big impact on range. Its value differs widely between samples.
3. It does not provide any insight into the pattern of distribution. Two distributions can have the same range but different patterns of distribution.

Range and Coefficient of Range in Different Series

(I) Calculation of Rnage in Individual Series

Example:

The scores of 15 students are shown in the data below:

Range and Coefficient of Range in Individual Series

Calculate the range and coefficient of range.

Solution:

In ascending order, the scores are 21, 25, 28, 30, 31, 32, 35, 40, 42, 49, 50, 51, 55, 60, 65

For the given values of scores, the Largest Item (L) = 65 and the Smallest Item (S) = 21

Range = L- S

= 65 – 21

Range = 44

$Coefficient~of~Range=\frac{L-S}{L+S}$

$Coefficient~of~Range=\frac{65-21}{65+21}$

$=\frac{44}{86}$

Coefficient of Range = 0.51

(II) Calculation of Range in Discrete Series

The values of the largest (L) and smallest (S) items in a discrete series should not be confused with the largest and smallest frequencies. They represent the largest and smallest values of the variable. Therefore, the range is determined without taking into account their frequencies by subtracting the smallest item from the largest item.

Example:

Find the range and coefficient of the range of the following distribution:

Range and Coefficient of Range in Discrete Series

Solution:

Range (R) = Largest item (L) – Smallest item (S)

= 9-3

Range = 6

$Coefficient~of~Range=\frac{L-S}{L+S}$

$Coefficient~of~Range=\frac{9-3}{9+3}$

$=\frac{6}{12}$

Coefficient of Range = 0.5

(III) Calculation of Range in Continuous Series

There are two ways to compute the range and coefficient of range for continuous frequency distributions:

1. First Method: Calculate the difference between the lower limits of the lowest-class interval and the upper limit of the highest-class interval.

2. Second Method: Calculate the difference between the midpoints of the lowest-class interval and the highest-class interval.

Note: Both methods will provide different results. However, both answers will be accurate.

Example:

Calculate the range and coefficient of range.

Range and Coefficient of Range in Continuous Series

Solution:

Range and Coefficient of Range by the First Method:

Range (R) = Largest Item (L) – Smallest Item (S)

= 70 – 10

Range = 60

$Coefficient~of~Range=\frac{L-S}{L+S}$

$Coefficient~of~Range=\frac{70-10}{70+10}$

$=\frac{60}{80}$

Coefficient of Range =0.75

Range and Coefficient of Range by the Second Method:

Range (R) = Mid-point of the Highest Class – Mid-point of the Lowest Class

= 65 – 15

Range = 50

$Coefficient~of~Range=\frac{L-S}{L+S}$

$Coefficient~of~Range=\frac{65-15}{65+15}$

$=\frac{50}{80}$

Coefficient of Range = 0.625

Quartile Deviation

What is Quartile Deviation?

Quartile Deviation or Semi-Interquartile Range is the half of difference between the Upper Quartile (Q₃) and the Lower Quartile (Q₁). In simple terms, QD is the half of inter-quartile range. Hence, the formula for determining Quartile Deviation is as follows:

$Quartile~Deviation=\frac{Q_3-Q_1}{2}$

Where,

Q₃ = Upper Quartile (Size of $3[\frac{N+1}{4}]^{th}$ item)

Q₁ = Lower Quartile (Size of $[\frac{N+1}{4}]^{th}$ item)

What is the Coefficient of Quartile Deviation?

As Quartile Deviation is an absolute measure of dispersion, one cannot use it for comparing the variability of two or more distributions when they are expressed in different units. Therefore, to compare the variability of two or more series with different units it is essential to determine the relative measure of Quartile Deviation, which is also known as the Coefficient of Quartile Deviation. It is studied to make the comparison between the degree of variation in different series. The formula for determining the Coefficient of Quartile Deviation is as follows:

$Coefficient~of~Quartile~Deviation=\frac{Q_3-Q_1}{Q_3+Q_1}$

Where,

Q₃ = Upper Quartile (Size of $3[\frac{N+1}{4}]^{th}$ item)

Q₁ = Lower Quartile (Size of $[\frac{N+1}{4}]^{th}$ item)

Calculation of Quartile Deviation in Different Series

1. Individual Series:

Example:

With the help of the data given below, find the interquartile range, quartile deviation, and coefficient of quartile deviation.

