Standard deviation vs variance: the difference

Standard deviation vs variance: the short answer

Both numbers measure spread: how far a typical value sits from the mean. They are tied together by one operation. The variance is roughly the average of the squared distances from the mean (for a sample, the sum of those squared distances divided by $n-1$), and the standard deviation is the square root of the variance.

That square root is the whole difference. Squaring the distances to build the variance also squares the units, so a variance of rainfall in millimeters comes out in millimeters squared, which you cannot picture. Taking the square root undoes that and returns the standard deviation to millimeters, the same unit as the data, which is why the standard deviation is the one you report and interpret.

What variance measures

The variance answers one question: on average, how far are the values from their mean, once each distance is squared? For a sample of size $n$ with mean $\bar{x}$ (read "x-bar"), the sample variance is

Here $x_i$ is a single data value, $\sum$ means add across all the values, and $s^2$ is the symbol for the sample variance. You subtract the mean from each value to get a deviation, square each deviation so that distances below and above the mean both count as positive, add them up, and divide.

The divisor is $n-1$, not $n$, for a sample. Dividing by $n-1$ corrects a known tendency of the sample to look less spread out than the population it came from, and it is the version on the AP formula sheet. The result is always in squared units, which is what makes the raw variance hard to interpret on its own.

What standard deviation measures

The standard deviation is the typical distance of a value from the mean, in the data's own units. You get it by taking the square root of the variance:

Because it lives in the original units, the standard deviation slots straight into statements you can reason about. A standard deviation of 3 millimeters says a typical day's rainfall sits about 3 millimeters from the average day.

The standard deviation is also the ruler for the empirical rule and for z-scores, where a value is described as so many standard deviations above or below the mean. You cannot use the variance that way, because its squared units do not match the data. This is why nearly every summary and every later inference procedure reports the standard deviation, or the related standard error, rather than the variance.

The differences side by side

Every row traces back to the square root. The variance is the building block; the standard deviation is the version you can read in context.

When to use which

Report the standard deviation whenever you are describing or comparing distributions, because it is in the same units as the data and pairs naturally with the mean. It is what you quote alongside a mean, what feeds z-scores, and what appears in confidence intervals and test statistics later in the course.

The variance shows up mostly as an intermediate step and in theory. It adds cleanly in ways the standard deviation does not, which is why formulas for combining spread are written in terms of variances. In this course you compute the variance mainly as the stop on the way to the standard deviation, so if a question asks you to interpret spread in context, convert to the standard deviation first. You can check any calculation with the standard deviation calculator.

The classic mix-up and how to avoid it

The usual error is reporting the variance where the standard deviation belongs, then interpreting it as if the units matched the data. Saying "a typical day is about 10 millimeters from average" when 10 is the variance in millimeters squared is wrong; the typical distance is $\sqrt{10} \approx 3.16$ millimeters. The number and its unit are both off.

Three habits prevent it. First, track units: if a spread is quoted in squared units, it is a variance, so take the square root before interpreting. Second, remember the direction of the link, since the standard deviation is the square root of the variance and the variance is the standard deviation squared. Third, when in doubt report the standard deviation, because it is the interpretable one. Building spread by hand in the standard deviation guide or the descriptive statistics sandbox makes the two-step relationship stick.