This probability rule determined by the research of the two mathematicians Bienayme' and Chebyshev, explains the variability of data about its mean when the distribution of the data set is not known. The rule states that the percentage of data observations lying within standard deviations of the mean is at least
The above rule holds good regardless of the shape of the data set. The formula applies to differences greater than one standard deviation about the mean, and k being greater than 1.
If a particular random phenomenon follows the pattern of bell-shaped model which is usually a normal distribution, we precisely know how any element is likely to be closed to or far from its mean. But, if the population model is not known, the Bienayme' - Chebyshev rule applies. This rule tells how likely any particular value can lie within a specified distance from the mean. If a data set takes any shape, at least [1 - (1/2)^{2}] 100 = 75% of the observations are contained within two standard deviations of the mean. In turn, at least 89% of all observations are contained within three standard deviations of the mean.
This rule helps managers to understand the information content of data when no known distribution can be assumed.
Rectangular distribution: It is used where every observation has the same frequency/probability. The inner rectangle contains at least 75% of the observations.
Non-symmetrical distribution: When the two halves of a distribution about the mean are not mirror images of each other then it is termed as a non-symmetrical distribution. The shaded area below contains at least 75% of the observations.