There is only one set of data so there is only one mean. Therefore there is no standard error of the mean.
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Standard error is random error, represented by a standard deviation. Sampling error is systematic error, represented by a bias in the mean.
The standard error increases.
The standard error of the underlying distribution, the method of selecting the sample from which the mean is derived, the size of the sample.
Standard error of the mean (SEM) and standard deviation of the mean is the same thing. However, standard deviation is not the same as the SEM. To obtain SEM from the standard deviation, divide the standard deviation by the square root of the sample size.
Standard error of the sample mean is calculated dividing the the sample estimate of population standard deviation ("sample standard deviation") by the square root of sample size.