Question

Please review the question and answer below and determine if what's being said is accurate (and why, or why not).

What is the distribution of sample means?

The distribution of the sample mean (also known as the sampling distribution of the mean) is the collection of sample means for all the possible samples. For example, a random sample of 100 automobiles were sent to 10 different testing plants to determine the number of defects that the automobile had. The testing plants results came back with a range of means between 10 and 20 defects with an average of 14. More specifically, 90% of the of the sample means were between 13 and 15 which is 1 standard deviation away from the mean. This type of result will tell an analyst that there’s a high likelihood that any car that comes off the production line will have 13 to 15 issues 90% of the time and what they should do next.

When computing the distribution of sample means it’s important to remember that sample means are variable then there’s a high likelihood that the samples will have different means. It’s also common knowledge that the standard deviation of the sampling distribution of the mean is also referred to as the standard error of the mean. In other words, the standard error of the mean is a measure of the dispersion of sample means around the population mean.

Answer 1

What is written is correct. May be you can add a few of these points ...

By “Sampling distribution of the mean”, we mean the distribution (or probability distribution) of the means of several samples drawn from a population.

How individual observations in a population behave depends on the population from which they are drawn. If we draw a sample of individuals from a normally distributed population, the sample will follow a normal distribution. If we draw a sample of individuals from a population with a skewed distribution (where many observations have values on the higher side or have values on the lower side), the sample values will display the same skewness. Whatever the population looks like - normal, skewed, bimodal- a sample of individuals taken from the population will display the same characteristics. This is only to be expected. Samples of individual observations from a population had to share the characteristics of the parent population.

It is observed during trials that if we draw a sufficiently large sample (typically > 30), the histogram of the individual observations resembles the histogram of the population from which the sample was drawn. In each case, the individual observations are spread out in a manner akin to the population histogram. The sample means, however, remain tightly grouped. This can be explained as follows:. In each sample, we get observations from throughout the population. The larger values keep the mean from being very small while the smaller values keep the mean from getting very large. There are so many observations, some large, and some small, that the mean ends up being "average". If the sample contained only a few observations, the sample mean might vary considerably from sample to sample, but with a large number of observations the sample mean doesn't get a chance to vary as much.

Thus, for large samples, the way the sample means vary about the population mean, that is, the sampling distribution of the sample means is independent of the shape of the parent distribution and is normal.