Questions
4–6 questions per board paper
Difficulty
Medium
Importance
Essential for Class 11/12 Statistics units
Overview
Measures of Dispersion quantify the spread or variability of a dataset around its central tendency, essential for understanding data reliability. Mastering these metrics is critical for board exams as they bridge the gap between descriptive statistics and analytical interpretation.
Range and Quartile Deviation
Range is the simplest measure of dispersion representing the difference between extreme values, while Quartile Deviation (Semi-Interquartile Range) focuses on the middle 50% of the data. These measures are robust against outliers but limited in their ability to describe overall data distribution.
- Range = L - S (Largest - Smallest value)
- Coefficient of Range = (L - S) / (L + S)
- Quartile Deviation (QD) = (Q3 - Q1) / 2
- Coefficient of Quartile Deviation = (Q3 - Q1) / (Q3 + Q1)
- Interquartile Range = Q3 - Q1
Mean Deviation
Mean Deviation measures the average of the absolute differences between each data point and the mean or median. Unlike standard deviation, it ignores the algebraic signs of deviations, making it a straightforward measure of average spread.
- MD about Mean = Σ|x - x̄| / n
- MD about Median = Σ|x - M| / n
- Coefficient of Mean Deviation = MD / Mean or Median
- Sum of absolute deviations is minimized when taken about the Median
Standard Deviation and Variance
Standard Deviation is the most widely used measure of dispersion as it accounts for the magnitude of deviations by squaring them, overcoming the limitations of Mean Deviation. It is the square root of the variance and is essential for higher-level inferential statistics.
- Variance (σ²) = Σ(x - x̄)² / n
- Standard Deviation (σ) = √Variance
- Root Mean Square Deviation = √[Σ(x - A)² / n]
- Coefficient of Variation (CV) = (σ / x̄) * 100
- SD is independent of change of origin but dependent on change of scale
Lorenz Curve
The Lorenz Curve is a graphical representation used to visualize the distribution of wealth or income inequality within a population. The further the curve bows away from the 45-degree line of perfect equality, the greater the dispersion or inequality.
- Diagonal line represents perfect equality
- Area between diagonal and curve indicates extent of dispersion
- Used frequently in economics to measure income disparity
- Requires cumulative percentages of both population and income
Formula Sheet
Range = L - S
QD = (Q3 - Q1) / 2
MD = Σ|x - A| / n
Variance (σ²) = Σfx² / N - (Σfx / N)²
Standard Deviation (σ) = √[Σf(x - x̄)² / N]
CV = (σ / x̄) * 100
Exam Tip
Always verify if the question asks for the measure of dispersion itself or its coefficient; calculating the coefficient requires dividing the dispersion measure by the average.
Common Mistakes
- Forgetting to take absolute values while calculating Mean Deviation, leading to the sum becoming zero.
- Confusing Standard Deviation with Variance by forgetting the final square root step in manual calculations.
- Applying the change of origin property to the Coefficient of Variation, which is incorrect since CV is a relative measure.
More Revision Notes
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