What does the standard error of the mean measure that the sample standard deviation does not?
answer
- two different questions, one formula apart
- spread of people versus precision of an estimate
- only one of them shrinks with n
- divide by the square root of n
basics
~20 sThe standard error of the mean measures how precisely a sample mean estimates the population mean; the sample standard deviation measures how spread out individual observations are. Only the standard error shrinks as the sample grows.
solid answer
~50 sThey answer different questions. The sample standard deviation `s` describes variability between individual observations: how far a typical patient, user or exam score sits from the sample average. The standard error of the mean describes variability of the *statistic*: how much the sample mean itself would bounce around if you drew a fresh sample of the same size. Numerically `SE = s / sqrt(n)`, so the standard error is always the smaller of the two and keeps shrinking as `n` grows, while `s` converges to the population spread and stops moving. That difference decides what you report: to say how variable patients' cholesterol is, quote the standard deviation; to say how well pinned down the average is, quote the standard error. Papers that write `190 +/- 4 mg/dL` meaning the standard error when readers assume the standard deviation make the population look far more homogeneous than it is.
go deeper
Be ready to state both definitions in one breath and write SE = s / sqrt(n) without hesitating. Know that the standard error is always smaller and that it, not the standard deviation, shrinks when you collect more data.
Explain the mechanics: variances of independent observations add, so the variance of the mean is sigma squared over n and its standard deviation is sigma over root n. Be able to convert between the two given n.
Show judgment about reporting. Say which quantity belongs on which chart, catch a paper or dashboard that plots standard errors while implying spread, and flag when clustered or repeated-measures data makes a naive standard error far too small.
Own the convention. Decide what your organisation's metric reports and experiment readouts display by default, insist the label and sample size travel with every plus-or-minus, and push back when tight error bars are used to imply a homogeneity the data never showed.
## Two different questions Almost every confusion here dissolves once you notice that a standard deviation and a standard error answer different questions. - **Standard deviation**: *How different are individual members of this group from one another?* It is a description of the data (or of the population the data came from). - **Standard error**: *How precisely does my sample statistic pin down the unknown population quantity?* It is a description of an **estimate**, not of the people. The standard error is not a special kind of standard deviation of the data. It is the standard deviation of a **sampling distribution** - the imaginary distribution you would get by repeating the whole study many times and recording the sample mean each time. ## The definitions Given a sample of `n` observations with sample mean `xbar` and sample standard deviation `s`: - `s` estimates the population standard deviation `sigma`, the typical distance of one observation from the population mean. - The standard error of the mean is `SE = sigma / sqrt(n)`, estimated in practice by `s / sqrt(n)` because `sigma` is unknown. ## Where the formula comes from For independent observations, variances of sums add. Writing the sample mean as `(X1 + X2 + ... + Xn) / n`: ``` Var(sum) = n * sigma^2 Var(mean) = Var(sum) / n^2 = sigma^2 / n SD(mean) = sigma / sqrt(n) ``` The division by `n` inside the variance is the entire story: averaging cancels independent noise, and the standard error is the square root of what is left. For large `n` the sampling distribution of the mean is also approximately normal, which is what lets people attach the familiar bell-curve reasoning to a standard error. ## Only one of them shrinks This is the sharpest diagnostic. Suppose adult cholesterol truly has `sigma = 40 mg/dL`. | n | typical `s` | `SE = s / sqrt(n)` | |---|---|---| | 25 | about 40 | 8.0 | | 100 | about 40 | 4.0 | | 400 | about 40 | 2.0 | | 10,000 | about 40 | 0.4 | Collecting more patients does not make patients more alike, so `s` hovers near 40 forever. It does make the **average** more reliable, so `SE` marches toward zero. If someone tells you their standard deviation got smaller because they collected more data, they are describing a standard error. ## The reporting trap A clinical paper reports `mean total cholesterol 190 +/- 4 mg/dL (n = 100)`. Two readings are possible: - If `4` is the **standard deviation**, the study population is remarkably uniform - almost everyone between roughly 182 and 198. - If `4` is the **standard error**, the underlying spread is `s = 4 * sqrt(100) = 40 mg/dL`, a completely ordinary, wide population, and `4` only says the average is well estimated. Same two numbers, wildly different clinical picture. Because the standard error is always the smaller number, quoting it makes error bars look tight - which is why it gets quoted even when the standard deviation is what the sentence needs. Good practice is to state explicitly which one is shown, plus `n`, so a reader can convert between them: `s = SE * sqrt(n)`. ## Which one should you report? - **Describing the population or predicting an individual** - how variable are response times, how heavy are the patients, what range should a new observation fall in: **standard deviation**. - **Describing how well you know a summary quantity** - the average revenue per user, a treatment effect, a coefficient: **standard error**. - Error bars on a chart of group means are usually meant to support comparisons between groups, so a standard error (or an interval built from it) is appropriate; error bars meant to show how heterogeneous each group is should be standard deviations. ## Standard errors are not only for means Any statistic computed from a sample has a sampling distribution and therefore a standard error - a proportion, a difference between two group means, a regression slope. The mean is simply the case with the cleanest formula. What generalises is the concept: **standard error = the standard deviation of the statistic across hypothetical repeated samples**, and it is the currency that later machinery - test statistics and interval estimates - is denominated in. ## Assumptions worth naming `s / sqrt(n)` assumes observations are independent and identically distributed. If your 400 responses come from 20 offices whose employees answer alike, the effective sample size is far below 400 and the naive standard error is too small - a real and common way for reported precision to be a fiction.
- A paper prints 'cholesterol 190 +/- 4 mg/dL, n = 100' - how do you tell whether 4 is a standard deviation or a standard error?Ask which one is plausible. If 4 were the standard deviation, nearly every patient would fall within about 8 mg/dL of 190, which no real cholesterol population does. Read as a standard error it implies `s = 4 * sqrt(100) = 40 mg/dL`, an ordinary spread. Reputable reporting labels the quantity and gives `n` so readers can convert with `s = SE * sqrt(n)`.
- Does a large sample standard deviation always imply a large standard error?No. The standard error depends on both spread and sample size, `s / sqrt(n)`. A very heterogeneous population measured on 100,000 people can have a tiny standard error, while a tight population measured on 6 people can have a large one. Only the ratio matters, which is why sample size must always be reported alongside.
- When should error bars on a chart show the standard deviation rather than the standard error?Show the standard deviation when the point is how variable individuals are - dose response across patients, latency across requests, score spread within a class. Show the standard error when the point is whether group averages differ, since it reflects estimate precision. Label which is plotted; unlabelled error bars are uninterpretable and routinely overread.
The standard deviation is how much the runners in a race differ from one another; the standard error is how much the team's average finishing time would wobble if you drew a new roster of the same size.
saying these in an interview costs you the question
- Says standard error is just another name for standard deviation
- Claims a bigger sample makes individual observations less variable
- Quotes a standard error when describing how spread out patients are
- Reports plus-or-minus values without saying which quantity or giving n
- Thinks the standard error describes the spread of the population