Measures of Dispersion

Measures of Dispersion: Introduction & Range - The Spread Spectrum

• Measures of Dispersion (MD): Quantify the spread or variability of data around a central tendency. Indicate data homogeneity/heterogeneity.
• Range: Simplest and crudest measure.
  • Definition: Difference between the highest ($H$) and lowest ($L$) observed values.
  • Formula: $Range = H - L$
  • Merits: Easy to calculate and understand.
  • Demerits: Highly affected by extreme values (outliers); ignores data distribution between extremes.

⭐ Range is highly susceptible to sampling fluctuations and is not based on all observations.

Measures of Dispersion: Quartiles, IQR & QD - Middle Ground Masters

• Quartiles: Values dividing ranked data into four equal parts.
  • $Q_1$ (Lower Quartile): 25th percentile.
  • $Q_2$ (Median): 50th percentile.
  • $Q_3$ (Upper Quartile): 75th percentile.
• Interquartile Range (IQR): Difference between $Q_3$ and $Q_1$; measures spread of middle 50% of data.
  • Formula: $IQR = Q_3 - Q_1$.
  • Robust to outliers.
• Quartile Deviation (QD): Half the IQR; also called semi-interquartile range.
  • Formula: $QD = (Q_3 - Q_1)/2$.
• Key for constructing Box & Whisker plots, which visually represent data distribution and presence of outliers.

⭐ IQR is a robust measure of spread, preferred over range for skewed data or data with outliers, as it is not affected by extreme values in the dataset's tails.

Measures of Dispersion: Mean Deviation - Average Absolutes

• Definition: The average of absolute deviations of observations from a central tendency measure (mean, median, or mode).
• Formula (from arithmetic mean $\bar{X}$): $MD = \frac{\sum |X_i - \bar{X}|}{N}$
  • $|X_i - \bar{X}|$: Absolute deviation of an observation $X_i$.
• Characteristics:
  • Based on all observations.
  • Absolute values used, ignoring signs of deviations.
  • Less affected by extreme values compared to Standard Deviation.
  • Simpler to understand and compute than SD.

⭐ Mean Deviation is least when deviations are taken about the median.

Measures of Dispersion: Variance & Standard Deviation - The Golden Standards

• Variance: Average of squared differences from the Mean.
  • Population Variance ($\text{Population Var or } \text{Var(X) or } \text{MSD or } \text{Mean Square Deviation}$): $\sigma^2 = \frac{\sum (X - \mu)^2}{N}$
  • Sample Variance ($s^2$): $s^2 = \frac{\sum (X - \bar{X})^2}{n-1}$ (uses $n-1$ for unbiased estimate)
• Standard Deviation (SD): Positive square root of Variance. $\sigma = \sqrt{\text{Variance}}$ or $s = \sqrt{\text{Variance}}$.
  • Most widely used & reliable measure; units same as data.
  • Foundation for normal distribution interpretation. 
• Normal Distribution & Empirical Rule:
  • Mean $\pm 1$ SD: covers approx. 68% of observations.
  • Mean $\pm 2$ SD: covers approx. 95% of observations.
  • Mean $\pm 3$ SD: covers approx. 99.7% of observations.
  • 📌 Mnemonic: Remember 68-95-99.7 for 1, 2, 3 SDs!

⭐ Coefficient of Variation (CV) = $(\frac{SD}{\text{Mean}}) \times 100%$. It's a relative measure of dispersion, useful for comparing variability between datasets with different units or widely differing means.

Measures of Dispersion: Coefficient of Variation - Relative Ruler

• Definition: A standardized, unitless measure of relative variability, expressed as a percentage.
• Formula: $CV = (SD / \text{Mean}) \times 100%$
  • SD: Standard Deviation
  • Mean: Arithmetic Mean
• Interpretation:
  • Higher CV $\rightarrow$ $\uparrow$ relative variability.
  • Lower CV $\rightarrow$ $\downarrow$ relative variability.
• Use: Compares dispersion of datasets with different units or significantly different means.

  ⭐ CV is preferred over SD for comparing variability between groups with widely different means or different units of measurement (e.g., comparing variability in height (cm) and weight (kg)).

High-Yield Points - ⚡ Biggest Takeaways

• Range: Simplest measure; highly affected by outliers.
• Interquartile Range (IQR): Spread of middle 50% of data; not affected by outliers.
• Variance: Average of squared deviations from the mean; units are squared.
• Standard Deviation (SD): Square root of variance; most common measure; 68-95-99.7 rule applies for normal distribution.
• Coefficient of Variation (CV): Relative measure of dispersion (SD/Mean) × 100; compares datasets with different units.
• Standard Error (SE): Measures precision of sample mean (SD/√n); decreases with ↑ sample size (n).