variance
/ VAIR-ee-ans /
If the standard deviation is the typical distance of a reading from the average, the variance is simply that distance squared — the standard deviation before you take the final square root. It is the more raw, mathematical measure of how spread out a set of numbers is, even if it is harder to picture directly.
Formally, the variance is the average of the squared gaps between each measurement and the mean. Squaring does two things: it makes every gap positive so they cannot cancel out, and it gives extra weight to the readings that stray furthest. Because the gaps are squared, the variance comes out in squared units, which is why chemists usually report its square root, the standard deviation, instead.
It matters because variances, unlike standard deviations, simply add together when independent sources of error combine, which makes the variance the natural currency for tracking how uncertainty builds up through a calculation. The caveat is interpretive: its squared units make it awkward to read off directly, so it tends to work behind the scenes while the standard deviation is what gets reported.
If weighing contributes a standard deviation of 0.3 mg and dilution another of 0.4 mg, the combined uncertainty is found by adding their variances, 0.09 plus 0.16, then taking the square root to get 0.5 mg.
Independent error sources add as variances, not as standard deviations.
Variance equals the standard deviation squared, and standard deviation equals the square root of the variance; the two carry the same information, one in squared units, the other in the original units.