# Why are Fibonacci numbers used for estimating effort?

What is the reason for using Fibonacci numbers to estimate effort for Agile projects?

I know this is not strictly testing related, but testers are still part of the planning and understanding of this is useful when estimating testing effort.

• Is it a 5 or a 6? 5 or 8? 8 or 13? If the buckets are farther apart it helps choose the complexity bucket easier vs. consecutive numbers. It’s not about the numbers per se - they’re a proxy for a “fuzzy notion of complexity” keeping them “far apart” avoids arguing about n or n+1.
– PhD
Commented Mar 5, 2020 at 16:33
• Following what PhD wrote, as the complexity increases, the resolution gets harder and it becomes more difficult to pinpoint the exact amount of difficulty between useful discrete levels. Fibonacci produces useful bucket sizes that reflect that. Commented Mar 7, 2020 at 19:18

Spoiler: No scientific reason.

Fibonacci grows very fast, so people will have fewer options before reaching enormous values; thus it incentivizes breaking work down in smaller pieces.

If the smallest typical work takes 1 hour, a big piece may take 8, 9, 10, ..., 16, ... even 32 hours.

However, if the smallest piece of work takes 1 story point, and they grow in Fibonacci, a big piece may take only either 8, or 13, or 21, and only if you stretch much 34 story point - 4 options to pick rather than 24.

• Fibonacci does not grow exponentially (though it comes close after 13/21, approaching the golden ratio, about 1.6180339887498948482). The ratio between any two consecutive numbers would need to be a constant (which is not the case for Fibonacci). A second derivative (or equivalent) greater than zero does not imply exponential. For instance, it could be square or cubic (which is not exponential). Commented Mar 6, 2020 at 15:06
• Hate to spoil your spoiler, but there really is a scientific reason. Commented Mar 6, 2020 at 19:06
• @PeterMortensen see math.stackexchange.com/questions/2981007/… although fib is not an exponential sequence as there isn't the constant ratio that requires, there are exponential functions which bound it above and below for large enough values, and therefore it has exponential growth. You seem to have confused the two terms. Commented Mar 7, 2020 at 11:57

Fibonacci series is just one example for estimation efforts. Some teams also use series as below:

``````1, 2, 5, 8, 20, 40, 100, ....

1, 2, 4, 8, 16, 32, 64, ....
``````

The idea is to use an exponential scale for estimating efforts.

The reason is the larger the story point, the more uncertainty there is around it and the less accurate the estimate will be.

• None of those are exponential (the ratio between any two consecutive numbers would need to be a constant). The second is polynomial (square growth). They are non-linear, with a second derivative (or equivalent) greater than zero. There is probably a word for it. Commented Mar 6, 2020 at 14:57
• @PeterMortensen: The word you are looking for is "superlinear": it grows faster than a line. Commented Mar 6, 2020 at 14:58
• @PeterMortensen: The Fibonacci Sequence does not take the form of an exponential b^n, but it does exhibit exponential growth. Essentially, the Fibonnaci series approximately increases by a ratio (i.e. the golden ratio), meaning you can approximate its growth using an exponential x, which means it has the textbook (but approximate) velocity of exponential growth. Not mathematically precise but more than close enough for it to be called exponential in a non-mathematical context. Commented Mar 6, 2020 at 15:21
• @PeterMortensen the second sequence is 2^n, which by definition is exponential. Polynomial (square) growth would be 1,4,9,16,25,36... Commented Mar 6, 2020 at 15:58
• The first is also exponential because it is bounded below by 2^n Commented Mar 7, 2020 at 2:49

They reflect that the degree of uncertainty grows as you look further out and at bigger tasks with more dependencies.

For example, today you can be reasonably confident about how much effort is needed for a small task. You may be highly confident that you can finish it within a day, and, critically there is little uncertainty about the factors involved. So it is easy to say this should be a 3 rather than a 2, or a 1 rather than a 2 with a reasonably high degree of certainty. Humans are involved and they have many variable factors to consider.

When you look further out, and at bigger tasks, uncertainty becomes MUCH greater. Many other dependencies are involved and many of them have uncertainties.

Given this, one learns that trying to be accurate about the future this way, for exampling estimating that a task will take 18 days instead of 17 is likely to be a foolish endeavour because, 18 days out you are unlikely to have the information to make an accurate estimate. This is why we use Fibonacci. When you get to a task that you consider to be a 13 there is a lot of uncertainty and it is best to consider the 'next' level of effort to be 21 as that is about the level of accuracy you can practically use ahead of time.

Another real world example of how uncertainty increases further out in the future - whether you go shopping in 1 day or in 2 days is something you have a high degree of certainty about. If you are out of milk then there is a big difference between 1 more day and 2 more days! However, today, you don't know what your situation will be in a couple of weeks, right? So that's why it jumps from say 8 to 13 - somewhere in those 5 days you will need more milk again, but which specific day in that range you can't predict right now.

## Estimation is hard. Particularly about the future.

Common estimating methods include numeric sizing as well like 1 to 10 or sizes like XS, S, M, L, XL, XXL, XXXL or Fibonacci sequence 1, 2, 3, 5, 8, 13, 21, 34, etc.

The reason for using the Fibonacci sequence is to reflect the uncertainty in estimating larger items. A high estimate usually means that the story is not well understood in detail or should be broken down into multiple smaller stories. Smaller stories can be estimated in greater detail. It would be a waste of time to discuss if it is 19, 20 or 25 the story is simply too big.

Important is that the team shares a common understanding of the scale it uses so that every member of the team is comfortable with it.

