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Put simply, the standard error of the sample mean is an estimate of how far the sample mean is likely to be from the population mean, whereas the standard deviation of the sample is the degree to which individuals within the sample differ from the sample mean. If the population standard deviation is finite, the standard error of the mean of the sample will tend to zero with increasing sample size, because the estimate of the population mean will improve, while the standard deviation of the sample will tend to approximate the population standard deviation as the sample size increases.

When do we use S.E.M. and when do we use S.D.?

Quote Source: Wikipedia

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  • $\begingroup$ Possible duplicate: stats.stackexchange.com/questions/32318/… $\endgroup$ – Stefan Feb 24 '18 at 4:01
  • $\begingroup$ @Stefan, my question is asking why we would use one over the other, not about the intrinsic properties of each method of statistical analysis. Hence, an answer to my question "when we use SEM...SD" would include a scenario which necessitates/calls for the use of either SEM or SD. $\endgroup$ – Jonathan Smith Feb 24 '18 at 13:04
  • $\begingroup$ Jonathan I tried to reword some of the answers from linked post and hope this makes things more clear. $\endgroup$ – Stefan Feb 24 '18 at 18:21
  • $\begingroup$ I disagree with Wiki's use of the term "population mean". I believe they mean "null hypothesized mean". That's not even a necessary concept, since a CI can be constructed without specifying a null hypothesis. $\endgroup$ – AdamO Feb 27 '18 at 18:18
  • $\begingroup$ You should give a link to the article. Possibly en.wikipedia.org/wiki/Standard_error ? $\endgroup$ – Acccumulation Mar 23 '18 at 19:51
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Suppose you want to know the average height of adults in a given country. Also suppose that if you were able to get all height measurements, the distribution of the data would follow a normal distribution. This distribution has then two important features, that is the mean $\mu$, which is the center of the distribution, and the standard deviation $\sigma$, which is a measure of spread around the center of that distribution. In this scenario, one standard deviation around the mean would capture in 68% of the data points; two standard deviations would capture 95% of the data points; and three standard deviations would capture 99.7% of the data points (see here).

However, in almost all cases, you are not able to measure all members of a given population, instead and you have to rely on taking random samples from that distribution to estimate how far the sample mean is from the true population mean $\mu$. If this is your goal, then you calculate the standard error of the mean. One standard error of the mean is then the interval in which the true population mean would fall 68% of the time if sampling was repeated over and over again. Usually in statistics a 95% confidence interval is used, which you can get by multiplying the standard error with 2 (see link above). Given the formula for the standard error of the mean, it is also apparent that if the sample size goes up, the interval tends to zero and you are closing in on your population mean $\mu$ (as in your quote above). Thus, the standard error of the mean is a tool in inferential statistics, that is inferring from the distribution of a random sample (observed data) to properties of an underlying unknown distribution, or the population.

The standard deviation on the other hand is used to describe the variability in the observed data only (i.e. the sample) without making any inferences with respect to properties of the underlying unknown distribution. The standard deviation is commonly used in descriptive statistics.

Now depending on whether you want to infer properties of an unknown distribution from a random sample (which is what we are mostly interested in when doing statistics), or whether you want to simply describe the variability in your sample, you should use the standard error of the mean and the standard deviation, respectively.

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The standard error of the mean is:
a parameter.
calculated based on the believed population SD and the sample size
a property of the distribution
used when you are considering the parameters to be known (e.g., you are calculating what a probability would be under a null hypothesis)
a fixed number

The standard deviation of the sample is:
a statistic
empirically measured
a property of the sample
used when you want to describe the actual data
a random variable that follow a chi-square distribution, assuming that the samples are IID standard normal (or, more precisely, the sample variance is so distributed)

So, for instance, suppose someone tells you that a distribution has a standard deviation of 1. You take a sample of 16 measurement. The SEM is $1/\sqrt{16}$ = .25 . You then find the sample mean, find the deviations, square them, and take the average, and then take the square root. That’s the SD of the sample. You can then compare that to .25. If they’re close, then the assertion that the population SD is 1 is reasonable. If they’re far apart, then it’s likely false.

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