Midvariance

TL;DR

• Quantifies how spread out values are around the median, not the mean.
• Produces a single numeric value by averaging squared deviations from the median.
• Useful for comparing the spread of different datasets.

Definition

Midvariance is a statistical measure that indicates the average spread or dispersion of a set of data around the median. It is calculated by taking the average of the squared differences between each data point and the median value.

$$\text{Midvariance} = \frac{1}{n}\sum_{i=1}^{n} (x_i - \tilde{x})^2$$

where ( \tilde{x} ) is the median and (n) is the number of observations.

Explanation

Midvariance summarizes how far, on average, observations lie from the median by squaring each deviation from the median and then averaging those squared deviations. Because it references the median, midvariance reflects dispersion relative to the dataset’s middle value.

Examples

Example 1 — Simple numeric dataset

Consider the dataset of 10 observations of a variable X: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10].
The median value of this dataset is 5.5, and the midvariance can be calculated as follows:

(1-5.5)^2 + (2-5.5)^2 + (3-5.5)^2 + (4-5.5)^2 + (5-5.5)^2 + (6-5.5)^2 + (7-5.5)^2 + (8-5.5)^2 + (9-5.5)^2 + (10-5.5)^2 = 35.25

The midvariance of this dataset is 35.25, which indicates that the observations are relatively evenly spread around the median value.

Example 2 — Student heights

Dataset with median height 5 feet and heights: 4 feet, 5 feet, 5 feet, 5 feet, 6 feet, 6 feet, 6 feet, 6 feet, 7 feet, 8 feet.
The midvariance calculation for this dataset would be:

(4-5)^2 + (5-5)^2 + (5-5)^2 + (5-5)^2 + (6-5)^2 + (6-5)^2 + (6-5)^2 + (6-5)^2 + (7-5)^2 + (8-5)^2 = 8

The midvariance of this dataset is 8, indicating a relatively small spread of heights around the median value of 5 feet.

Use cases

• Identifying and analyzing the variability of a dataset.
• Comparing the spread of different datasets.

Related terms

• Median