# sampling distribution of the mean

Let a population, then we extract samples of size n, each one with its own mean. Let Xn the random variable that links every sample to its mean. We can study its distribution called sampling distribution of the mean.

When a population has any distribution, we use:

Central limit theorem: if we take a simple random sample of size n with mean μ and standard deviation σ (n great enough, n ≥ 30), the sampling distribution of the mean Xn approximates to a normal distribution

The standard deviation of its distribution is usually called standard error

If we have a normal distribution N(μ,σ), then we have the same situation:

Generally the standard deviation of the population is unknown. Then, we approximate this parameter by using the standard deviation of the sample, if n is great enough (n ≥ 100).

Example: The heights of 1200 students of a secondary school are distributed in a normal distribution with mean 1.72 and SD 0.09 m. If we take 100 samples of 36 students each, calculate:

a) Mean and SD expected in the sampling distribution of the mean

b) How many samples are expected to have a mean between 1.68 and 1.73 m?

c) How many samples are expected to have a mean lower than 1.69 m?

a)

b)

Then, the number of samples is: 0.7448·100 ≈ 74 samples

c)

Then, the number of samples is: 0.0228·100 ≈ 2 samples

Exercise: In the population, IQ scores are normally distributed with a mean of 100 and standard deviation of 15. If we repeatedly pulled random samples of 25 individuals from the population and measured their IQ, How many samples are expected to have a mean between 95 and 105?

Solution: 22.61≈ 23 samples