Question

Why do sample means have less standard deviation than population standard deviation? Provide an explanation for the difference in standard deviation between sample means and population standard deviation.

Answer 1

Sample means have less standard deviation than population standard deviation because the standard deviation of a sample mean is divided by the square root of the sample size, making it a more precise estimate of the population mean.

Step-by-step explanation:

When calculating the standard deviation of a sample mean, it is divided by the square root of the sample size, while the standard deviation of a population is calculated using the entire population. This leads to the sample mean having a smaller standard deviation compared to the population standard deviation.

For example, let's consider a population with a mean of 100 and a standard deviation of 20. If we take a sample of size 10, the standard deviation of the sample mean would be 20 / √10, which is approximately 6.32. This is smaller than the population standard deviation of 20.

The smaller standard deviation of the sample mean indicates that the sample mean is a more precise estimate of the population mean, as it is less affected by individual data points.