# sampling distribution of the mean

Let a population, then we extract samples of size n, each one with its own mean. Let X_{n} 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 X_{n} approximates to a normal distribution

The standard deviation of its distribution is usually called **standard error**

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).

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

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

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