# SEM Interpretation

Standard error of the mean (SEM) represents the accuracy of the mean. Here's my question/doubt. Does higher the SEM mean higher the accuracy of the mean? To be more precise, what indicates more accuracy of the mean? sem = 3.5 or sem = 1.5?

This may seem odd but it keeps on pestering me for a very long time. Thank you in advance.

The SEM quantifies how far your estimate of the mean is likely to be from the true population mean. So smaller means more precise / accurate. In that sense, SEM=1.5 indicates that your sample mean is a more accurate estimate of the population mean than if SEM was 3.5.

• NO, that’s not what the s.e.m. is. It is the square root of the variance divided by the square root of n.
– DWin
Commented Apr 5 at 4:32
• @DWin, would you care to elaborate? I would like to understand what's wrong about my answer or my wording. You are of course correct that the standard error of the mean is calculated as s/√(n), but the question was how to interpret the SEM with respect to the estimated mean. Specifically, whether a smaller or a larger SEM indicates that the estimated mean is more "accurate" / "precise". SEM quantifies uncertainty in the estimate of the mean (doi.org/10.4103%2F2229-3485.100662), thus I think it's fair to say that a smaller SEM indicates a more accurate estimate of the population mean. Commented Apr 6 at 7:55
• The true mean and the true standard deviation are unknown. The correct formulation must state that under the assumption that the true mean is equal to the estimate and the true SD is the same as the estimate and that the sample is drawn from a Normal distribution that the interval formed by the mean -/- 1.96 SEM will contain the true mean 95% of the time. It’s all conditional on a variety of assumptions and is not a statement about where the “truth“ will be found.
– DWin
Commented Apr 15 at 3:02

Usually, the sample mean is used as an estimator for the population mean and is defined as: $\bar{Y}= \frac{1}{n}(Y_1 + ... +Y_n)$. If the sample is random, the single $Y_i's$ will be random and in turn the estimator $\bar{Y}$. It is random because it differs from one randomly drawn sample to another. The probability distribution of the estimator is called sampling distribution of $\bar{Y}$ and can be described by its expectation and variance. The standard deviation of $\bar{Y}$ is called standard error of the mean. As such it describes the amount of variation around the expectation of the sample mean.

A SEM closer to 0 would therefore indicate that the possible values for $\bar{Y}$ are clustered closer around the expectation of the sample mean. Since the sample mean is unbiased (its expectation equals the population mean) a SEM closer to 0 will indicate that possible values for $\bar{Y}$ are closer to the population mean. In this sense, the estimator as a statistic is more precise/accurate if the SEM is smaller. The realized value for your estimator, the estimate, is not necessarily more "accurate" compared to an estimate of an estimator with a higher SEM. But the probability that it is closer to the population mean is higher.