# Power Laws, Pareto Distributions, and Performance

… a core concept used in Quantitative Methods and Atlas104

Figure 1.

### Concept description

Wikipedia (reference below) describes a power law as “a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one quantity varies as a power of another.”

One type of power law distribution is the Pareto distribution, named after Italian economist Vilfredo Pareto. Wikipedia (reference below) notes that:

“Pareto originally used this distribution to describe the allocation of wealth among individuals since it seemed to show rather well the way that a larger portion of the wealth of any society is owned by a smaller percentage of the people in that society. He also used it to describe distribution of income. This idea is sometimes expressed more simply as the Pareto principle or the “80-20 rule” which says that 20% of the population controls 80% of the wealth.”

Different values of the coefficients in a Pareto distribution will produce a “90-10 rule” or a “70-30 rule.”

Power law (including Pareto) distributions are not Normal Distributions.

Figure 2. Power law L-curve vs. normal distribution bell curve

###### L-curves vs. bell curves

Power law distributions are sometimes called L-curves to contrast with the bell curves associated with normal distributions, as depicted in the frequency distribution in Figure 2.

Note that this is a different “L” than seen in Figure 1, where the variable (performance) is on the vertical axis and the high part of the “L” depicts the high performance of high performers and illustrates the “80-20 rule.” The frequency distribution in Figure 2 has the variable on the horizontal axis. If the variable in Figure 2 is performance, then the high part of the “L” represents the high number of low performers.

###### Superstars and hyper performers – when distribution of performance follows a power law

The distribution of performance in many fields does not follow a normal distribution but appears to be closer to a power law distribution.

For example, in 2012 Ian Clark (reference below) examined research performance by political scientists in Ontario universities:

“… anyone can see that some professors are better at research than others, in that they win more peer-reviewed grants and generate more publications that are cited by more scholars. … To illustrate, here is the distribution of scores on the h-index (an index that attempts to measure the impact of a published work) for 21 randomly selected associate professors of political science in three Ontario universities: 16, 14, 10, 10, 9, 9, 7, 6, 6, 5, 5, 5, 4, 4, 4, 2, 2, 2, 1, 1, 0. Clearly, the majority of the research contribution in this sample is produced by a minority of the professors. [This is consistent with] a power law distribution such that 70 percent of the research of any population of professors is produced by 30 percent of the faculty.”

Also in 2012, Ernest O’Boyle Jr. and Herman Aguinis (reference below) published their results of 5 studies involving 198 samples including 633,263 researchers, entertainers, politicians, and amateur and professional athletes. They found (Figure 3) that:

“Results are remarkably consistent across industries, types of jobs, types of performance measures, and time frames and indicate that individual performance is not normally distributed – instead, it follows a Paretian (power law) distribution.”

Figure 3. Frequency distribution of individual performance
(O’Boyle and Aguinis, 2012)

O’Boyle and Aguinis (pp. 111-113) highlight a number of ethical and management issues raised by their findings:

“Our results lead to some difficult questions and challenges in terms of practice, policy making, and societal issues because they have implications for discussions around equality and merit (Ceci & Papierno, 2005). There are several areas within OBHRM [organizational behaviour and human resource management] such as employee training and development and compensation that rely on the assumption that individual performance is normally distributed, and any intervention or program that changes this distribution is seen as unnatural, unfair, or biased (Schleicher, & Day, 1998). In evaluation, interventions are deemed successful to the extent that all those who go through them experience improved performance. But, if training makes the already good better and leaves the mediocre and poor performers behind, then this is usually seen as an indication of program faultiness. The Matthew effect (Ceci & Papierno, 2005; Merton, 1968) states that those already in an advantageous position are able to leverage their position to gain disproportionate rewards. It is disproportionate because the perception is that their inputs into a system do not equal the outputs they receive. Training programs that especially benefit elite performers are seen as unfair because they artificially alter the normal curve that is the “natural” distribution of performance. The Matthew effect has been found in a variety of settings (e.g., Chapman & McCauley, 1993; Judge & Hurst, 2008; Sorenson & Waguespack, 2006). Likewise, compensation systems such as pay for performance and CEO compensation are an especially divisive issue, with many claiming that disproportionate pay is an indicator of unfair practices (Walsh, 2008). Such differences are seen as unfair because if performance is normally distributed then pay should be normally distributed as well.

