Ergodicity in psychology
This article is about the concept ergodicity, and how this is relevant within the context of psychological research. Understanding what ergodicity is, and knowing what the consequences are when ergodicity does not hold, are central to appreciating some of the discussions about the need for ILD research in psychology. It is also important to understand ergodicity in relation to your study, as assumptions that underlie ergodicity affect how psychological processes can be best measured and analyzed, and whether results can be generalized to other cases and time points.
Ergodicity is a powerful concept: When it holds, you can generalize results from the population to the individual case and vice versa. Hence, when something is ergodic, it means that studying a population informs you about the processes that operate within a person; in that case, it is not necessary to do ILD research. However, if ergodicity is absent, such generalizations across levels are not without risk; then, trying to generalize from the population to a process operating within a person may result in incorrect conclusions. In that case, ILD research is essential to learn more about the process, as well as about possible individual differences therein.
In this article, you will find: 1) a brief background explaining where the concept of ergodicity comes from and why it was introduced in psychology; 2) the specific conditions that underlie ergodicity; 3) a discussion of the consequences of ergodicity in the context of psychological research; and 4) the implications of non-ergodicity for psychological research.
1 Background
Ergodicity is a concept that originally comes from the field of thermodynamics, and was brought to psychology by Peter Molenaar. He used the concept to critique the one-sided focus on [nomothetics] in psychological research, which is based on studying individual differences in a large sample typically at a single or a few points in time.
By introducing psychologists to ergodicity, and arguing that meeting the required conditions for it is extremely unlikely in psychology, Molenaar tried to raise awareness about the limitations of nomothetic research. He used this to advocated for a stronger focus on the individual through the use of [idiographic research] based on time series analysis: ``The unavoidable consequence of the ergodic theorems is that psychometrics and statistical modeling as we now know it in psychology are incomplete. What is lacking is the scientific study of the individual […]’’ (p.216, Molenaar, 2004).
Since the introduction of the concept of ergodicity in psychology, some researchers have set out to investigate whether ergodicity holds for a particular psychological process (e.g., Fisher et al., 2018; Golino et al., 2025). These studies show that, while there may be some similarities across some people some of the time, the general conclusion should be that individuals tend to differ at least to some extent; this means ergodicity does not exist for these processes.
2 Ergodicity conditions
For a process to be ergodic it needs to be
- stationary, which implies there are no structural changes over time; and
- [homogeneous], which implies each individual from the population is characterized by the same distributional properties.
Both conditions are described briefly below; more information about each concept can be found on their separate pages.
2.1 Stationarity in the context of ergodicity
Stationarity implies that it does not matter when we measure a process exactly, as the properties (e.g., means, variances, covariances) will be the same. It is a condition that applies to a single case (i.e., N=1); within the context of ergodicity, it has to hold for each and every case.
Omar wants to know how people manage their frustration while learning to differentiate in a math class. He wonders whether to study a single person more intensively over time, or that he can just observe a large group of individuals at a single point in time and then generalize the results to the process that takes places within individuals. The latter would have the advantage that it requires less time to gather the data; however, Omar realizes that it would require ergodicity to hold, because only then is it possible to make cross-level generalizations.
For ergodicity to hold, a process has to be stationary. When thinking about the process here, Omar assumes that it is a learning process in which an individual becomes more and more skilled in solving differentiation problems; this may happen in small gradual steps, but perhaps also in more sudden jumps when the person masters a certain concept or trick. As a result of this learning curve, the person may experience more frustration in the beginning versus less frustration later on. Hence, the process of experiencing frustration is probably not stationary, unless the difficulty of the math problems that a person has to solve changes in accordance to their ability to solve them. Although this may be the case for some people, this is unlikely to be exactly true for all people. Hence, stationarity is not met here, and therefore ergodicity will not hold.
Most—if not all—processes will not be stationary across the entire lifespan. Many psychological processes are characterized by some form of development at least during some phases of life; moreover, some process are also characterized by decline towards the end of life, making them non-stationary. However, for specific stretches of time, certain processes may be (approximately) stationary.
