Intended for healthcare professionals

Rapid response to:


The economic impact of pandemic influenza

BMJ 2009; 339 doi: (Published 20 November 2009) Cite this as: BMJ 2009;339:b4888

Rapid Response:

Benford's Law to evaluate the detection and reporting of cases during the A(H1N1) influenza outbreak

Benford's Law to evaluate the detection and reporting of cases during
the A(H1N1) influenza outbreak

Since the beginning of the A(H1N1) pandemic, a question concerning both
the general population as well as the scientific community is whether the
health systems are adequately responding to this worldwide challenge. It
is argued that the question is not easily answered because there are no
established criteria for making an adequate evaluation. Fortunately,
Benford’s Law, also called the “Law of Anomalous Numbers” and the “First-
digit Law,” is a method that can help overcome this obstacle. This law
states that for a determined set of numbers, those whose leading digit is
the number 1 will appear more frequently than those numbers that begin
with other digits; the other digits appear with decreasing frequency. This
can be expressed formally as P(d) = log[1+(1/d)] d= {1,2,…, 9}, where for
a series of numbers, P(d) is the probability that a digit will be the
leading number.1,2 While Benford’s Law has been shown to be useful to a
variety of topics,1,2 one of the most frequent current applications is for
the detection of financial fraud.3

In the case of the A(H1N1) influenza outbreak, the number of
laboratory-confirmed cases can be of use to discover whether or not the
detection and reporting processes performed well. If the daily incidence
follows the distribution described by Benford’s Law, there is evidence of
good performance. As a preliminary empirical test, data reported to WHO
by 135 countries until July 6, 2009 were analyzed (see Figure 1). Note the
similarity between the observed and theoretical data. These data suggest
that the system for detecting and reporting cases functions well.

Benchmarking among countries using this simple method will enable the
identification of certain patterns which, with information from other
sources, will help answer the question regarding the performance levels of
health systems.


1. Benford F. The law of anomalous numbers. Proc Am Philos Soc

2. Hill TP. The first digit phenomenon. Amer Sci 1998;86:358-363.

3. Durtschi C, Hillison W, Pacini C. The effective use of Benford’s
law to assist in detecting fraud in accounting data. J Forensic Account

Competing interests:
None declared

Figure 1. First digits frequencies for the Benford distribution and the world incidence of A(H1N1) influenza until July 6, 2009.

Competing interests: No competing interests

04 December 2009
Alvaro J. Idrovo
Av. Universidad 655, CP. 62100. Cuernavaca, Morelos, Mexico
Center for Health Systems Research, National Institute of Public Health