# Representing uncertainty using significant figures

We forecast 43,279 new cases

2022-04-24

It’s not uncommon for modellers to produce estimates written to many significant figures, despite the fact that these estimates are highly uncertain. They might be built on poor quality data, or even no data at all. This practice can lose credibility with consumers of statistics who – rightly or wrongly – consider significant figures to be a reflection of uncertainty1.

For example, suppose we wish to give a point estimate $$\hat \theta$$ of some parameter. One way to represent uncertainty is the number of significant figures that $$\hat \theta$$ is displayed to. For example, supposing you told me a number was 1000, then I’d have a idea about the precision you’re talking about That idea would be different to if you had told me that the number was 1100, 1140, 1144 or 1144.3.

If $$\hat \theta$$ is displayed to $$n$$ significant figures, then we might suppose that our brains construct pseduo-credible intervals for $$\theta$$ given by $$[L_n, U_n]$$ which narrow as $$n$$ grows larger. I’d be interested to find out if any psychological research has been done into this landscape. For example, how does the credible interval size $$U_n - L_n$$ vary with $$n$$? To what extent does perception of uncertainty vary by individual, background or culture?

Given an understanding of this landscape, are there any thought-out guidelines on how we should use significant figures to communicate uncertainty? Such a representation of uncertainty will have limitations as compared with a credible interval $$[L, U]$$:

1. For each point estimate $$\hat \theta$$ only a small number of possible pseudo-credible intervals $$[L_n, U_n]$$ are possible.
2. There is no way to represent asymmetric uncertainty.
3. If $$\hat \theta$$ happens to be followed by many zeros even when a high number of significant figures are used, then it will mistakenly be interpreted as having high uncertainty.

Although in reality we are not limited to only providing a single point estimate with which to represent uncertainty, in spoken language it is rare to hear intervals. Instead, we often use words such as “about” or “roughly” – though these words, like using significant figures, are very crude.

In the text of a scientific paper, or an application developed to support academic work, we have the opportunity to present uncertainty as we wish. This might include a credible interval, though I worry about the extent to which this information may be ignored and more in some ways irrelevant information like the number of significant figures being focused on. Perhaps a practical recommendation might be to use both of these tools, and not restrict ourselves to one or the other, by providing point estimates and confidence intervals both rounded to a suitable number of significant figures. Even if some information is lost in this way, it is unlikely that policy would change significantly as a result of small changes to estimates.

I got in touch with someone working at UNAIDS for their suggestion on how estimates should be displayed. They provided the following Excel formula:

IF(X > 1000000, ROUND(X, -5), IF(X > 100000, ROUND(X, -4), IF(X > 10000, ROUND(X, -3),
IF(X > 1000, ROUND(X, -3), IF(X > 100, ROUND(X, -2), "<100")))))

With the warning that I’m not an Excel native, my understanding of this formula is that ROUND(X, -n) rounds to the nearest $$10^n$$, the result being to give most numbers to two significant figures, and any number under 100 is just presented as “<100”2. In practice, there isn’t rounding here which adapts to the uncertainty. I can certainly imagine the benefit isn’t worth the added complication, or another explanation could be that having measures of uncertainty for all estimates can’t be relied upon.

As a final note, I heard the book “Making Numbers Count: The Art and Science of Communicating Numbers”, and the concept of subitizing, mentioned on a recent More or Less episode which I think could be worth a read. I’d be interested if anyone has other recommendations in this space – let me know!

1. From a Bayesian point of view we might claim that it’s the posterior distribution which represents uncertainty, not how a point estimate happens to be written. However, I’m sure we are all guilty of foregrounding point estimates when communicating results on occasion.↩︎

2. It was also suggested that words can be used to add clarity, e.g. 43 thousand instead of 43,279.↩︎