When we talk about statistics, most students are familiar with the measures of central tendency such as mean, median, and mode. But these only tell us about the center of the data. To understand how data is spread out, we need the measures of dispersion.
This blog explores dispersion in detail — from definitions and formulas to solved and unsolved examples, and even implementation using R, Python, and Excel.
What is Dispersion?
Dispersion refers to the degree of variation or spread of data values around a central value (like the mean). If data values are tightly clustered, dispersion is low; if they are widely scattered, dispersion is high.
In real-world applications — from finance and economics to engineering and medicine — understanding dispersion is crucial for decision-making.
Types of Measures of Dispersion
1. Range
- Definition: Difference between maximum and minimum value.
- Formula:
R=Xmax−XminR = X_{\text{max}} – X_{\text{min}}R=Xmax−Xmin
- Quick Insight: Easy to compute but very sensitive to outliers.
2. Quartile Deviation (Semi-Interquartile Range)
- Based on the spread of the middle 50% of data.
- Formula:
QD=Q3−Q12QD = \frac{Q_3 – Q_1}{2}QD=2Q3−Q1
- Useful in skewed distributions.
3. Mean Deviation (Average Absolute Deviation)
- Measures the average distance from the mean, ignoring direction.
- Formula:
MD=∑∣Xi−Xˉ∣NMD = \frac{\sum |X_i – \bar{X}|}{N}MD=N∑∣Xi−Xˉ∣
4. Variance and Standard Deviation
- Variance: Average of squared deviations from the mean.
- Standard Deviation: Square root of variance, more interpretable in real units.
- Formulas:
σ2=∑(Xi−Xˉ)2N,σ=σ2\sigma^2 = \frac{\sum (X_i – \bar{X})^2}{N}, \quad \sigma = \sqrt{\sigma^2}σ2=N∑(Xi−Xˉ)2,σ=σ2
These are the most widely used measures, especially in research and analytics.
5. Coefficient of Variation (CV)
- Standard deviation expressed as a percentage of the mean.
- Formula:
CV=σXˉ×100CV = \frac{\sigma}{\bar{X}} \times 100CV=Xˉσ×100
- Useful for comparing variability across different datasets.
Solved Examples
Example 1:
The marks of 5 students are: 10, 15, 20, 25, 30. Find the standard deviation.
Solution: Xˉ=10+15+20+25+305=20\bar{X} = \frac{10+15+20+25+30}{5} = 20Xˉ=510+15+20+25+30=20 σ=(10−20)2+(15−20)2+(20−20)2+(25−20)2+(30−20)25=50=7.07\sigma = \sqrt{\frac{(10-20)^2 + (15-20)^2 + (20-20)^2 + (25-20)^2 + (30-20)^2}{5}} = \sqrt{50} = 7.07σ=5(10−20)2+(15−20)2+(20−20)2+(25−20)2+(30−20)2=50=7.07
Thus, the standard deviation is 7.07.
Example 2:
The daily wages of 100 workers are normally distributed with mean = 500 and SD = 50. Find the CV.
Solution: CV=50500×100=10%CV = \frac{50}{500} \times 100 = 10\%CV=50050×100=10%
Unsolved Problems (for practice)
- Calculate the quartile deviation for the following dataset: 5, 7, 9, 10, 12, 14, 15, 20.
- Find the mean deviation about the mean for the data: 8, 12, 15, 20, 25.
- For the series 2, 4, 6, 8, 10, compute variance and standard deviation.
Applications with R, Python, and Excel
In R
data <- c(10, 15, 20, 25, 30)
mean(data)
sd(data)
var(data)
In Python (using NumPy)
import numpy as np
data = [10, 15, 20, 25, 30]
mean = np.mean(data)
std_dev = np.std(data)
variance = np.var(data)
print(mean, std_dev, variance)
In Excel
- Use
=AVERAGE(range)for mean - Use
=STDEV.P(range)for population standard deviation - Use
=VAR.P(range)for variance
Why Students Must Master Dispersion
- University exams: Core topic in undergraduate statistics.
- Data science: SD and variance form the foundation of probability, regression, and machine learning.
- Research & business: CV and MD are widely applied in economics, finance, and risk analysis.
Final Thoughts
Understanding measures of dispersion is just as important as knowing averages. While averages tell us the center, dispersion reveals the story of variability — which is often where the real insights lie.
To explore in detail, check out our premium notes with solved and unsolved problems (with LaTeX-formatted explanations and R/Python/Excel implementations). Perfect for university students and exam preparation.