Solution:

Q₁ = $Size~of~[\frac{N+1}{4}]^{th}~item=Size~of~[\frac{7+1}{4}]^{th}~item=Size~of~2^{nd}~item$

Q₁ = 140

Q₃ = $Size~of~3[\frac{N+1}{4}]^{th}~item=Size~of~3[\frac{7+1}{4}]^{th}~item=Size~of~6^{th}~item$

Q₃ = 268

Interquartile Range = Q₃ – Q₁ = 268 – 140 = 128

Quartile Deviation = $\frac{Q_3-Q_1}{2}=\frac{268-140}{2}=64$

Coefficient of Quartile Deviation = $\frac{Q_3-Q_1}{Q_3+Q_1}=\frac{268-140}{268+140}=\frac{128}{408}=0.31$

Interquartile Range = 128

Quartile Deviation = 64

Coefficient of Quartile Deviation = 0.31

2. Discrete Series:

Example:

From the following table giving marks of students, calculate the interquartile range, quartile deviation, and coefficient of quartile deviation.

Solution:

Q₁ = $Size~of~[\frac{N+1}{4}]^{th}~item=Size~of~[\frac{199+1}{4}]^{th}~item=Size~of~50^{th}~item$

Q₁ = 68

Q₃ = $Size~of~3[\frac{N+1}{4}]^{th}~item=Size~of~3[\frac{199+1}{4}]^{th}~item=Size~of~150^{th}~item$

Q₃ = 88

Interquartile Range = Q₃ – Q₁ = 88 – 68 = 20

Quartile Deviation = $\frac{Q_3-Q_1}{2}=\frac{88-68}{2}=10$

Coefficient of Quartile Deviation = $\frac{Q_3-Q_1}{Q_3+Q_1}=\frac{88-68}{88+68}=\frac{20}{156}=0.12$

Interquartile Range = 20

Quartile Deviation = 10

Coefficient of Quartile Deviation = 0.12

3. Continuous Series:

Example:

Calculate interquartile range, quartile deviation, and coefficient of quartile deviation from the following figures:

Solution:

Q₁ = $Size~of~[\frac{N}{4}]^{th}~item=Size~of~[\frac{100}{4}]^{th}~item=Size~of~25^{th}~item$

Q₁ lies in the group 20-30

l₁ = 20, c.f. = 24, f = 29, i = 10

$Q_1=l_1+\frac{\frac{N}{4}-c.f.}{f}\times{i}=20+\frac{25-24}{29}\times{10}=20+0.34$

Q₁ = 20.34

Q₃ = $Size~of~[\frac{3N}{4}]^{th}~item=Size~of~[\frac{3\times100}{4}]^{th}~item=Size~of~75^{th}~item$

Q₃ lies in the group 30-40

l₁ = 30, c.f. = 53, f = 24, i = 10

$Q_1=l_1+\frac{\frac{3N}{4}-c.f.}{f}\times{i}=30+\frac{75-53}{24}\times{10}=30+9.16$

Q₃ = 39.16

Interquartile Range = Q₃ – Q₁ = 39.16 – 20.34 = 18.82

Quartile Deviation = $\frac{Q_3-Q_1}{2}=\frac{39.16-20.34}{2}=9.41$

Coefficient of Quartile Deviation = $\frac{Q_3-Q_1}{Q_3+Q_1}=\frac{39.16-20.34}{39.16+20.34}=\frac{18.82}{59.5}=0.31$

Interquartile Range = 18.82

Quartile Deviation = 9.41

Coefficient of Quartile Deviation = 0.31

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Calculation of Range | Quartile Deviation | Coefficient – Measures of Dispersion | Absolute | Relative | Statistics