Hence, when you are setting up your study it is important to think about the time span of the study, as well as the granularity of the temporal lens that you plan to adopt. As you zoom in or out (i.e., change the time span of your study), this is likely to have consequences for the degree to which the process you observe can be considered (approximately) stationary or not. Moreover, even while focusing on the same time span, when you increase the granularity you may also start to see non-stationarity where there seemed to be stationarity when using a less detailed temporal lens. In addition to theorizing about stationarity of a process at a particular timescale and over a particular time span, you may also consider stationarity itself a topic of investigation.
2.2 Homogeneity
Homogeneity implies that each person in the population must be characterized by the same covariance structure, skewness, etc., but also by the same mean over time.
Bernhard and Ellen want to investigate whether a child puts in more effort in an academic task, when they belief they have more control. They therefore obtain time series data from multiple children, who report for each academic task in class how much effort they put in, how they performed, and how much control they think they have.
When analyzing the data per child, most children are characterized by a pattern in which putting in more effort is followed by a better performance, which is then followed by experiencing more control. For some of these children, experiencing more control is followed by putting in more effort; that is, they seem motivated by experiencing control (and the reverse, demotivated when experiencing a lack of control).
But for other children, experiencing more control is followed by putting in less (rather than more) effort; that is, they seem to feel they can relax a bit when the experience control (and the reverse, try harder when experiencing a lack of control).
Both patterns are understandable, but they represent quite distinct dynamics; if these dynamics reflect a causal process, then the way to intervene would be quite different for these two groups.
The difference that Bernhard and Ellen found across individuals implies that there is no homogenous process. As a consequence, ergodicity does not hold, results obtained at the population level (e.g., based on cross-sectional research) cannot be generalized to the level of the individual where the process is actually taking place.
This example is based on a study that was performed by Schmitz & Skinner (1993).
Bruno wants to study the relation between alcohol and anxiety. He studies several individuals using a daily diary to asks them in the evening how many alcoholic beverages they had, and how anxious they feel. He finds out that each person responds with the same decrease in anxiety after having an alcoholic beverage, and he therefore concludes this is a general pattern.
This makes Bruno wonder whether the process is ergodic. To find out, he studies a large sample of individuals, and asks them at one evening how many alcoholic beverages they had, and what their level of anxiety is. When considering the relation, he sees that individuals who had more alcoholic beverages actually feel more anxious than individuals who drank less.
On closer inspection of his initial daily diary study, Bruno notices that, although the dynamics within individuals were the same across individuals, the individuals had very different means over time on the two variables. He thus realizes that the process is not fully homogeneous; hence, it is not ergodic.
From the large body of empirical research using ILD, we know that homogeneity is not likely to be found in psychological research. Oftentimes individuals are characterized by their own mean over time; this actually forms the root of the [within-between] distinction. Moreover, individuals may also have different degrees of variability over time, represented by differences in their variances, or different degrees of skewness in their scores. Also, the correlations (or covariances) between different variables (or between a variable and itself at a different time point) may differ across people. When such individual differences are present, we cannot generalize the results obtained for one person to another person. This also implies that generalizing from the population to each and every individual in the population, and vice versa, is not possible here.
Whether homogeneity is present or not, can also be considered a research question. To study this, you need to have ILD of at least two persons, such that you can compare their distributional properties. But even when two individuals are characterized by the same means, variances, skewness, etc., this does imply that this will be the case for other individuals as well. Hence, having ILD from a large sample of individuals may be required for a more thorough investigation.
3 When ergodicity holds
If a process is ergodic, it does not matter when we measure it, or for what cases (e.g., persons, dyads) we measure it: We will always obtain the same distributional characteristics, that is, the same means, variances, covariances, etc.. Subsequently, the results we obtain when analyzing these data—for instance, factor loadings and measurement error variances when doing factor analysis, or regression coefficients and residual variances when doing regression analysis—will also be the same over time and cases.
The implications of ergodicity are far-reaching; if it holds, we can: 1) generalize results across levels (i.e., between the population and the individual); 2) generalize results across cases (i.e., from individual to individual, or from dyad to dyad); and 3) generalize results across time.
3.1 Generalize across levels
To understand what is meant by generalizing across levels, consider Figure 1. For illustrative purposes, it shows only a small part of a population (i.e., 10 persons) over a small number of time points (i.e., 10 time points).
The purple selection shows an example of cross-sectional research: It is based on observing a large sample of individuals at a single point in time. The red selection shows an example of N=1 research: It is based on observing a single person at many time points.
