Doorgaan naar hoofdcontent

The Fanciful Number 2.9013, Plus or Minus Nothing

Can we capture any aspect of human psychology in a single number?

In 1956 a paper entitled The Magical Number Seven Plus or Minus Two saw the light. It was destined to become a classic in cognitive psychology. In it, George A. Miller argued that human short-term memory is limited to seven units of information. The 7 was just an approximation, which is why Miller added the cautious plus-or-minus-two.

In 1993 my former colleague Anders Ericsson wrote a much-cited paper in which he argued that minimally 10 years of deliberate practice are required to achieve expertise in a domain, an idea that was popularized by Malcolm Gladwell.

It is rare to tie psychological performance to a number, but Miller’s proposal is modest, as is Ericsson’s. The numbers are not claimed to be exact and are derived in a straightforward way from the data.

There is another number in psychology that makes a lot less sense: the number 2.9013. It represents the ratio between positive emotions and negative emotions that an individual or a group expresses within a given period of time. How did this number get to be so exact, with no less than four decimals? I mean it’s not 3 plus or minus 2 but it’s exactly 2.9013! The number is based on a set of differential equations used to model fluid dynamics.

According to a paper by Frederickson and Losada (2005) there is a tipping point in the ratio between positive and negative emotions. Below the tipping point (e.g., if your ratio is 2.8593) you are languishing and above the tipping point you are flourishing (up to 11.6346, after which things take a turn for the worse again). 2.9013 has become known as the positivity ratio. 

There is only one problem. The math behind 2.9013 and 11.6346 has recently been demonstrated by Brown and colleagues to be entirely fanciful. To make matters worse, the reasoning that the math gave rise to was also flawed.

Fredrickson & Losada's (2005) Butterfly Plot
The paper by Frederickson and Losada has been cited 970 times on Google scholar (as of July 19, 2013), which is a very high number for a psychology article. Hindsight is 20/20 and I was not aware of the positivity ratio before I read the Brown et al. article—so we must take what I’m saying here with a grain of salt—but I found it shocking that so many researchers had apparently forgotten to put on their critical thinking caps when they cited this article. Why did they fall for the positivity ratio? (I don't mean the general idea, which seems plausible enough, but the exact ratio and its origin in fluid dynamics.) Here are some reasons why this might have happened.

(1) Physics envy. At heart, all social scientists want to be natural scientists and so if natural scientists use differential equations to do cool stuff, they want to use them too.

(2) A desire for simplicity. How nice when everything and everyone can be reduced to a single number. We all have our BMIs and IQ,s if we are baseball hitters our RBIs and OBPs, if we are chess players our ELO-ratings, if we are researchers our H-Indexes, if we are earthquakes our Richter scale scores, and if we are scientific journals our impact factors. So why can’t we have a positivity ratio as well?

(3) A desire to see order in nature. The universe has yielded to scientific inquiry. Just as the elements let themselves be herded beautifully into the Periodic Table, so human emotional well-being can be cordoned off between two numerical boundaries.

(4) The desire to see all things connected. The positivity ratio was derived from a model of fluid dynamics. Apparently, the same set of equations that can be used to describe convective flow in fluids can be used to derive the upper and lower limits of emotional well-being. This gives a sense of profundity: the researchers have really hit on something fundamental to human existence here. It appears that Losada was so taken with the correspondence between fluid dynamics and emotional dynamics that he took his own metaphor literally, which resulted in a bizarre line of reasoning. He used a parameter in his model that expresses the ratio between buoyancy and viscosity in fluids. In the performance of the teams of workers he had observed, he noticed an interaction between what he described (entirely metaphorically of course) as buoyancy and viscosity. High-performance teams operated in a buoyant atmosphere whereas low-performance teams could be characterized as being stuck in a viscous atmosphere highly resistant to flow. 

In her response to the Brown et al. article, Frederickson swiftly relinquishes the tipping point and distances herself from Losada’s mathematical modeling. She notes that Losada did not want to respond to the Brown et al. article. This left her to defend the questionable math for which she was not responsible, so it is perfectly understandable that her positivity ratio dipped significantly below 2.9013 and that she threw her erstwhile co-author under the bus.

Frederickson argues that there are three components to her 2005 paper with Losada: (1) theory, (2) empirical evidence, and (3) mathematical modeling. Without the mathematical modeling, the first two remain. Frederickson is right that it is common for psychology to only have theory and empirical data.  But one has to wonder whether the work would have had the impact it has had without the mathematical model. One also has to wonder what the relevance of the model to the theory is if it can be jettisoned so easily and why it was included in the first place.

So what lessons can we draw from this latest kerfuffle in social psychology? We should be cautious whenever someone claims a psychological phenomenon can be captured in a number, especially when it has a lot of decimals. And we should keep our physics envy in check. We should also be wary of “deep truths” and of our inclination to see connections everywhere.

See also my next post on this topic.


  1. Like you, I was also not familiar with this paper/ideas until the critique was published.

    However, I think most people citing the paper were *not* citing it because of the specific point estimates and were instead citing it for other reasons (e.g., the general idea that positive emotions should outweigh negative emotions for flourishing).

    The original paper has been cited 970 times. Of those 970 times only six papers cite the 2.9013 value (or < 1% of the citations), 125 papers cite the 2.9 value (or ~13% of the citations*), 2 papers cite the 11.6346 value (or < .05% of the citations), and 27 papers cite the 11.6 value (or ~3% of the citations).

    I also tried searching for "11.5" but this only referred to non-positivity-ratio-related-numbers and for "3" and "11" but these referred to too many other things to be really analyzed (e.g., page numbers).

    *some of the citations are not in reference to the 2.9 or 11.6 etc in the original paper and instead are Means or SDs of other number (e.g., the SD of age was 2.9).

    1. Very nice analysis, Mark. I suspected that most people were not citing the paper because of the specific point estimate. Of course this is what must have made the paper appealing to people but they probably couldn't find a use for the point estimate in their own work. It is like a nipple on a breastplate as a character in a George R.R. Martin novel would say.

  2. I assume you have some data to back up your contention about the desires that exist at the heart of all social scientists? :)

  3. :) Let's chalk this one up to poetic license.

  4. Ummm, h-indexes aren't particularly useful for measuring researchers, and RBIs are next to useless for measuring hitters. BMIs can list anorexic body-builders as obese.

    So maybe simple numbers for complex phenomena are a bad idea in general. (Complex numbers, on the other hand...)

    1. I wrote a piece in the psychologist about this general point in the context of cut-off points in statistics:


Een reactie posten