In statistics, the concept of the shape of a probability distribution arises in questions of finding an appropriate distribution to use to model the statistical properties of a population, given a sample from that population. The shape of a distribution may be considered either descriptively, using terms such as "J-shaped", or numerically, using quantitative measures such as

skewness In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined. For a unimo ...

and

kurtosis In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kur ...

. Considerations of the shape of a distribution arise in statistical

data analysis Data analysis is a process of inspecting, cleansing, transforming, and modeling data with the goal of discovering useful information, informing conclusions, and supporting decision-making. Data analysis has multiple facets and approaches, en ...

, where simple quantitative descriptive statistics and plotting techniques such as

histograms A histogram is an approximate representation of the distribution of numerical data. The term was first introduced by Karl Pearson. To construct a histogram, the first step is to " bin" (or " bucket") the range of values—that is, divide the ent ...

can lead on to the selection of a particular family of distributions for modelling purposes. Standard deviation diagram

Descriptions of shape

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 and multi-modal.Yule & Kendall (1950): Chapter 4 — Frequency Distributions A

bimodal distribution In statistics, a multimodal distribution is a probability distribution with more than one mode. These appear as distinct peaks (local maxima) in the probability density function, as shown in Figures 1 and 2. Categorical, continuous, and dis ...

would have two high points rather than one. The shape of a distribution is sometimes characterised by the behaviours 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 In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu i ...

is usually regarded as having short tails, while an

exponential distribution In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant averag ...

has exponential tails and a

Pareto distribution The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto ( ), is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actu ...

has long tails.

Notes

References

:*Yule, G.U., Kendall, M.G. (1950) ''An Introduction to the Theory of Statistics'', 14th Edition (5th Impression, 1968), Griffin, London. :*den Dekker A. J., Sijbers J., (2014)
Data distributions in magnetic resonance images: a review
, ''Physica Medica'' {{Statistics, descriptive Theory of probability distributions

Descriptions of shape

See also

Notes

References