Surfstat.australia: an online text in introductory Statistics

SUMMARISING AND PRESENTING DATA

CHARACTERISTICS OF A DISTRIBUTION

Some features which characterise a frequency distribution are: modality, symmetry, central tendency and variability. As central tendency and variability are particularly important concepts, they will be discussed separately in the following sections.

Modality and Symmetry

Any value of a data variable or random variable at which the frequency curve or probability curve reaches a peak is called a mode. Most distributions in practice have one peak and are described as "unimodal". A distribution with two peaks is called "bimodal".

An extreme portion of a distribution, where the relative frequency becomes low either to the left or to the right, is called a tail.

A distribution is said to be symmetric if the relative frequency (or probability) is the same the same distance either side of its centre, m. Mathematically, the distribution of X-m is then the same as the distribution of m-X. The mean and median of a symmetric distribution are equal.

The most important symmetric distribution is the normal distribution, which is unimodal and therefore is symmetric about the mode. For symmetric unimodal distributions, the mean, median and mode are all the same.

An asymmetric frequency distribution is skewed to the left if the lower tail is longer than the upper tail, and skewed to the right if the upper tail is longer than the lower tail. Distributions of positive-valued random variables values are often skewed right.

An example of positive skewness
(skew to the right)

An example of negative skewness
(skew to the left)

Progress check

  1. A symmetric distribution has
  2. In a symmetric distribution, which are equal?


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