Variance vs standard deviation: why the units change the meaning

Before calculating either measure, decide what the listed values represent. For a population of $N$ values with mean $\mu$, the variance and standard deviation are $\sigma^2=\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^2,\qquad \sigma=\sqrt{\sigma^2}.$ The square on $\sigma$ is part of the notation: variance is the square of the standard deviation.

For a sample used to estimate a larger population, the familiar estimator is $s^2=\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2.$ The denominator changes because the purpose changes. This article uses a complete population, so it divides by $N$. Naming that choice prevents two correct formulas from appearing to disagree.

Take the population $2,4,6$. Its mean is $\mu=4$, so the deviations from the mean are $-2,0,2$. Their sum is zero, as it must be when deviations are measured from the arithmetic mean. Squaring gives $4,0,4$.

The population variance is $\sigma^2=\frac{4+0+4}{3}=\frac{8}{3}\approx2.67.$ The population standard deviation is therefore $\sigma=\sqrt{\frac{8}{3}}\approx1.63.$ These are not two competing estimates. The second is exactly the square root of the first, expressed on a different scale.

Adding the raw deviations would always give zero, so positive and negative distances would cancel. Squaring prevents that cancellation, keeps every contribution non-negative, and gives larger deviations more influence. It also produces formulas that work smoothly in algebra, probability, optimisation, and statistical modelling.

Squaring is not the only possible way to measure spread. Absolute deviations also avoid cancellation. The choice depends on what the measure needs to do: squared deviations are especially useful when large errors should count more and when algebraic structure matters.

If the data are measured in hours, each deviation is measured in hours, but each squared deviation is measured in square hours. Variance therefore has units of $\text{hours}^2$. That is mathematically consistent, yet it is not the scale on which the original observations are read.

Taking the square root returns standard deviation to hours. In the example, a standard deviation of about $1.63$ data units can be compared directly with the values $2,4,6$ and their mean $4$. It should not be described as the average absolute distance from the mean; that is a different measure.

For the same population, the mean absolute deviation from the mean is $\frac{|{-2}|+|0|+|2|}{3}=\frac{4}{3}\approx1.33.$ It also uses the original data units, but it combines distances linearly rather than squaring them first.

The two measures answer related questions with different emphasis. Mean absolute deviation is a direct average of distances. Standard deviation is the square root of an average of squared distances, so unusually large deviations affect it more strongly. Neither number should replace the other without considering the purpose of the analysis.

Both measures must be non-negative and equal zero only when every value is identical. Adding the same constant to every observation changes the mean but not the deviations, so variance and standard deviation stay unchanged. Multiplying every observation by a factor $a$ multiplies variance by $a^2$ and standard deviation by $|a|$.

For example, scaling $2,4,6$ by $10$ gives $20,40,60$. The variance becomes $800/3\approx266.67$, exactly $100$ times the original variance, while the standard deviation becomes about $16.33$, exactly $10$ times the original standard deviation. If a calculation violates those relationships, revisit the deviations, denominator, or square root.

Use variance when squared deviations are part of the model, when variances need to be combined, or when the algebra is the main concern. Use standard deviation when the result must be discussed beside the original observations. Use mean absolute deviation when a direct average distance is the intended summary. Always report whether the calculation describes a population or a sample.

A reliable study routine is to write the mean, list the deviations, check that they sum to zero, square or take absolute values according to the chosen measure, and inspect the units at the end. You can practise that sequence with your own data in Euler’s tutor, or explore more worked explanations in Euler Learn.