Abstraction and the problem with percentages

How do you relate to the numbers on your report? Can you visualise what it is trying to tell you? I’m going to suggest that for most of you, it will be a no.

How big does a number have to be so it becomes abstract? One thousand, one million, more? We’ll it is a lot lower that you think. Lets try a simple exercise to show you what I mean. Close your eyes and image a black background, and then a a white dot. (You can use whatever colours you prefer). That dot is now sitting in your imagination, now try adding one more dot at a time, and try to figure out at what point your ability to hold those dots in your imagination stops.

It should be around 15 dots, so that shows you what one version of abstraction is, there are other types. In the real world using your normal vision at what point can you look at a number of items, for example boxes before you stop to stop and count them. It is about 5 to 6.

Did you count or do you see 3 chairs?

So there are limits to our cognitive process to visualise and count numbers. As mentioned in my other blog posts, for example Your Brain & Report Design Part 1 – The Attentive vs the Intuitive Process, which goes through the how you have two path ways, the intuitive and the attentive process, looking at some items and being able to see the number and not count it, would come under the intuitive process (Don’t forget it can be error prone). Where as stopping and counting the items would be under the attentive process.

One good example of the movement of the attentive process to the intuitive process, is the understanding and recognition of dice patterns. Do you count the dots any more or just see the pattern? If you have very young children watch them as they grow up, to see the point they stop counting the dots on the dice and start seeing the pattern.

Reducing Abstraction

So that is one side of abstraction, our limit to visualise objects and quickly count things. The other side is the ability to relate to them, to turn them from an abstract concept to a number that has meaning to us. The number one offender is the percentage, but why?

In the book ‘Thinking, fast and slow’ by Daniel Kahneman, he shows the impact of something called frequency format and how it affects how we relate to a number. For example:

  • 5%

It is nice simple low value, it is 5 out of 100. But as been mentioned above, we would find it hard to imagine 100 items, 5 should not be an issue. We just see the small number. Let’s change it to something more relatable:

  • 1 in 20

Which do you find that you relate to more? The percentage or the ratio?

There is some body of evidence that when parts manufacturers, moved from expressing part failure rates as percentages, to a ratio, there was a lot more attention to the issues in the manufacturing process to drive those numbers down. But why?

Materialising the number as a ratio, moves it from an abstract concept of ‘how likely’ to the intuitive ‘how many’. You see this in adverts to make things personal to you. Does the poster say 16% of men may get prostate cancer? No it will say 1 in 6 men in their lifetime.

So moving from a ‘how likely’ to ‘how many’ number enables the person to understand, relate and visualise a lot more easily than an abstract percentage. However it can have its draw backs.

As part of the impact into frequency format, psychologists conducted a survey, were the question was asked which is the more dangerous disease:

1 – A disease that kills 1,286 people out of 10,000

2 – A disease that kills 24.91% of the population

The majority of people answered 1, the one that kills 1,286 people is the more deadly. If you stop to work it out, it is about half as deadly at around 12%. So what happened? People found it easier to relate to the number and make it personal.

Cat Food & Perfect Ratios

Mr Schrodinger, I think the box is too small!

If you are of a certain age and from the UK, you may recall the adverts for Whisker Cat Food, which had the marketing line of ‘8 out of 10 owners said their cats preferred it’.

Lets break it down with three examples, that all have the same outcome:

  • 8 out of 10 owners said their cats preferred it
  • 4 out of 5 owners said their cats preferred it
  • 80% owners said their cats preferred it

So we have expressed it in three ways, two ratios and a percentage. We can ignore the percentage, we have already discounted the percentage from our ability to relate to it. Which leaves us with two last options:

  • 8 out of 10 owners said their cats preferred it
  • 4 out of 5 owners said their cats preferred it

Which do you more relate to? Is there a good ratio to use? Yes, the power of 10’s. Use were you can ratios in multiples of 10s. Also round up numbers where you can. For example:

  • 3.75 in 15

3.75 is 1/4 quarter of 15, but it should be expressed as:

  • 3 in 10

Where we have rounded up 2.5 to 3. It makes a bit more sense doesn’t it!

Framing the number – Positive or Negative?

This is to make you think about the context of how you present the result. So, lets say you go to the doctor, and you have to have an operation, or an injection or something that will may have some sort of an impact on you. Does the doctor say:

  • 9 out of 10 people have no complications after this surgery
  • 1 in 10 people have complications after this surgery

Which would you prefer to hear? So framing is presenting the value in a way to make it a bit more or less personal. So for the example of manufacturing defects, would you use:

  • 9 in 10 success rate
  • 1 in 10 failure rate

Which should be focusing on? Well in this use case the 1 in 10 failure, that makes it more relatable to the situation.

As mentioned on abstraction and relating the numbers to you, if the Doctor is really good they’ll say ‘90% of people have no complications after surgery’

When to use a percentage?

As every good IT consultant will say, ‘It depends’. Something to consider:

5% Cash back on purchases sounds a lot better than 5p back in every £1 spent! You know it is a small amount , and easy to compare across other offers. The higher the percentage the better the deal. For cloud services 99.999% uptime, sounds a lot better than 0.9 seconds downtime per day. The more/less decimals the better/worse the service level. So when measuring your service level you are comparing the difference, not ‘how many’, you can see if it has reached the acceptable agreed service level or not.

Other uses of percentages

Sometime around 2014, there was survey of 10,000 secondary school children. They were asked what drugs they had tried. A few year later they did the same survey, again asking the same questions. The media was then full of news items like, ‘100% increase in cocaine usage in school children’, and ‘A resent survey has shown that drugs usage has doubled in a few years’ and so on. Looking at the data, it showed that it had increased 1 student, to 2. So technically it was correct. However not quite the sensation the newspapers would lead you to be.

In ‘Thinking, fast and slow’ it talks about that there is some evidence that in the US legal system, defense or prosecution lawyers will use one or the other.

  • Defense – DNA evidence in capital cases is wrong in 1 in 1,000 cases
  • Prosecution – DNA is right in over 0.01% of cases

You can now see what they are doing to relate, or disassociate the numbers to the jury.

Rounding up

You should have some insights on abstract numbers, and how by using ratios, you can move numbers to something a bit more intuitive and a lot more relatable. I always say when I present talks about subject like this, that for the intuitive and the attentive process, one is not better than the other. We just want to help people process the information quicker, and I will say it for ratios and percentages, one is not better than the other, just consider it for your use case.


Cat Image: https://unsplash.com/photos/mrTydVjg04o

Cashback: https://www.vectorstock.com/royalty-free-vector/cash-back-5-percent-money-refound-concept-badge-vector-28061818

Chair Picture: Jon Lunn (Me)

Thinking, Fast and Slow – Daniel Kahneman – Chapter ’Rare Events’

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