Why variance is in squared units (and SD is not)

Why variance comes out in squared units

One step in the formula does all of it: the squaring.

A deviation is a difference from the mean, so it carries the same unit as the data. Square that deviation and you square the unit along with the number. Centimeters become centimeters squared, dollars become dollars squared, minutes become minutes squared. Every later step in the variance formula, adding the squared deviations and dividing by a plain count, leaves the unit exactly where the squaring put it.

So the variance ($s^2$ for a sample, read "s squared") is measured in the square of the data's unit. The standard deviation ($s$) is its square root, which cancels the squaring and lands back in the data's own unit. That is the whole story, and the rest of this page is that story with real numbers attached.

Follow the unit through the formula

Here is the sample variance, with $x_i$ a single data value, $\bar{x}$ ("x-bar") the sample mean, and $n$ the number of values:

Track the unit at each stage for heights measured in centimeters.

Two rows carry the weight. Squaring changes cm into $\text{cm}^2$, and dividing by $n-1$ changes nothing, because $n-1$ is a count with no unit attached. Divide centimeters squared by 4 and you still have centimeters squared.

That is also why the mean itself is in cm and needs no fixing. The mean adds values and divides by a count, so nothing ever gets squared.

A concrete example: five heights in centimeters

Five students are measured, in centimeters: 160, 165, 170, 175, 180.

The mean is $\frac{850}{5} = 170$ cm. The deviations are $-10$, $-5$, $0$, $5$, and $10$ cm. Square each one and the units go with them: 100, 25, 0, 25, and 100 $\text{cm}^2$. Those add to 250 $\text{cm}^2$.

A variance of 62.5 $\text{cm}^2$ is a correct number, and it is also a number nobody can picture. Centimeters squared is an area unit. Nothing about these five students is an area, so there is no physical thing 62.5 $\text{cm}^2$ describes.

Take the square root and the interpretation appears:

A typical height in this group sits about 7.91 cm from the mean of 170 cm. That sentence works because 7.91 is in centimeters, the same unit as the heights it is being compared with.

Why the square root is the fix, not a trick

Square roots undo squares for units in exactly the way they undo squares for numbers. If $s^2$ is in $\text{cm}^2$, then $\sqrt{s^2}$ is in $\sqrt{\text{cm}^2} = \text{cm}$. Nothing is being handled loosely here; it is the same algebra that turns an area of 9 $\text{m}^2$ into a side length of 3 m.

This is why almost everything downstream uses the standard deviation rather than the variance. A z-score subtracts the mean and divides by the standard deviation, and that division only produces a clean unitless number because the top and the bottom share a unit. The empirical rule counts standard deviations out from the mean along the same axis the data live on. Both would break if you fed them a variance.

So the division of labor is simple. Variance is what the algebra is built from, and the standard deviation is what you report and interpret. Standard deviation vs variance sets the two side by side, and what standard deviation actually means unpacks the interpretation sentence.

Check it by changing units: centimeters to meters

Convert the same five heights to meters: 1.60, 1.65, 1.70, 1.75, 1.80. Nothing about the students changed, only the ruler.

Run the same arithmetic and the sample variance comes out as 0.00625 $\text{m}^2$, while the standard deviation is about 0.0791 m.

Now check the conversion. Since 1 cm = 0.01 m, one $\text{cm}^2$ equals $0.01 \times 0.01 = 0.0001$ $\text{m}^2$. So

which matches. The standard deviation converts the ordinary way instead: $7.91 \text{ cm} \times 0.01 \frac{\text{m}}{\text{cm}} = 0.0791 \text{ m}$.

That difference is the point. Rescale your data by a factor $a$ and the variance is multiplied by $a^2$ while the standard deviation is multiplied by $|a|$. Double every value and the variance quadruples, from 62.5 to 250, while the standard deviation only doubles, from about 7.91 to about 15.81. A quantity whose value depends on the square of the scale factor is, by definition, living in squared units.

Use the units as an error check

Units catch mistakes faster than re-reading arithmetic does. Three habits worth keeping.

• Any spread you interpret in context should be in the data's own unit. If you find yourself writing "a typical student is about 62.5 from the mean" and the 62.5 came out of a variance calculation, you have skipped the square root. The typical distance is $\sqrt{62.5} \approx 7.91$ cm.
• Adding a constant to every value cannot change either one. Add 10 cm to all five heights and the deviations are untouched, so the variance stays at 62.5 $\text{cm}^2$ and the standard deviation stays at about 7.91 cm. Shifting the data moves the center, not the spread.
• Label the variance when you write it down. Writing 62.5 $\text{cm}^2$ rather than a bare 62.5 makes it much harder to hand the number to a z-score or an empirical-rule statement by mistake.

One more consequence worth knowing. Variances of independent quantities add, and standard deviations do not, which is why formulas that combine spread are written with $s^2$ and $\sigma^2$ and end with a square root at the very last step. Squared units are the price of that convenient algebra.

Once the number is back in the data's units, the next question people ask is whether it is a large one. That has no answer without context, which is a higher standard deviation better takes apart.