The "shape of a distribution" refers to the shape of a probability distribution. It most often arises in questions of finding an appropriate distribution to use in order to model the statistical properties of a population, given a sample from that population. The shape of a distribution will fall somewhere in a continuum where a flat distribution might be considered central; and where types of departure from this include:
- mounded (or unimodal)
- u-shaped
- j-shaped
- reverse-j-shaped
- multi-modal
The shape of a distribution is sometimes characterized by the behaviors of the tails (as in a long or short tail). For example, a flat distribution can be said either to have no tails or to have short tails. A normal distribution is usually regarded as having short tails, while a Pareto distribution has long tails. Even in the relatively simple case of a mounded distribution, the distribution may be skewed to the left or skewed to the right (with symmetric corresponding to no skew).
As previously mentioned, the overall shape of a sampling distribution is expected to be symmetric and approximately normal. This is due to the fact, or assumption, that there are no outliers or other important deviations from the overall pattern. This fact holds true when we repeatedly take samples of a given size from a population and calculate the arithmetic mean for each sample.
An alternative to the sample mean is the sample median. When calculated from the same population, it has a different sampling distribution to that of the mean and is generally not normal; although, it may be close for large sample sizes.
The Normal Distribution
Sample distributions, when the sampling statistic is the mean, are generally expected to display a normal distribution.