How To Really Treat Percentages With Negative Numbers

When do you really need to treat percentages with negative numbers? “Not often. If not, never!” This might be the typical person’s answer. But what if I told you that this “special” requirement comes up far more often than you might imagine. Most of the time, the issue goes under the radar or is hidden in plain sight.

It is often the case that you and I are the end-users of such calculations rather than the ones doing them. Therefore, we are often oblivious to the fact. However, if we are not careful or if we do not understand what it is that we are reading/using, it may lead us to make poor decisions in potentially critical situations.

Before I proceed any further, let me take a moment to explain what exactly is the requirement/issue that I am discussing here.

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What are Percentages with Negative Numbers?

Our typical understanding of percentage calculations is reserved for positive numbers only. A percentage is nothing but a way of expressing proportions or fractions.

For example, if 750 out of 1200 seats in a theater have been filled, we say that (750/1200)*100 = 62.5% of the seats have been filled.

Consequently, 37.5% (100%–62.5%) of the seats are still available. It gives us a simplified view of the situation; we need not be bothered by the total number of seats in the theater. All that we need to focus on is:

1. How much of the theater is full?

2. And how much of it remains available?

So far, so good. But now, let me introduce you to a hypothetical situation. You are running a business that offers three products: A, B, and C. You had launched all three products simultaneously and are interested in working out the numbers at the end of the first month. Let us say that at the end of the first month, the return from each product line looks as depicted below:

As you can see, Product A returned a total of 600 monetary units, Product B: 400, and Product C: minus 400. Needless to say, product C has resulted in a loss. And your system quantifies this loss as a negative number (-400). Your task now is to answer the following question:

What percentage has each product contributed to the bottom line of the business at the end of the month?

This simple situation may be hypothetical, but is a very frequent phenomenon that occurs in many different contexts of life (more on that later). I am mentioning this to differentiate the fact that this is not one of those math-textbook problems that are as close to reality as the fact that Santa Claus exists.

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Taking the Standard Approach to Treat Percentages with Negative Numbers

Let us start with our usual approach to calculating percentages. As the first step, we could compute the total revenue.

In our case, this happens to be 600 monetary units. Now, to compute product A’s percentage contribution to the total, we could divide product A’s return by the total return and multiply the result by 100:

(600/600)*100 = 100%

That does not look right, does it? It can’t be true that product A has contributed 100% to the total return at the end of the month. We know for a fact that Product B delivered a profit and Product C delivered a loss.

If we chose to add the percentage contributions of Product A and Product B, it would exceed 100%. So, what is that we are doing wrong here, and how should we proceed.

Renowned mathematician and Professor, Jordon Ellenberg offers the following slogan to live by when faced with such a problem:

“Don’t talk about percentages of numbers when the numbers might be negative.”

– Jordon Ellenberg.

I would love to live by this slogan. But real life won’t let me! If my investor or client demands to see a percentage number, I cannot quote Jordon and back off. I “have” to produce a number, and make sure that it is as reasonable a number as possible. So, how should we proceed now?

How to Really Treat Percentages with Negative Numbers

You see, the challenge lies with the question we are asking:

What percentage has each product contributed to the bottom line of the business at the end of the month?

When we are dealing with negative numbers, this question is not sufficient. We need to know ‘what kind of percentage’ we need to calculate as well. That question, in turn, leads to a couple of other questions:

1. What is the reason we need to see the percentage contributions of each product?

2. Is it helpful to differentiate profitable products from non-profitable ones?

Let us say that the investor/client is interested in learning which ideas are working and which are not. For this purpose, it makes sense to calculate percentages in terms of absolute values. This would work out to be as follows:

Now, we could calculate the percentage contributions of individual products to the absolute total revenue as follows:

1. Product A: (600/1400)*100 = 42.86%

2. Product B: (400/1400)*100 = 28.57%

3. Product C: (400/1400)*100 = 28.57%

You might be sceptical about Product C contributing 28.57% to the total absolute revenue. This is where the second question we asked comes into the picture (Is it helpful to differentiate profitable products from non-profitable ones?).

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Context-Driven Segmented Calculation

When we use words such as “revenue” or “profit”, we tend to expect positive numbers and this introduces a bias into calculations and the corresponding interpretations. Instead, we could define the net absolute total revenue as “net change”, and segregate profitable products’ contributions from the non-profitable ones.

The corresponding report/calculation would turn out as follows:

Such a calculation structure gives the percentage contributions of each product to the bottom line whilst preserving information about which product is doing well and which isn’t.

Percentages with Negative Numbers in Daily Life

What we have seen is just an exemplary illustration of a requirement to calculate percentage figures with negative numbers. Such a requirement goes well and beyond this example.

In areas of book-keeping and operations research (for instance), such challenges are a common phenomenon. You might think that you are off the hook as long as you do not work in these professions. But think again!

If you are the end-user of any information produced by such ventures, you will have to be careful about what it is that you are looking at. As we have just seen from the illustration, it is possible to tell a different story just by calculating the numbers differently. There is no clear right or wrong; the calculation method depends on the purpose of the numbers produced.

In other words, there is room for manipulation or misunderstanding. So, if you are dealing with numbers that can be negative (like profit/loss, debt/credit, etc.,) and you see percentage figures based out of them, be sceptical and curious.

Try and understand what information is objectively the best for your decision-making process and evaluate if this information is available or derivable from the information you have at hand.

On the other hand, if you are the one producing the information, try to handle the calculation as responsibly as possible by providing different versions of percentage calculations for different use-cases and/or end-users.

To conclude, when you are treating percentages with negative numbers, exercise caution and evaluate carefully!

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Further reading that might interest you: How To Really Deal With The Friendship Paradox? and What Is So Special About 69! ?

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