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I am confused after learning about the different terms.

I understood Standard error of the means to be the Standard Deviation of the sample means, whilst Sampling error is the Standard Deviation within one sample.

Am I understanding it correctly? Or have I oversimplified the comparisons or made a mistake in my understanding of the two concepts?

Nick Cox
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user10433947
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2 Answers2

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You have misunderstood the concept of sampling error. Sampling error is the error that is incurred when the statistical characteristics of a population are estimated from a sample of the population due to the choice of sample.

As a concept this is distinct from the standard error, which you understand correctly.

To demonstrate the distinction a little more clearly, consider a population that contains a single member with some characteristic C. Now imagine that you wish to measure the average C within this population. As the population only contains a single member, only one sample is possible, the sample that measures the single existing member. As such there can be no sampling error due to there being no choice in the sample to take.

Despite this it is still be possible to take repeated samples from this population, each of which could have its own unique 'measurement' error. As such these repeated measurements could produce differing estimates of C and as such have a standard error greater than 0.

Nick Cox
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Ryan
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  • your last sentence mentions standard error. Did you mean sampling error? – user10433947 Apr 16 '19 at 12:55
  • No, I meant standard error. In my example there is zero sampling error (as a census was taken) and positive standard error. This is caused by the estimate also containing a random non-sampling error (measurement error) that creates differences between each run of the census. – Ryan Apr 16 '19 at 13:39
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    If there's only 1 member of this population, you can't get any sampling error, no matter what attribute you measure. Sampling error and measurement error are not the same. – gung - Reinstate Monica Apr 01 '21 at 14:08
  • Sampling error (sampling variation) is often used loosely to refer to the fact that repeated samples differ. In practice whenever there is also measurement error what you see in sample data is a combination of the fact that (e.g.) different individuals appear in any given sample AND whatever errors of measurement occur. In practice even if you have only ONE sample, it is often contaminated by measurement error. Whether that is considered as sampling error is an open question. After all, we usually believe that -- given measurement error -- a different sample would be, hmm, different. – Nick Cox Aug 11 '23 at 09:27
  • In short, standard error (of the mean) is whatever is defined by the formula used to calculate it. I doubt that there is a consistent, rigorous, formal definition of sampling error. – Nick Cox Aug 11 '23 at 09:28
  • "the sample that measures the single existing member. As such there can be no sampling error due to there being no choice in the sample to take." I find this example confusing. The measurements can be seen as taking a sample from a (hypothetical) distribution of measurement errors. The situation is still of a type that a sample is used to infer properties of a population. The 'single member population' that is the object of interest doesn't make that the sampling process hasn't multiple members. – Sextus Empiricus Aug 11 '23 at 09:30
  • I use as a teaching example a set of measurements by a group of students of a certain distance. Is that one sample or a repeated sample? Note also the scientific commonplace that each individual chunk of material (soil, rock, blood, water, whatever) is called a sample by people who work with it. – Nick Cox Aug 11 '23 at 09:37
  • @NickCox I would say it depends on what your population of interest is. for example, If you are truly just interested in what that one distance is, then i would see this as a repeated sample. If you are interested in studying the distribution of measurements themselves on the other hand then you would be taking a larger sample from the distribution of all possible errors. i feel both would be valid in their own contexts. – Ryan Aug 24 '23 at 16:59
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The standard error is the (estimated) standard deviation of the sampling/measurement error.

The actual sampling error of a particular sample is often unknown, but based on the distribution of values within the sample one might infer the standard deviation of the distribution of the sampling error. Related: Statistics question: Why is the standard error, which is calculated from 1 sample, a good approximation for the spread of many hypothetical means?

Sextus Empiricus
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