Sponsorship impression error bars belong on the report you send the sponsor

The post-show sponsor report says 12,600 impressions against the hall entrance banner. Somebody built that number three weeks ago from a footfall count, a movement assumption and a visibility factor, and by the time it reached the deck all three had collapsed into one confident integer with no decimal point and no caveat.

Sponsorship impression error bars are the missing half of that figure. The estimate is a product of three uncertain inputs, so the uncertainty compounds, and the honest version of 12,600 is a range wide enough that the sponsor should probably be told about it before they build next year's budget on the point estimate.

Where the three estimates come from

Take the banner. The impression figure is traffic multiplied by pass frequency multiplied by a visibility factor.

Traffic through the entrance zone, say 6,000 people. That comes from a counter, and counters have documented error. Pass frequency, say 3.5 passes per person across the show, which comes from an assumption about how many times an attendee re-enters the hall. Visibility factor, say 0.60, meaning six in ten of those passes are assumed to happen in a position and at an angle where the banner is actually in view.

Six thousand times 3.5 is 21,000. Times 0.60 gives 12,600. Every one of those three inputs has a range around it, and the question is what the range around 12,600 looks like.

Suppose you are prepared to say the traffic count is good to plus or minus 15 per cent, pass frequency to plus or minus 30 per cent, and the visibility factor to plus or minus 40 per cent. Those are bounds, stated honestly. Nothing here depends on them being exactly right, only on somebody writing them down.

Multiplying the extremes gives you the wrong band

The instinct is to multiply the low ends together and the high ends together.

Low: 0.85 times 0.70 times 0.60 gives 0.357. Applied to 12,600, that is 4,498. High: 1.15 times 1.30 times 1.40 gives 2.093, which is 26,372. So the band is roughly 4,500 to 26,400, a spread of nearly six to one.

That calculation is correct as an absolute worst case and it is too pessimistic to be useful. It assumes all three errors go the same direction at full magnitude simultaneously, which for three independent inputs is a corner of the probability space almost nothing lands in. Hand that band to a sponsor and they will conclude, reasonably, that you have no idea.

How wide is the band if you do it properly?

The standard treatment is in the Guide to the Expression of Uncertainty in Measurement, published as JCGM 100:2008 by the Joint Committee for Guides in Metrology, whose member organisations include BIPM, ISO, IEC and OIML. Two clauses do all the work.

Clause 4.3.7 covers the case where you can only state bounds. If you know a quantity lies somewhere in an interval and have no reason to prefer any value inside it, you treat it as a rectangular distribution, and the standard uncertainty is the half-width of the interval divided by the square root of three. So plus or minus 15 per cent becomes a standard uncertainty of 8.66 per cent, 30 per cent becomes 17.32 per cent, and 40 per cent becomes 23.09 per cent.

Clause 5.1.6 covers the case where the result is a product of its inputs, which is exactly our situation. For a product, the relative combined uncertainty is the square root of the sum of the squared relative uncertainties. So take 8.66 squared, which is 75.0, plus 17.32 squared, which is 300.0, plus 23.09 squared, which is 533.1. That sums to 908.1, and the square root is 30.1 per cent.

That 30.1 per cent is one standard uncertainty. Clause 6.3.1 says a coverage factor of 2 to 3 is normal, and 2 gives roughly 95 per cent coverage, so multiply by two for 60.3 per cent. Applied to 12,600, the expanded uncertainty is 7,594, and the interval runs from 5,006 to 20,194.

Report 12,600 with a band of 5,000 to 20,200. Still wide. Considerably narrower than the six-to-one worst case, and derived by a method a sponsor's analytics team will recognise on sight.

Which input is costing you the most?

This is the part worth the effort, and almost nobody does it.

The three squared terms were 75.0, 300.0 and 533.1, summing to 908.1. Traffic contributes 8.3 per cent of the total variance. Pass frequency contributes 33.0 per cent. The visibility factor contributes 58.7 per cent, more than the other two combined.

That single line tells you where to spend money next year. Buying a better people counter would shave the 15 per cent traffic bound down to maybe 5 per cent, and the combined uncertainty would fall from 30.1 per cent to 29.2 per cent, which is nothing. Halving the uncertainty on the visibility factor, from 40 per cent to 20 per cent, drops the combined figure to 22.6 per cent, and the reported band tightens from 5,000 to 20,200 down to 6,900 to 18,300.

So the cheapest way to make your sponsorship numbers more credible at this show is a small piece of observational work on how many people actually have the banner in view when they pass, which is a morning with two clipboards. The counter upgrade the ops team keeps asking for would barely move it. Deriving that visibility factor properly is its own subject and belongs elsewhere in this cluster.

What goes on the sponsor report

A single line, in the same place every year.

Estimated impressions, 12,600, with a 95 per cent interval of 5,000 to 20,200. Built from an entrance count of 6,000 plus or minus 15 per cent, an assumed 3.5 passes per attendee plus or minus 30 per cent, and a visibility factor of 0.60 plus or minus 40 per cent. Largest source of uncertainty, the visibility factor.

Four sentences. They do three things a bare 12,600 cannot. They let a sponsor compare your figure against a media buy where intervals are normal. They tell your own sales team not to promise 12,600 to next year's buyer. And they make the following year's improvement visible, because if the band narrows, somebody did work.

The AMEC Barcelona Principles 3.0, published in 2020, put this under their seventh principle, which holds that communication measurement and evaluation are rooted in integrity and transparency to drive learning and insights. A figure with no stated method fails that test whether or not the figure is right.

The IAB's Digital Out-Of-Home Measurement Guide, published in July 2025, is stricter still on the same point. It describes the MRC standards as requiring that impression counts rest on validated methodologies accounting for the location, orientation and characteristics of the display inventory as well as the movement patterns of the surrounding audience, with validation extending to the external data sources used in the estimate and periodic audits to keep it honest. The same guide separates a gross impression from an opportunity to see and both of those from a likelihood to see, which is a reminder that even the word impression needs defining before the number after it means anything. The related question of ranking assets once you have those figures is handled in putting every asset in one CPM column.

Where this stops

Error propagation handles random uncertainty in inputs you have identified. It does nothing about bias, and impression estimates are usually biased in one direction.

If your pass frequency assumption came from a sales conversation, it is high. If the visibility factor was chosen because 0.60 looked reasonable, it is high. If the traffic counter double-counts people who step backwards through the beam, that is high too. Three optimistic inputs produce an optimistic point estimate, and putting a symmetric band around it just gives you a symmetric band around the wrong centre. The interval is honest about precision and silent about accuracy.

The second limit is that this whole exercise assumes the three inputs are independent. They are not, quite. A day with unusually heavy traffic is also a day with more crowding, which lowers the share of passes with a clear sightline, so traffic and the visibility factor are negatively correlated and the true combined uncertainty is a little smaller than 30.1 per cent. Estimating that correlation costs more than it is worth at this level of precision, and the effect makes your band conservative, which is the right direction for a number going to a customer. Counting the underlying physical impressions at all is a separate discipline covered in what a physical impression count can and cannot claim, and none of it protects you against adding the same attendee up nine times across a package. All three sit under the same exhibitor analytics work.

Pick the single largest sponsorship asset on your last post-show report, write down the three inputs behind its impression figure and a plus-or-minus bound for each, and run the two clauses above. If the resulting band is wider than the number is impressive, you have found this year's measurement project, and the variance split has already told you which input it is.