Sample Moments and Their Standard Errors

You compute a strategy's annualized return over its 3-year live track record and get 14%. Someone else ran a version with a slightly different universe and got 11%. Is the first strategy actually better, or is the 3-percentage-point gap just the kind of wobble you'd expect from two different 3-year windows of the same underlying process? You can't answer that from the 14% and the 11% alone — you need to know how much a sample mean computed from that few years of data typically wobbles around the true long-run mean. Every moment you compute from a finite sample — mean, variance, skewness, kurtosis — is itself a random variable with its own spread, and quoting the number without its uncertainty is like reporting a distance without units.

An analogy: polling with a small sample

If you poll 20 random voters and 55% say they'll vote yes, that 55% is your best guess at the true population support — but it's a guess built from just 20 people, and a different 20 people would likely give you 48% or 61% instead. Poll 2,000 people and the same style of estimate lands much closer to the true value every time, because averaging over more people cancels out more of the individual noise. The number itself (55%) tells you the point estimate; the sample size tells you how much to trust it. Sample moments in finance work identically: the mean and variance you compute from a sample are your best guesses at the true, unobservable population values, and their reliability depends on how much data went into them.

The moments and their standard errors, one symbol at a time

Given $n$ observations $x_1, \dots, x_n$, the sample mean is $\bar{x} = \frac{1}{n}\sum_i x_i$ and the sample variance is $s^2 = \frac{1}{n-1}\sum_i (x_i - \bar{x})^2$ — the familiar plain-English average and spread. Higher moments describe the shape: skewness measures lopsidedness (a long left tail, typical of equity returns, gives negative skew), and kurtosis measures how fat the tails are relative to a normal distribution (finance data is almost always fatter-tailed than normal, called excess kurtosis).

The key extra piece is the standard error (SE) of each estimate — how much that estimate would vary if you redrew a fresh sample of the same size. For the sample mean, assuming roughly independent observations:

$$SE(\bar{x}) = \frac{s}{\sqrt{n}} .$$

In plain English: the uncertainty in your estimated average shrinks with the square root of the sample size — quadrupling your data only halves your uncertainty, which is why a few extra months of backtest data buys you much less confidence than it feels like it should. For the sample variance (large-sample approximation, assuming near-normal data):

$$SE(s^2) \approx s^2 \sqrt{\frac{2}{n-1}} .$$

In plain English: variance estimates are noisier, relatively, than mean estimates for the same $n$, and this approximation gets worse the further the true distribution is from normal — with fat tails, the variance estimate's standard error can be considerably larger than this formula suggests, because a single extreme observation moves $s^2$ a lot.

Skewness and kurtosis have standard errors too, and they shrink even more slowly with $n$ — for a normal distribution, $SE(\text{skew}) \approx \sqrt{6/n}$ and $SE(\text{kurtosis}) \approx \sqrt{24/n}$, which is a plain-English way of saying: you typically need a lot more data to pin down the shape of a distribution's tails than to pin down its center.

Worked example 1: the mean and its SE from ten years of returns

Ten years of monthly returns ($n = 120$), sample mean $\bar{x} = 0.8\%$ per month, sample standard deviation $s = 4.5\%$ per month. The standard error of the mean is

$$SE(\bar{x}) = \frac{4.5\%}{\sqrt{120}} = \frac{4.5\%}{10.95} \approx 0.41\% .$$

A rough 95% confidence interval for the true monthly mean is $\bar{x} \pm 2 \times SE \approx 0.8\% \pm 0.82\%$, i.e., roughly $[-0.02\%, 1.62\%]$. Even with ten full years of monthly data, the true mean return is uncertain enough that you can't confidently rule out zero — a stark illustration of why "the mean return was positive" is a much weaker claim than it sounds, and why track records shorter than this are even less conclusive about whether a strategy has genuine edge.

Worked example 2: comparing two variance estimates

Strategy X: $n=60$ trades, $s_X^2 = 4\%^2$. Strategy Y: $n=250$ trades, $s_Y^2 = 4\%^2$ — the same point estimate of variance, but from very different sample sizes. Their standard errors:

$$SE(s_X^2) \approx 4 \sqrt{\frac{2}{59}} = 4 \times 0.184 \approx 0.74, \qquad SE(s_Y^2) \approx 4 \sqrt{\frac{2}{249}} = 4 \times 0.0896 \approx 0.36 .$$

Even though both strategies report the identical variance of $4\%^2$, Strategy X's estimate is roughly twice as uncertain as Strategy Y's, purely because it's built from a quarter as many observations. If you were deciding which volatility estimate to trust more when sizing a position, Strategy Y's number deserves more weight even though the raw figures are identical — a distinction invisible if you only look at the point estimates and never compute their standard errors.

What this means in practice

Every headline statistic in a strategy write-up — mean return, volatility, skew, Sharpe ratio — is a sample moment with an implicit standard error that determines how much weight it deserves. Reporting a Sharpe ratio of 1.5 from an 18-month backtest without its standard error invites overconfidence, because the true Sharpe could plausibly be well below 1 or well above 2 given how few independent return periods went into the estimate (see Sharpe ratio standard error). Skewness and kurtosis, needing the most data to estimate reliably, are exactly the moments quants most often over-trust from short samples — a "fat left tail" detected in two years of daily data might just be two or three bad days, not a structural feature of the strategy's return distribution.