Standard Error

Every estimate computed from a sample carries uncertainty, because a different sample would have produced a different number. The standard error puts a size on that uncertainty. Formally, it is the standard deviation of a statistic's sampling distribution: if the same study were repeated many times, the standard error describes how much the results would scatter around the true value.

The idea applies to any statistic. Regression coefficients, sample proportions and differences between means all have standard errors. In each case a small standard error means the statistic pins down the parameter tightly, and a large one means the data leave considerable room for doubt.

For the sample mean the formula is simple. If the population standard deviation σ is known, the standard error is σ / √n. In practice σ is rarely known, so it is replaced by the sample standard deviation s, giving s / √n. Suppose a fund's monthly returns over 36 months have a standard deviation of 4%. The standard error of the mean monthly return is 4% / √36, about 0.67 percentage points.

The square root has a practical consequence. Going from 36 to 144 months, four times the data, only halves the standard error. For noisy series such as equity returns, even several decades of data leave the mean return estimated with surprisingly wide error bands.

The two terms are easy to confuse because both are standard deviations. The standard deviation of the data measures how dispersed the individual observations are around their mean; it is a property of the population being sampled and does not shrink as more observations are collected. The standard error measures how dispersed the estimate would be across samples; it shrinks as the sample grows.

The distinction has real consequences. A fund with 20% annual volatility does not become less risky because ten more years of data are available. What the extra data change is the analyst's confidence in the estimated average return, which is a statement about the estimate, not about the fund.

Confidence intervals and hypothesis tests about a mean are built on the standard error. A confidence interval takes the sample mean and adds and subtracts a critical value from the normal or t-distribution multiplied by the standard error. A t-statistic divides the gap between the sample mean and the hypothesized mean by the standard error; the larger the result, the stronger the evidence against the hypothesis.

Candidates should compute the standard error from the standard deviation and sample size, choose between σ and s, and remember that the denominator of the test statistic is the standard error, not the standard deviation. Swapping the two is one of the most frequent errors in inference questions.