101 LearnersLast updated on June 4th, 2025
Measures of Dispersion
Measures of dispersion are positive real numbers that are used to measure the dispersion of data around a central value. These measures will help to understand how widely spread out or scattered the given data is. To describe the data variability, we can use measures of dispersion. In this topic, we will explore the various metrics of dispersion.
What are the Measures of Dispersion?
Measures of dispersion help to analyze the spread of data within a dataset. It indicates the extent to which a data varies from the average. The five measures of dispersion include range, variance, standard deviation, mean deviation, and quartile deviation.
We can use these metrics to identify whether a given dataset is uniform or diverse. The dispersion value is zero when all data points are the same. However, the measures of dispersion increase, as the differences between data points increase.
Difference Between Measures of Dispersion and Central Tendency
In statistics, data can be described by fundamental concepts such as measures of central tendency and measures of dispersion.
Measures of dispersion quantify data variability, while central tendency describes its average behaviour. There are five common measures of dispersion. They are range, variance, standard deviation, mean deviation, and quartile deviation. While mean, median, and mode are the three common measures of central tendency. Measures of dispersion are always non-negative and increase as data variability rises.
The value of central tendency can be any real number, which is based on its data distribution. To understand how data values deviate from the average, measures of dispersion are helpful, while measures of central tendency find a single representative value to summarize the given data.
What are the Types of Measures of Dispersion?
The measures of dispersion are classified into two main categories. They are absolute measures of dispersion and relative measures of dispersion. These main types are further divided into different subcategories. The various parameters of these types share the same unit. Now let us examine the two main types of measures of dispersion.
Absolute Measure of Dispersion and the Formulas
The absolute measure of dispersion is a measure of dispersion that is quantified and expressed in the same units as the data. Meters, kilograms, and dollars are some examples of the absolute measures of dispersion that are represented in the same units as the data. For instance, when the standard deviation of BVT company’s salary distribution is $700, that indicates salaries vary by around $700 from the mean. Here are some of the absolute measures of dispersion:
Range: It is an absolute measure of dispersion that can be defined as the difference between the distribution’s maximum and minimum values. To calculate the range, the formula we can employ is:
Range = H - S
Here, H is the highest value and the S is the smallest value in the dataset.
Mean deviation: It is the difference between each data point and the mean, calculated as the arithmetic mean. The formula for finding the mean deviation is:
Mean deviation = ∑n1 (x - x̄) / n
Here x̄ denotes the mean, median, or mode of the dataset and it is the central value.
n is the number of values and x is the individual data point.
Standard deviation: It is the square root of the mean of the squared deviations from the average value of the dataset. Here there are two types of standard deviation, they are population standard deviation and sample standard deviation. Hence, they have separate formulas.
For population standard deviation:
σ = √ σ2
For sample standard deviation:
s = √ s2
Variance: It is the mean or average of the squared deviations from the mean of the provided dataset. The formula for population variance is:
σ2 = ∑ (xi - μ)2 / N
The formula for calculating sample variance is:
S2 = ∑n1 (xi - x̄)2 / (n - 1)
Here, xi refers to each value of a dataset.
x̄ is the mean.
n is the number of values in the dataset.
Interquartile range: It is defined as the difference between the upper quartile (Q3) and the lower quartile (Q1). It has the formula of Q3 - Q1. Here, Q3 is the upper or the third quartile.
Q1 is the lower or first quartile.
The Relative Measures of Dispersion and the Formulas
To compare separate datasets with different units, relative measures of dispersion are used. They are represented as percentages and ratios also, these measures are unitless. Here, some of the few relative measures of dispersion are listed below:
Coefficient of Range: In a dataset, the coefficient of range is calculated as the ratio of the difference between the maximum and the minimum values to the sum of the two values. The formula for calculating the coefficient of range is:
(H - S) / (H + S)
Coefficient of Variation: It is expressed as a percentage. In a dataset, the coefficient of variation (CV) is the ratio of the standard deviation to the mean. The formula for the coefficient of variation (CV) is:
CV = (Standard deviation / Mean) × 100
Coefficient of Mean Deviation: It is the ratio of the mean deviation to the central value, from which it is used to calculate. The following is the formula:
Coefficient of Mean Deviation = Mean deviation / x̄
Here, x̄ is the central value, from which the mean deviation is used to calculate.
Coefficient of Quartile Deviation: The coefficient of quartile deviation is calculated as the ratio of the difference between the first and third quartiles to their sum. The formula for finding the coefficient of quartile deviation is:
Q3 - Q1 / Q3 + Q1
Real-life Applications of Measures of Dispersion
To know how much the given data deviates from a central value point, the fundamental concept of measures of dispersion is used. The real-world significance of the measures of dispersion is limitless.
- Business professionals and marketing teams use measures of dispersion to assess the risk of starting a new business, investing in mutual funds, or trading in stock markets. For instance, they can assess the risk of stocks with low standard deviation.
- Teachers and education analysts use measures of dispersion to evaluate student performance. For example, if the student’s performance is consistent with low dispersion, the teachers can identify that there is no need for any new tutorial methods.