“Our results put the usual conceptions and definitions of fairness and bias, which are based on the norm of normality, into question and lead to some thorny and complicated questions from an ethical standpoint. How can organizations balance their dual goals of improving firm performance and also employee performance and well-being (Aguinis, 2011)? Is it ethical for organizations to allocate most of their resources to an elite group of top performers in order to maximize firm performance? Should separate policies be created for top performers given that they add greater value to the organization than the rest? Our results suggest that practitioners must revisit how to balance the dual goals of improving firm performance and employee performance and well-being as well as determine the proper allocation of resources for both elites and nonelites.

“Beyond concepts of ethics and fairness, a Paretian distribution of performance has many practical implications for how business is done. As we described earlier, a Pareto curve demonstrates scale invariance, and thus whether looking at the entire population or just the top percentile, the same distribution shape emerges. For selection, this means that there are real and important differences between the best candidate and the second best candidate. Superstars make or break an organization, and the ability to identify these elite performers will become even more of a necessity as the nature of work changes in the 21st century (Cascio & Aguinis, 2008b). Our results suggest that practitioners should focus on identification and differentiation at the tails of the distribution so as to best identify elites.”

###### Additional papers on power laws and performance by Aguinis et al.

Herman Aguinis is a prolific researcher and provides downloadable pdfs on his website, http://www.hermanaguinis.com/pubs.html.

His recent publications on the implications of the power law distribution of performance include:

Joo, H., Aguinis, H., & Bradley, K. J. in press. Not all non-normal distributions are created equal: Improved theoretical and measurement precision. Journal of Applied Psychology. doi: 10.1037/apl0000214 [pdf version

Aguinis, H., Martin, G. P., Gomez-Mejia, L. R., O’Boyle, E. H., & Joo, H. in press. The two sides of CEO pay injustice: A power law conceptualization of CEO over and underpayment. Management Research, The Journal of the Iberoamerican Academy of Management. doi: 10.1108/MRJIAM-02-2017-0731 [pdf version]

Aguinis, H., O’Boyle, E., Gonzalez-Mulé, E., & Joo, H. 2016. Cumulative advantage: Conductors and insulators of heavy-tailed productivity distributions and productivity stars. Personnel Psychology, 69: 3-66. [pdf version]

Crawford, G. C., Aguinis, H., Lichtenstein, B., Davidsson, P., & McKelvey, B.  2015. Power law distributions in entrepreneurship: Implications for theory and research. Journal of Business Venturing, 30: 696-713. [pdf version]

Aguinis, H., & O’Boyle, E. 2014. Star performers in twenty-first-century organizations. Personnel Psychology, 67: 313-350. [pdf version]

O’Boyle, E., & Aguinis, H. 2012. The best and the rest: Revisiting the norm of normality of individual performance. Personnel Psychology, 65: 79-119. [pdf version]

###### Sources

Wikipedia, Power law, at https://en.wikipedia.org/wiki/Power_law, accessed 11 June 2017.

Wikipedia, Pareto distribution, at https://en.wikipedia.org/wiki/Pareto_distribution, accessed 11 June 2017.

Ian D. Clark (2012), How to get better research – and teaching – from universities, Policy Options, Vol. 34, No. 1, December 2012-January 2013, pp. 48-50 at http://policyoptions.irpp.org/magazines/talking-science/clark/, accessed 11 June 2017.

O’Boyle, E., & Aguinis, H. (2012), The best and the rest: Revisiting the norm of normality of individual performance. Personnel Psychology, 65: 79-119. [pdf version]

Josh Bersin (2014), The Myth Of The Bell Curve: Look For The Hyper-Performers, Forbes.com, at https://www.forbes.com/sites/joshbersin/2014/02/19/the-myth-of-the-bell-curve-look-for-the-hyper-performers/#e2f3f5c6bca0, accessed 11 June 2017.