This Demonstration shows an amazing result, the central limit theorem: by sampling a population with an unknown distribution, either finite or infinite, the sampling distribution of the mean value of each sample will be normal, with estimated mean

close to the distribution mean

, and variance

close to the distribution variance

divided by the square root of the sample size

. This result requires that the sample size be large [1].

We consider four different probability distributions for

as a function of

between the interval

and

; outside this interval, the probability is set to 0. The

are

,

,

and one that is discrete;

is the constant that enables the integral of the probability distribution to be normalized; that is,

.

Use the "trials

*T*" slider to set the number of samples, each one of size

, sampled from the corresponding

.

The top plot shows the plot of the selected

; a red line indicates the position of the probability distribution

. On the right, the values of

and

of

are indicated.

The bottom plot shows the histogram obtained from the sample; a Gaussian distribution with parameters

and

is overlaid on the histogram. A red line indicates the position of the probability distribution

. On the right, the values of

and

of the

are indicated.