Rounding numbers - different ways to do it (ref 64)

Often, people show too many digits. “Income grew £119,591,234 to £624,512,645, an increase of 23.685%.” We want to do mental arithmetic on those numbers – it helps us grasp them – but all those digits hinder us. Try computing last year’s income – it’s £624,512,645 less £119,591,234. Tough.

Instead, round numbers, e.g.: “Income grew £120m to £625m, an increase of 24%”. Easier to mentally juggle. Less intimidating. Easier to remember. More engaging.

However, there is more to rounding than meets the eye. Figure 1 shows income for five competitors (Xy, Pq, etc). Unrounded numbers. Not nice. Figure 2 shows the numbers to the nearest £m - so far, so good - but then rounds them five different ways.

Column B – to one decimal place: still too many digits, even for Gh’s income of £22.9m. Remember: this is competitor income, so we seek general indications of magnitude, not exact figures – £23m is fine.

Column C – to no decimal place: better, but still too many digits for Xy (£3,623m).

Column D – to three digits: Xy is now £3,620m, not £3,623m. Better - but still not ideal. Not many people can mentally juggle three digits - after all, to see how much bigger Xy is than Pq, we struggle to compute 3,620 less 753 – so we instead shorten and simplify, and compute 3,600 less 750. Much easier.

Column E – to two digits: it’s better for some numbers – Xy’s income is now £3,600m, not £3,623m. But now we have big rounding errors. Study Mn and Kl’s income – in column D, they differ by almost 10%, but in column E, they’re both shown as £120m. It’s an unacceptable rounding error. (What’s ‘acceptable’? See later.)

So... rounding to three digits retains too many digits, but rounding to two digits can introduce rounding errors. What now?

Column F – variable rounding: that is, round to two digits to aid mental arithmetic, but when big rounding errors arise, round to three digits. Study Mn and Kl - now shown as 124 and 116.

All sorted. Variable rounding. It's great. Try it one day.

A free download to help you: OK, I admit... in Excel, it’s tricky to variably round. My website shows how, albeit it's a bit contrived - click here, enter my Downloads page, and check out Download 027 (Chapter 3). In PowerPoint and MS Word though, it’s easier than not rounding. It’s easier to type that income ranges from ‘£3.6bn to £23m’ than from ‘£3,623,145,794 to £22,876,431’. Easier for us when preparing the report. Easier for others when they see it.

Time for concerns on variable rounding:

'But £3,600m isn't accurate.' Nor is £3,623m. The real number is £3,623,145,794 – and a few pence. So the issue isn’t: “Should we show inaccurate numbers?”, but: “How inaccurate is OK?”. Answer: mostly, it’s better to be approximate and understood, than accurate and confusing. I’ve read that rounding errors of 3% or less are usually acceptable – but I’m not totally convinced. It depends on context. More on this later.

'But £3,600m and £23m are inconsistently rounded.' No, they're more consistently rounded - consider the two numbers’ rounding errors… showing both numbers to no decimal places gives widely diverse rounding errors for each number. With variable rounding, rounding errors are much more similar.

'£3,600m looks estimated – and numbers won’t add up.' Surely £3,623m looks more… definitive. In which case, put a note alongside: “Numbers rounded for clarity; small rounding differences may arise; exact numbers available”. As for not adding up, sometimes that doesn’t matter. For the five competitors’ income, I’d show their Total as £4.6bn. Rough – but wrong. The real total is £4,638,844,616. And a few pence.

Here's when not to round:

If dealing with small differences: if you’re a currency trader, you’d lose lots if you round.

If doing repeated multiplications: don’t round in the middle of compound interest computations, otherwise large rounding errors accumulate. 

If you’ve a control check to make: if accountants round numbers too soon, the trial balance won’t balance. Ensure it balances first, then round after.

If the numbers are to be used by others for analysis: people use figures from the Office of National Statistics (ONS) to produce ‘derived’ figures (e.g. one figure divided by another). The ONS can’t round numbers too much, otherwise derived numbers are wrong.

There are numerous benefits to rounding, and I've several real-life - and slightly jaw-dropping - stories to illustrate them.

But let's stop there. I've covered what I need to for now.


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