Dispersion

TL;DR

• Quantifies how spread out values are within a dataset.
• Measured by statistics such as range, interquartile range (IQR), standard deviation, variance, mean absolute deviation (MAD), and coefficient of variation (CV).
• Different measures respond differently to extreme values; some (e.g., IQR, MAD) are more robust to outliers.

Definition

Dispersion refers to the spread of data or information within a group or population — the variation or deviation of individual values from the average or mean.

Explanation

Dispersion is assessed using two related categories of statistics:

• Measure of dispersion: quantifies the amount of spread or scattering of data within a group. Common examples are range, interquartile range (IQR), and standard deviation.
• Measure of variability: quantifies the amount of change or fluctuation in the data. Examples include coefficient of variation (CV), mean absolute deviation (MAD), and variance.

Standard deviation and variance use squared deviations from the mean and are computed by aggregating squared differences between each value and the mean, then dividing by the number of values. Standard deviation is the square root of that average squared deviation.

$$\sigma = \sqrt{\frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^2}$$

Variance is the average of the squared deviations:

$$\sigma^2 = \frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^2$$

Coefficient of variation expresses dispersion relative to the mean (often as a percentage):

$$\text{CV} = 100 \times \frac{\sigma}{\bar{x}}$$

Mean absolute deviation is the average of the absolute differences between each value and the mean.

Examples

Dataset: {3, 5, 7, 8, 9, 10, 11}

• Range:

  • 11 - 3 = 8

• Interquartile range (IQR):

  • (8 - 5) = 3

• Standard deviation:

  • Standard deviation of the data set {3, 5, 7, 8, 9, 10, 11} is 3.5

• Coefficient of variation (CV):

  • 100*(3.5/7.5) = 46.7%

• Mean absolute deviation (MAD):

  • (2+2+0.5+0.5+0.5+1+1)/7 = 1.5

• Variance:

  • (4+4+0.25+0.25+0.25+1+1)/7 = 2.5

Use cases

• Useful for analyzing and comparing data sets in order to understand the distribution of values and make informed decisions.

Notes or pitfalls

• Range is a simple measure but is influenced by extreme values or outliers.
• Interquartile range and mean absolute deviation are more robust measures because they are not influenced by extreme values or outliers.
• The source notes variance as commonly used and describes it as not influenced by extreme values or outliers, similar to range and interquartile range in the original text.

Related terms

• Range
• Interquartile range (IQR)
• Standard deviation
• Coefficient of variation (CV)
• Mean absolute deviation (MAD)
• Variance