If ergodicity holds, these two approaches result in data that are characterized by the same means, variances, skewnesses, etc.. Moreover, if we have a multivariate focus, then covariances etc. are also identical across these approaches. Hence, when studying the cross-section, we can generalize the results to any particular individual from the population. And similarly, when studying a single individual, we can generalize the results to the entire population.
That is, generalizing across levels is a two-way street: It concerns generalization from the population to the individual, but also from the individual to the population.
3.2 Generalize across cases
To understand what generalization across cases implies, consider Figure 2.
When ergodicity holds, this implies that the results that we obtained for a particular person can be generalized to every other person from the same population. Hence, studying the process within one person informs us about the process in any other member of the population.
Such a generalizable result can be considered a general law, as it applies equally to each and every individual in the population. It has been argued that if we want to establish general laws, this should be done by taking such a replicated N=1 approach—as opposed to the common held belief that nomothetic research helps us to establish general laws (Grice, 2004; Hamaker, 2025). If you define a general law as a rule that applies to each and every case of a population, you have to either assume ergodicty, or actually study it through using ILD.
3.3 Generalize across time
To understand what is meant by generalizing across time, consider Figure 3.
When ergodicity holds, it does not matter at what time we observe the population: It will always be characterized by the same distributional properties (i.e., same means, variances, covariances, etc.). Hence, while individual cases may be characterized by variation over time, and thus have different scores at different time points, the population level features will not vary over time.
Generalizability across time also applies within an individual: When we take an N=1 time series approach (see the red selection in Figure 1), and we compare the first half of the series with to the second half, this also results in the same distributional properties (i.e., same means, variances, covariances, etc.).
Clearly, developmental processes, which are characterized by for instance a smooth trend or a more sudden change within a person over time, do not adhere to this form of generalizability. Hence, such processes cannot be ergodic.
4 What if ergodicity does not hold?
When there is no ergodicity, this places limitations on how results can be generalized: We cannot generalize results form the population to the individual or vice versa; errors stemming from such generalizations are known as the [ecological fallacy]. Moreover, we cannot generalize results from one individual to another, nor can we generalize results across time.
When the process you observed is non-stationary, you should use an analysis technique that can account for the particular kind of non-stationarity that characterizes the data. Much of this is dependent on the time span during which we want to study the process, the time frame of our measurements, and the temporal granularity of our measurements.
When a process is non-homogenous, we may decide to focus our attention on whether individuals differ from each other in for instance their mean, the variability, the dynamics, or something else; subsequently, we can then investigate how such individual differences are related to each other, and to other person features. We may also investigate whether such differences are continuous or categorical; in the latter case we can consider whether we can find subgroups of individuals with homogenous processes.
ILD research is ideally suited for studying these issues, as it allows us to investigate processes where they are operating (i.e., within a case over time), and does not require us to make strong assumptions about similarity across time, cases, and levels. Moreover, ILD research allows us to actually study ways in which processes are non-stationary, ways in which people are not homogenous, and ways in which results at different levels differ from each other.
5 Takeaway
Whether a particular process is ergodic or not, can be a research question. But what is most important to realize here is that when a construct is non-ergodic, this has major implications for what we can and cannot conclude about it based on particular measurements and analysis approaches.
In general, psychological researchers should not just assume ergodicity, but either investigate whether it holds, or use an approach that does not require the assumptions of stationarity and/or homogeneity. When operating from the idea that a process may be non-ergodic, we should consider the possible consequences of this when deciding how we measure the process, how we model the process, as well as when we use the results to draw conclusions regarding our theory about the process.
6 Further reading
We have collected various topics related to ergodicity for you to read more about below.
- Intra-individual versus inter-individual correlation
- [Within/between distinction]
- Ecological fallacy
Acknowledgments
This work was supported by the European Research Council (ERC) Consolidator Grant awarded to E. L. Hamaker (ERC-2019-COG-865468).
References
Citation
@article{hamaker2025,
author = {Hamaker, Ellen L. and Schuurman, Noémi K.},
title = {Ergodicity in Psychology},
journal = {MATILDA},
number = {2025-05-23},
date = {2025-05-23},
url = {https://matilda.fss.uu.nl/articles/ergodicity-in-psychology.html},
langid = {en}
}