Variance vs Standard Error

Variance measures spread by adding up the squared distances from every value to the mean and then dividing. For a sample:

Read $s^2$ as s squared, where $x_i$ are the observations, $\bar{x}$ (x bar) is the sample mean and $n$ is the sample size. For a full population the symbol becomes $\sigma^2$ (sigma squared), the mean becomes $\mu$ (mu), and the divisor is the population size rather than $n - 1$.

Work through 6, 9, 10, 12, 13. The mean is 10, so the deviations are -4, -1, 0, 2 and 3. Squared, those are 16, 1, 0, 4 and 9, summing to 30. Treated as a sample, $s^2 = 30 / 4 = 7.5$. Treated as an entire population, $\sigma^2 = 30 / 5 = 6$. On small data sets the gap is far larger than a rounding difference, so the sample-or-population question has to be settled before the arithmetic starts.

Squaring is also what makes variance awkward to report. If those five numbers are minutes, the variance is 7.5 square minutes, a unit with no physical meaning. Taking the square root undoes that and returns the standard deviation, $s = \sqrt{7.5} = 2.74$ minutes, which is why AP Statistics quotes $s$ in almost every context and keeps $s^2$ for the algebra underneath.

Two properties fall straight out of the definition. Variance is never negative, and it equals 0 exactly when every value is identical to the mean, since that is the only way for all the squared terms to vanish. And because the distances are squared before they are added, one far-out observation contributes out of all proportion to the rest, so variance is not resistant. When outliers are present the interquartile range is the spread measure to reach for, and the median rather than the mean is the matching center.

Variance appears directly in one of finance's standard measures. An asset's beta is the covariance of its returns with the market's, divided by the variance of the market's returns, so the denominator is exactly the quantity defined here: beta.

The standard error is an estimate. It plays the same role for a statistic that the standard deviation plays for raw data, and it is called a standard error precisely because the quantity it reports is built from statistics rather than parameters. For a sample mean it is $SE_{\bar{x}} = \frac{s}{\sqrt{n}}$, using $s$, the sample standard deviation. For a sample proportion it is $SE_{\hat{p}} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$, using $\hat{p}$ (p-hat, the sample proportion). Put the parameters in instead, $\frac{\sigma}{\sqrt{n}}$ (sigma over the square root of n) and $\sqrt{p(1-p)/n}$, and you have the true standard deviation of the sampling distribution, which is the thing the standard error is estimating. That one swap, statistic in place of parameter, is the whole difference between the two names.

Suppose 25 measurements have a sample standard deviation of $s = 4.5$ grams. Then $SE_{\bar{x}} = \frac{4.5}{\sqrt{25}} = \frac{4.5}{5} = 0.9$ grams. Read the two numbers aloud: individual measurements sit about 4.5 grams from the sample mean, while the sample mean itself sits about 0.9 grams from the population mean across repeated samples of 25.

The misreading is "the standard error tells you how spread out the data are." It does not. That is $s$, and $s$ stays about the same as you collect more data, because a larger sample estimates the same population spread. The standard error shrinks toward zero with $\sqrt{n}$ because it describes a statistic, not an observation.

A second wrong reading is treating a small standard error as proof the estimate is close to the truth. It measures precision, not accuracy. A convenience sample of 4,000 produces a tiny standard error around whatever value its method is centered on, and no amount of precision repairs a center that is wrong.

One boundary worth memorizing: a one-sample z-test for a proportion puts the null value $p_0$ (p-naught) where $\hat{p}$ normally sits, giving $\sqrt{p_0(1-p_0)/n}$, because every quantity in a test is computed as though $H_0$ were true. Otherwise the standard error is the denominator of a standardized test statistic and the second factor in a margin of error.