Introduction
Understanding dispersion or variability in data is an important part of statistics. Dispersion tells us how much the data values differ from each other or from the average. Two broad ways of measuring dispersion are absolute measures and relative measures. Additionally, important tools like variance and coefficient of variation help in understanding the spread of data. Let’s explore both these sets of comparisons in detail.
a) Absolute Measures vs Relative Measures of Dispersion
1. Meaning
- Absolute Measures: These give the dispersion in actual units of data. They show the amount of variation in numerical form.
- Relative Measures: These give the dispersion in percentage or ratio form. They are unitless and help in comparing datasets of different units.
2. Units
- Absolute: Have the same units as the original data (e.g., rupees, meters, kilograms).
- Relative: No units — they are pure numbers or percentages.
3. Examples
- Absolute Measures: Range, Mean Deviation, Standard Deviation, Quartile Deviation
- Relative Measures: Coefficient of Range, Coefficient of Variation, Coefficient of Quartile Deviation
4. Use
- Absolute: Useful for analyzing dispersion within a single dataset.
- Relative: Useful for comparing two or more datasets with different units or scales.
5. Comparison Example
Suppose you have:
- Standard deviation of height = 5 cm (absolute measure)
- Coefficient of variation = 5% (relative measure)
The coefficient allows you to compare this with another variable like weight, even though units are different.
b) Variance vs Coefficient of Variation
1. Definition
- Variance: It is the average of the squared differences from the mean. It shows how much values deviate from the mean.
- Coefficient of Variation (CV): It is the ratio of standard deviation to mean, expressed as a percentage. CV = (σ/μ) × 100
2. Formula
- Variance (σ²) = Σ(x – μ)² / n
- CV = (Standard Deviation / Mean) × 100
3. Units
- Variance: Units are squared (e.g., cm², Rs²)
- CV: Unitless; represented in %
4. Use
- Variance: Used to measure the spread or dispersion of data.
- CV: Used to compare variability between different datasets, even with different units or magnitudes.
5. Example
Suppose:
- Dataset A: Mean = 50, SD = 10 ⇒ Variance = 100, CV = 20%
- Dataset B: Mean = 100, SD = 15 ⇒ Variance = 225, CV = 15%
Even though Dataset B has higher variance, its CV is lower, meaning it is relatively more consistent.
Conclusion
Understanding the difference between absolute and relative measures of dispersion helps in choosing the right statistical tools depending on whether you’re comparing data sets or analyzing one set in detail. Similarly, variance helps in understanding the spread of data in units, while coefficient of variation helps in comparing the stability of different datasets. These tools are widely used in finance, quality control, business planning, and data analysis.