Question

The distribution of sample means has the same mean as the underlying population, but the standard deviation differs. Use a real world scenario to explain why it makes sense the variation decreases as the sample size increases.

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Answer 1

We know that the sampling distribution of the sample means follows an approximately normal distribution and it has a mean equal to the population mean and the standard deviation is equal to the standard error of the population mean. (SD of sampling distribution of sample mean = ?/sqrt(n)). Also, if we increase sample size, the variation decreases. We have to see this fact by using one real-world scenario which is given below:

Suppose we are considering all students with age between 13 to 18 years in the particular school as the population under study. We select the study variable as heights of the students. We know that height follows an approximately normal distribution. Suppose the population of heights of students have an average of µ = 170 cm with the population standard deviation of ? = 20 cm.

Now, suppose we select different classes of students as samples with size 25 and find out the averages for the heights of the students in each class. (Consider the classes only with students of 13 to 18 years old.) Suppose, we get sample means as X1bar = 168, X2bar = 169.7, X3bar = 171.5, etc.

Now, if we calculate the mean of all sample means, we will get the value approximately equal to the µ = 170 cm. We are given a sample size = n = 25.

The standard deviation of the sample means will be equal to the ?/sqrt(n)).

Standard deviation of sampling distribution of sample mean = ?/sqrt(n))

A standard deviation of sampling distribution of a sample mean = 20/sqrt(25)

A standard deviation of sampling distribution of a sample mean = 20/5

A standard deviation of sampling distribution of a sample mean = 4 cm

Now, suppose we increase the sample size as 36, then

Estimate for mean for sampling distribution = 170 cm

Estimate for SD for sampling distribution = 20/sqrt(36) = 20/6 = 3.33 cm

Suppose, n = 81

Estimate for mean for sampling distribution = 170 cm

Estimate for SD for sampling distribution = 20/sqrt(81) = 20/9 = 2.22 cm

This means as we increase the sample size, standard deviation or variation decreases.