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HSSLIVE Plus One Economics Chapter 16: Measures of Dispersion Notes

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Measures of dispersion quantify the spread or variability within a dataset, complementing central tendency measures to provide a more complete understanding of data distribution. Range, the simplest measure, calculates the difference between the highest and lowest values but is affected by outliers. Quartile deviation examines the spread of the middle 50% of values, offering greater stability. Mean deviation measures the average distance of observations from the mean, while standard deviation—the square root of variance—provides a comprehensive measure of dispersion with important statistical properties. Relative measures like coefficient of variation allow for meaningful comparisons between distributions with different units or scales. Together, these tools help economists analyze data reliability, consistency, and distribution characteristics.

Chapter 16: Measures of Dispersion

Introduction

Measures of dispersion describe how spread out or scattered the values in a dataset are from the central tendency measures. While central tendency tells us about the typical value, dispersion tells us about the variability or consistency in the data.

Types of Measures of Dispersion

1. Range

The range is the simplest measure of dispersion, representing the difference between the maximum and minimum values in a dataset.

Formula: Range = Maximum value – Minimum value

Example: For the dataset: 5, 8, 12, 15, 20 Range = 20 – 5 = 15

Advantages:

  • Easy to calculate
  • Simple to understand

Disadvantages:

  • Only uses two extreme values
  • Highly influenced by outliers

2. Quartile Deviation (Semi-Interquartile Range)

The interquartile range (IQR) is the difference between the third quartile (Q3) and first quartile (Q1). Quartile deviation is half of the IQR.

Formula:

  • IQR = Q3 – Q1
  • Quartile Deviation = (Q3 – Q1) / 2

Example: For the dataset: 5, 8, 12, 15, 20 Q1 = 6.5, Q3 = 17.5 IQR = 17.5 – 6.5 = 11 Quartile Deviation = 11/2 = 5.5

3. Mean Deviation

Mean deviation is the average of the absolute differences between each value and the mean (or median) of the dataset.

Formula: Mean Deviation = ∑|x – x̄| / n

Example: For the dataset: 5, 8, 12, 15, 20 with mean = 12 Mean Deviation = (|5-12| + |8-12| + |12-12| + |15-12| + |20-12|) / 5 = (7 + 4 + 0 + 3 + 8) / 5 = 22/5 = 4.4

4. Variance and Standard Deviation

Variance is the average of squared differences from the mean. Standard deviation is the square root of variance.

Formula:

  • Variance (σ²) = ∑(x – x̄)² / n
  • Standard Deviation (σ) = √[∑(x – x̄)² / n]

Example: For the dataset: 5, 8, 12, 15, 20 with mean = 12 Variance = [(5-12)² + (8-12)² + (12-12)² + (15-12)² + (20-12)²] / 5 = (49 + 16 + 0 + 9 + 64) / 5 = 138/5 = 27.6 Standard Deviation = √27.6 ≈ 5.25

Coefficient of Variation

Coefficient of variation is the ratio of standard deviation to mean, expressed as a percentage. It allows comparing dispersion between datasets with different units or means.

Formula: CV = (Standard Deviation / Mean) × 100

Example: For our dataset with mean = 12 and standard deviation = 5.25 CV = (5.25/12) × 100 = 43.75%

Applications in Computer Science

In computer applications, measures of dispersion are used for:

  • Evaluating data consistency and reliability
  • Detecting anomalies and outliers
  • Quality control in software testing
  • Risk assessment
  • Performance evaluation of algorithms
  • Network traffic analysis
  • Machine learning model evaluation
  • Image processing for noise detection

Complete Chapter-wise Hsslive Plus One Economics Notes

Our HSSLive Plus One Economics Notes cover all chapters with key focus areas to help you organize your study effectively:

Economics: Indian Economic Development

Economics: Statistics for Economics

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