Sample mean: distribution and confidence interval

Explore the properties of a single sample taken at random from a distribution, sampling distributions of the sample mean and confidence intervals.

This applet has four main sections:

1. First the underlying distribution must be specified. Distributions included in this applet are the normal, beta and discrete.
   • The beta distribution is constrained to lie within a fixed interval (unlike the normal distribution which extends to infinity in both directions). The shape of the beta distribution can be symmetric about a central peak, and it can also be skewed, with the peak shifting to the left or right.
   • In this applet, a discrete distribution can be binary valued (0 or 1) or take values 1 to 5, 1 to 7 or 0 to 10. To assign probabilities to discrete distributions, enter your probabilities (which must lie between zero and one, and sum to one) at each value of the variable, starting with the lowest value and working up to the highest. Note that the probability of the last value is calculated automatically.
   The next step is to enter the sample size and click on Simulate, which creates 100 random samples from the underlying distribution.
2. The histogram illustrates one of the 100 samples.
3. The sample means histogram displays the histogram of all 100 sample means.
4. The confidence interval chart depicts confidence intervals for 20 of the 100 samples.