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This is an accepted version of this page This is the latest accepted revision, reviewed on 17 January 2025. Observation that in many real-life datasets, the leading digit is likely to be small For the unrelated adage, see Benford's law of controversy. The distribution of first digits, according to Benford's law. Each bar represents a digit, and the height of the bar is the percentage of ...
Benford's law, which describes the frequency of the first digit of many naturally occurring data. The ideal and robust soliton distributions. Zipf's law or the Zipf distribution. A discrete power-law distribution, the most famous example of which is the description of the frequency of words in the English language.
Benford's law is an observation that in many real-life sets of numerical data, the leading digit is likely to be small. [21] In sets that obey the law, the number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time.
A discrete probability distribution is the probability distribution of a random variable that can take on only a countable number of values [15] (almost surely) [16] which means that the probability of any event can be expressed as a (finite or countably infinite) sum: = (=), where is a countable set with () =.
The term law of total probability is sometimes taken to mean the law of alternatives, which is a special case of the law of total probability applying to discrete random variables. [ citation needed ] One author uses the terminology of the "Rule of Average Conditional Probabilities", [ 4 ] while another refers to it as the "continuous law of ...
For example, a ranked list of US metropolitan populations also follow Zipf's law, [8] and even forgetting follows Zipf's law. [9] This act of summarizing several natural data patterns with simple rules is a defining characteristic of these "empirical statistical laws".
In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be stable if its distribution is stable.
English: illustration of Benford's law, using the population of the countries of the world. The chart depicts the percentage of countries having the corresponding digit as first digit of their population (red bars). For example, 64 countries of 237 (=27%) have 1 as leading digit of the population.