Question

For a random variable, X, let \(\bar{X}\) be the sample average. The sample size is n. The mean and the standard deviation of X are μ and σ, respectively. The standard deviation of \(\bar{X}\) is
1. nσ 
2. σ 
3. \(\sigma \over n\)
4. \(\sigma \over \sqrt{n} \)

Answer 1

Correct Answer - Option 4 : \(\sigma \over \sqrt{n} \)

Explanation:

Sampling Distribution of Sampling Mean (\(\bar{X}\))

If repeated random samples of a given size n are taken from a population of values for a quantitative variable, where the population mean is μ and the population standard deviation is σ.

• The mean of all sample means (\(\bar{X}\)) is the population mean μ.

As for the spread of all sample means, the Central Limit theorem dictates the behaviour much more precisely than saying that there is less spread for larger samples. In fact, the standard deviation of all sample means is directly related to the sample size, n as indicated below.

• The standard deviation of all sample means (\(\bar{X}\)) is \(\mathbf{\sigma \over\sqrt{n}}\)

Note: the square root of sample size n appears in the denominator, the standard deviation does decrease as the sample size increases.