Researchers who acknowledge the uniqueness of human interactions among agents, and in particular the fact that human beings learn from events and adjust their choices and behaviors to take them into account, must challenge one of the key assumptions that underlies most statistical analysis. This is the assumption of ergodicity. The following is adapted from an appendix to an article by Hazy and Ashley (2011).

The ergodic assumption or *ergodicity*requires first that the random process being analyzed is stationary (defined below) meaning it doesn't change with time. There are stationary processes that are non-ergodic as well as stationary processes that are ergodic. However if a process is not stationary then it cannot be ergodic. In other words *stationarity *is a necessary although not sufficient condition for ergodicity (Bendat & Pierso. 1966). Consequently we first need to be able to determine whether or not the system properties we are studying are characterized by the concept of stationarity.

Stationarity is defined as follows: Suppose we compute the mean and/or higher moments as well as joint moments (auto correlation) at some point in time for a collection of sample functions (an ensemble). The mean and the auto correlation function could both be calculated (for mathematical details see Hazy & Ashley, 2011). It is assumed that each sample function is equally likely.

If the mean and the autocorrelation functions vary as time varies then the random process defined by the sample functions or ensemble is considered to be non-stationary. If, however, the mean and auto-correlation functions do NOT vary as time changes, then the random process is considered to be weakly stationary.

For the situation where all possible moments, e.g. mean, variance, skewness, kurtosis, etc., and the joint moments are time invariant, then the random process is considered to be strongly stationary. Furthermore for numerous applications, weak stationarity justifies the assumptions of strong stationarity, thus satisfying the time *invariant assumption*.

Ergocity also requires that for a stationary process its properties do not require computation over multiple sample functions in the ensemble. If the random process is stationary and the mean and auto-correlation functions defined above do not differ where computed over different sample functions, the random process is considered ergodic.

Although stationarity and ergodicity are taken to be true for many processes that involve human systems, it seems likely that this is not always, nor is it even usually the case. In numerous systems involving humans and the measurement of human performance, it is axiomatic that outcomes vary with time. Thus stationarity and therefore ergodicity are not necessarily valid, especially for longer time horizons or in different contexts. For example, one of the most measured system properties is quality, and many measurements of the quality of human performance are not necessarily stationary, that is the individual events themselves vary over time and space and are thus not necessarily comparable. For example many performance measurements in a hospital such as medication errors are much larger at the end of July and in August, than any other month.

Thus, it is not necessarily appropriate to use statistical methods that depend upon stationarity (or ergodicity) when comparing medication error samples, or any others for that matter, that were taken at various times of the year. There is no guarantee that the error introduced by the lack of stationarity does not impact the relevant statistics being reported.

The study of human interaction dynamics (HID) seeks to develop methods which accurately extend current methods to accommodate these concerns.

**Reference**

Bendat, J. S., & Pierso, A. G. (1966). *Measurement and Analysis of Random Data,* pp. 9-13. New York: John Wiley & Sons, Inc.

Hazy, J. K. & Ashley, A. (3011), Unfolding the future: Bifurcation in organzing forms and emergence in social systems. *Emergence: Complexity and Organization**, **13(3)**, *57-79.

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