The variance $\sigma^2$ and the standard deviation $\sigma$ are one quantity wearing two coats. Read aloud: sigma-squared equals sigma times sigma, and back the other way, sigma equals the square root of sigma-squared. $\sigma$ is the spread in the data's OWN unit — marks, millilitres, minutes — and $\sigma^2$ is that number squared, so it wears the unit squared too: marks², mL², min². That squared unit is the tell, and it is the only tell you need.

Which is why the headline slip of this lesson is handing in a variance where a standard deviation was asked for. A statistics package will happily print $\sigma^2 = 144$, and the answer the question wanted was 12. If the number you are about to write down cannot be compared with the data itself — if 144 marks of wander sounds absurd on a test out of 100 — you are holding the square, and the root is one step away.

The other two dialects need no square at all. $R = x_{\max} - x_{\min}$ — R equals x-max minus x-min, where $x_{\max}$ is the largest observation and $x_{\min}$ the smallest. And $IQR = Q_3 - Q_1$ — I-Q-R equals Q-three minus Q-one, the third quartile minus the first. Both are subtractions, both wear the data's own unit, and both are widths, never totals: nothing here is ever added.