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A variable has unit variance if its variance (spread around the mean) equals one.

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Explanation

‘Variance’ measures the spread of data points around the mean; achieving ‘unit variance’ involves scaling the data so that this spread equals one.

This scaling is done by dividing each data point by the standard deviation of the dataset. Standardising variance is useful in comparing datasets that might have the same mean but different levels of dispersion.

The formula for unit variance is:

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

Where:

• σ² is the variance (read this as ‘sigma squared’)
• n is the number of observations
• $x_i$ represents each data point
• μ is the mean of the dataset (pronounced ‘myoo’, as in ‘few’)

To achieve unit variance, each data point is divided by the standard deviation (σ, or sigma) of the original dataset.

Example

Consider a dataset detailing the sprint times of athletes in different events.

By scaling these times to have unit variance, a coach can more clearly see which times are more consistent relative to others, helping to identify athletes with the most stable performance.

Next reading:

Z-score standardisation