Statistics Interview Question: What is the distribution of the sample mean?

I have been studying statistics for a few years now and it has come to my awareness that some themes stay the same even though the exact wording may be a bit different. This interview question is, what is the distribution of the sample mean?

The distribution of the sample mean, also called the sampling distribution of the sample mean, is the probability distribution of all possible sample means from repeated sampling of a given size from a population. Its shape, centre and spread depend on the population distribution and the sample size, and it approaches a normal distribution of increasing sample size, regardless of the population’s distribution: this concept being known as the central limit theorem.

The central limit theorem states that a population with mean, mu, and standard deviation, sigma, and take samples of the population with replacement, the distribution of the sample means will be approximately normally distributed.

The mean of this distribution is equal to the population mean, mu, and its standard deviation , standard error, is the population standard deviation divided by the square root of the sample size, n:- σ/√ n

Key characteristics of the sample distribution of the sample mean are:-

• Mean of the sampling distribution of the sample mean is equal to the population mean.
• Standard deviation of the sampling distribution of the sample mean is known as the standard error, and is calculated as:- σ/√ n where sigma is the population standard deviation of the and n is the sample size.
• If the shape of the population distribution is normal, the sampling distribution of the sample mean will also be normal. If the population distribution is not normal, the sampling distribution will become approximately normal sized as the sample size increases.

The central limit theorem can be proven in Python, as seen below:-

As can be seen below, as the sample size increases, the standard deviation decreases and the shape of the distribution approaches that of a normal distribution:-