Sampling Distribution

There is no dependency of the CLT on the distribution of the parent population from which the random samples are retrieved! No matter what the distribution of the parent population, its sample distribution is always normally distributed for n>30!

3.1. Number of Samples [more — closer to the population mean] [ACCURACY]— As number of random samples increases the mean of the sample would be very close to that of the actual population. Here is the sampling distribution plotted by the method df.plot.density(). Number of samples taken in this case is 100. The population is of the size 75. Each sample is of size 5. This happens because more samples accommodate the diversity in the real data.

3.2. Size of each sample (n) [more the tighter] [CONFIDENCE]— More is the size of each random sample, of the sampling distribution, the standard deviation of the sampling distribution would be lower — i.e, the distribution will be tighter as shown below. The standard deviation of the sampling distribution is known as the Standard Error S.E of the mean (SEM) .

NOTE THAT STANDARD ERROR OF MEANS (SEM) IS NOT THE STANDARD DEVIATION OF ONE SAMPLE. RATHER IT IS THE STANDARD DEVIATION OF SAMPLING DISTRIBUTION OF THE MEANS OF ALL THE SAMPLES!!!!

This is the uncertainty involved in inferring the population mean from the sampling distribution mean. SEM measures how far the sample mean of the data is likely to be from the true population mean. The SEM is always smaller than the SD. In the situations to find the confidence intervals on the actual population from the sample, we always use SEM and not the SD of the sample!

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