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Why are deviations squared when calculating variance?

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sample space for a pair of dice — probability theory Sample space for a pair of dice.

In statistics, deviations are squared when calculating variance primarily to ensure that all differences from the mean contribute positively to the measure of variability.12

If you were to simply sum the raw deviations (the difference between each data point and the mean), positive and negative values would cancel each other out, often resulting in a sum of zero regardless of how spread out the data actually is. By squaring these differences, you eliminate the negative signs, ensuring that every data point’s distance from the mean is accounted for in the final calculation.1

This process is fundamental to determining the standard deviation, which is the square root of the variance. Because squaring the deviations changes the units of the data (e.g., squaring inches results in square inches), taking the square root of the variance at the end allows the standard deviation to be expressed in the same units as the original data, making it a more intuitive measure of spread.2