The Difference Between The distribution of A Sample and The Sampling Distribution

The distribution of a sample refers to the distribution of values observed in a single sample of data taken from a population. On the other hand, the sampling distribution refers to the distribution of values that would be obtained if we took many random samples from the same population and calculated a statistic( the mean or standard deviation) for each sample. 

For example, suppose we are interested in the proportion of adults in a population who own a smartphone.(Own, or Not) We randomly sample 100 adults from the population and find that 70 of them own a smartphone. The distribution of this sample of 100 adults is the binomial distribution, which describes the probability of obtaining different numbers of successes in a fixed number of trials. On the other hand, the sampling distribution for the proportion of smartphone owners would be the distribution of proportions that we would obtain if we repeated this process of sampling 100 adults and calculating the proportion of smartphone owners many times. In this case, if we assume that the true proportion of smartphone owners in the population is 0.6, then the sampling distribution would also be binomial with mean 0.6 and variance 0.24. However, the shape of the sampling distribution would be different from that of the distribution of the single sample. Specifically, the sampling distribution would be narrower and more symmetric than the distribution of the single sample, reflecting the fact that the variability due to sampling error is reduced when we take larger sample sizes.

To summarize, the distribution of a sample describes the values observed in a single sample of data, while the sampling distribution describes the distribution of a statistic that we would obtain if we took many random samples from the same population.