Overview
Statistics for the Class 11 curriculum focuses on quantifying the variability of data beyond simple averages. Mastering measures of dispersion like variance and standard deviation is essential for interpreting data sets and is a high-scoring component in board examinations.
Measures of Dispersion
Measures of dispersion describe how data points are spread out or clustered around the central tendency. Understanding the difference between absolute and relative measures is critical for solving descriptive statistics problems.
- Range = Maximum value - Minimum value
- Mean Deviation about Mean: (1/n) * Σ|xi - x̄|
- Mean Deviation about Median: (1/n) * Σ|xi - M|
- Absolute dispersion has the same unit as the original data
Variance
Variance is the arithmetic mean of the squares of deviations of data points from the mean. It provides a measure of how far each number in a set is from the mean and from every other number in the set.
- Variance (σ²) = (1/n) * Σ(xi - x̄)²
- Coefficient form: σ² = (1/n) * Σ(xi²) - (x̄)²
- Always non-negative
- Highly sensitive to outliers due to the squaring of deviations
Standard Deviation
Standard deviation is defined as the square root of the variance, bringing the dispersion value back to the same unit as the original data. It is the most commonly used measure of dispersion in both academic exams and professional research.
- Standard Deviation (σ) = √Variance
- Calculated as √[(1/n) * Σ(xi - x̄)²]
- Smaller values indicate data points are close to the mean
- Standard deviation is invariant under change of origin
Formula Sheet
σ² = Σ(xi - x̄)² / n
σ = √[Σ(xi²)/n - (Σxi/n)²]
Mean Deviation = Σ|xi - x̄| / n
Exam Tip
Always verify if the question asks for variance or standard deviation; students frequently calculate the correct intermediate variance but fail to perform the final square root operation.
Common Mistakes
- Forgetting to take the square root of the variance when calculating standard deviation.
- Confusing the formulas for population variance and sample variance (n vs n-1).
- Neglecting to calculate the absolute value of deviations when finding Mean Deviation.
More Revision Notes
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