- The manufacturing companies can use the measures of dispersion such as standard deviation to monitor the quality and consistency of their product manufacturing.
- Weather forecasters can apply the measures of dispersion to analyze the climatic change of a city over time. For instance, if the temperature range is highly inconsistent, meteorologists can forecast an unpredictable climatic change.
- In the fields of medicine and sports analysis, the measures of dispersion play a crucial role in evaluating the performance of drugs or athletes.
Common Mistakes and How to Avoid Them on Measures of Dispersion
In mathematics and statistics, students need to calculate and evaluate how much the given data is spread around its central value. For this purpose, understanding the concept of measures of dispersion is crucial to solving complex mathematical problems and making better conclusions. However, students make some common errors while performing the measures of dispersion. Here are some mistakes that are related to the measures of dispersion and their helpful solutions to avoid them.
Mistake 1
Confusion between samples and population variance.
Students should learn the proper formula and difference between samples and population variance. Otherwise, if they incorrectly apply the formulas, it will end up in wrong calculations and conclusions. Remember that, for population variance, students should divide the value by N, and for the sample variance, divide by N - 1.
Solved examples of Measures of Dispersion
Problem 1
Calculate the range for the given dataset: 4, 8, 10, 15, 28.
24
Explanation
To find the range for a dataset, we have to apply the formula for range:
Range = H - S
Here, H = 28
S = 4
Range = 28 - 4 = 24
The range for the dataset is 24.
Problem 2
Find the sample variance of the dataset: 1, 2, 4, 6.
4.92
Explanation
The formula for calculating sample variance is:
S2 = ∑n1 (xi - x̄)2 / (n - 1)
Here, n = 4
n - 1 = 4 - 1 = 3
x̄ = (1 + 2 + 4 + 6) / 4 = 13 / 4 = 3.25
Next, the variance is:
S2 = (1 - 3.25)2 - (2−3.25)2 - (4−3.25)2 - (6−3.25)2 / 3
S2 = (−2.25)2 - (−1.25)2 - (0.75)2 - (2.75)2 / 3
S2 = 5.0625 + 1.5625 + 0.5625 + 7.5625 / 3
S2 = 14.75 / 3
S2 = 4.9167
So, the sample variance is 4.92.
Problem 3
Find the interquartile range for the dataset 20, 25, 30, 35, 40, and 45.
15
Explanation
Here, we have to find the Q3 and Q1.
When the dataset contains an even number of values, it can be split into two equal parts:
The total number of data points is 6
Hence, n = 6
The lower half: 20, 25, 30
The upper half: 35, 40, 45
Then, we need to find the first quartile (Q1):
Q1 = 25 (the median)
Next, we can find the Q3:
Q3 = 40
So, we get the values. Now, we can substitute the values:
Interquartile range = Q3 - Q1
40 - 25 = 15
The interquartile range for the dataset is 15.
Problem 4
V Care company records the wages of two departments: Department A: Mean salary = $25,000, Standard deviation = $5,000 Department B: Mean salary = $30,000, Standard deviation = $6,000 Find which department has more relative variation using the coefficient of variation.
Both the departments have 20% as the relative variation.
Explanation
The formula for finding the coefficient of variation is:
CV = (Standard deviation / Mean) × 100
For the department A, the CV is:
CV = (5,000 / 25,000) × 100
CV = 0.2 × 100
CV = 20%
For the department B, the CV is:
CV = (6,000 / 30,000) × 100
CV = 0.2 × 100
CV = 20%
Hence, the A and B departments have the same relative variation (10%).
Problem 5
Calculate the coefficient of range for the dataset: 11, 21, 22, 34, 50.
0.639
Explanation
The formula for calculating the coefficient of range is:
(H - S) / (H + S)
Here, H = 50
S = 11
Now, we can substitute the values to the formula:
Coefficient of range = (50 - 11) / (50 +11)
Coefficient of range = 39 / 61 = 0.639
The coefficient of the range is 0.639
FAQs on Measures of Dispersion
1.Define measures of dispersion.
2.What are the types of measures of dispersion?
3.Are the range and coefficient of range the same?
4.What is the formula for interquartile range(IQR)?
5.Give some examples of the units of an absolute measure of dispersion.
6.How can children in Saudi Arabia use numbers in everyday life to understand Measures of Dispersion?
7.What are some fun ways kids in Saudi Arabia can practice Measures of Dispersion with numbers?
8.What role do numbers and Measures of Dispersion play in helping children in Saudi Arabia develop problem-solving skills?
9.How can families in Saudi Arabia create number-rich environments to improve Measures of Dispersion skills?
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Jaipreet Kour Wazir
About the Author
Jaipreet Kour Wazir is a data wizard with over 5 years of expertise in simplifying complex data concepts. From crunching numbers to crafting insightful visualizations, she turns raw data into compelling stories. Her journey from analytics to education ref
Fun Fact
: She compares datasets to puzzle games—the more you play with them, the clearer the picture becomes!
