Measure of Spread

Overview

The measure of spread describes how much variability or dispersion exists in a data set. It complements the measure of the centre by providing insights into the range, consistency, and overall distribution of the data.

Key Measures of Spread

Range:

• The simplest measure of spread.
• Calculated as:

$$\text{Range} = \text{Maximum Value} - \text{Minimum Value}$$

• Useful for quick assessments but sensitive to outliers.

Interquartile Range (IQR):

• The range of the middle 50% of the data.
• Calculated as: $\text{IQR} = Q3 - Q1$
• Where $Q1$ is the lower quartile (25th percentile) and $Q3$ is the upper quartile (75th percentile).
• Effective for reducing the impact of outliers.

Standard Deviation:

• Measures the average distance of each data point from the mean.
• Denoted as $\sigma$ (population) or $s$ (sample).
• Formula for population:

$$\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}$$

• Formula for sample:

$$s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$$

• Where:
  • $x_i$: Each data value.
  • $\mu$: Population mean ($\bar{x}$: Sample mean).
  • $N$: Population size ($n$: Sample size).

Outliers:

• Data points significantly different from others.
• Identified using:
  • IQR method: Values $< Q1 - 1.5 \times IQR$ or $> Q3 + 1.5 \times IQR$
  • Z-scores: Points with $|z| > 3$ are potential outliers.

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Worked Examples

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Summary

• The measure of spread quantifies variability in a data set.
• Range: Difference between the maximum and minimum values.
• Interquartile Range (IQR): Middle 50% range, useful for outlier detection.
• Standard Deviation: Measures average deviation from the mean.
• Outliers can distort spread and should be identified using IQR or Z-scores.
• Choose the appropriate measure based on data type and context.