• Could this be interpreted as estimate 13 is as hard as 5 and 8 combined? Commented Mar 5, 2020 at 8:43
• Absolutely not. Treat the numbers as measures of relative effort so you can choose between them. but do not treat them as math you can add. That way lies '5 people should do 7 days in 4 days' kinda approach which quickly fails in the complex world of software development where meeting fixed targets means ditching quality. Commented Mar 5, 2020 at 10:30
• Also known (mythical man month) as '9 women can't have 1 baby in 1 month' Commented Mar 5, 2020 at 10:31
• @MateMrše While there's no obligation that a 13 is the same hardness as a 5 and an 8, it is convenient if your team can do so. It makes analyzing your velocity much easier. You can talk in terms of points per week rather than some complicated vector. Personally, I have worked on one team where we chose to have colors of points which represented the types of efforts. Debugging, for instance is famously difficult to time box, so we tracked it differently than other tasks. Why? Because it was convenient for us to do so. Commented Mar 7, 2020 at 2:56

A notion I don't see in any of these answers is that in a simple 1-10 range, people can get bogged down in whether something is a 3 or is it really a 4? And what if another person thinks that it should be a 2 instead of a 3?

By using a Fibonacci sequence, you eliminate a bit of that "hair splitting".

This question is answered in a blog post by Jeff Sutherland (co-creator of Scrum). It's rooted in a US Department of Defense study on estimation.

Rand researchers then studied the effect of the numbers estimators can choose and found a linear sequence gave worse estimates than an exponentially increasing set of numbers. There are some recent mathematical arguments for this for those interested. The question then--if you want the statistically provable best estimate--is what exponentially increasing series to use. The Fibonacci is almost, but not quite exponential and has the advantage that it is the growth pattern seen in all organic systems. Why does the Fibonacci sequence repeat in nature?

Posted about this just recently. Reference to the original is at the bottom.

# Key Idea

Suppose you want to know how long a task will take. You take a look at the problem and estimate that it will be one hour's worth of difficulty/complexity. At the end of the hour, it's not done. You've realized something about the problem and it's as if you are starting over. What is a reasonable/minimal estimate of the time it will now take? Thinking for a moment, you realize it should be the amount of complexity you already knew about plus the complexity you just discovered. At this point, the old complexity was zero, so you add 1 and 0 to get your new estimate of 1. Continuing in this fashion you obtain the following Fibonacci Sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

Each step satisfies the following golden ratio since the sequence yields the most efficient next estimate

a+b / a = a / b

Or more explicitly

new_estimate_of_complexity = new_complexity + old_known_complexity

new_estimate_of_complexity / new_complexity = new_complexity / old_known_complexity

## Further Issues

But then you ask, does it mean that it is always as if I'm starting over? People will get upset if I have to tell them that I thought it was going to take 8 hours but now it will take 13 more. Thinking for a moment one realizes that if we only update the time, so that estimating 13 when we had 8 only adds 5 more hours, then it is as if the sequence is reversed.

..., 55, 34, 21, 13, 8, 5, 3, 2, 1, 1, 0

The same logic still applies. It's as if we started with one of these higher numbers and we're giving smaller estimates as time went on.

Furthermore, we can scale each number in the sequence by the same amount, say 8.

0, 8, 8 (2 days), 16 (four days), 24, 40 (one work week), 64, 104 (~ 2.5 weeks), 168 (~ one month), 272, 440 (a sprint), ...

This scaling number accounts for how long the problem solver needs to understand and reason about the problem. On average, per individual and across individuals, 8 hours is usually the best starting point, since it strikes a balance between all of the following issues:

• Risk of giving estimates that are too short.
• Risk of giving estimates that are too long.
• Accounting for interruptions.
• Error in estimating the complexity of a problem.
• The problem statement changing midstream.
• Etc.

## What it is not

It is not about uncertainty in the task. Quite the opposite, it's about what we know. Known complexity.

It's not about human perception, but it does have the side effect of influencing our perceptions. Yes, 13 really does seem bigger than 8 without being too much bigger.

It's not about guessing, which people commonly assume when they guess at the scaling factor. Clearly, identifying this number could be made more rigorous.

Scaling is not the same as using a different multiple such as 2. This would effectively subsume the Fibonacci Sequence, which we know is accounting for what is known about the complexity of the problem. Other sequences are making assumptions about the unknown as well as the known, which is why they are inefficient.

Splitting into smaller stories in order to reset the clock, is not necessary nor beneficial. Can anyone make this problem smaller than what it really is? Does anyone have a time machine so I can go back and make up my hours? No on both counts. But when you reset the clock, you are hiding how risky of a problem it happens to be. A problem, that is legitimately estimated to be 8 hours, is the same as one with 8 hours after resetting from 440 hours.

## References

• "Thinking for a moment, you realize it should be the amount of complexity you already knew about plus the complexity you just discovered. At this point the old complexity was zero, so you add 1 and 0 to get your new estimate of 1. Continuing in this fashion you obtain the following Fibonacci Sequence:" I don't understand. Fibonacci sequence is "the old number plus the one before that". You're saying that "the old complexity plus the complexity you just discovered" is the same. Why would the complexity you just discovered be equal to your previous estimate of total complexity?
– JiK
Commented Mar 6, 2020 at 12:57

A reason that wasn't yet mentioned is that it supports well to split a task / user story into two (non-equal) smaller ones - an 8 splits into 5 and 3, etc.
Of course, the same is true for combining them (if the numbers were consecutive).

• An 8 might also split into two 5s or two 3s or a 5 and a 2, etc. The subtasks should be estimated separately and not forced to comply with the original estimate. They probably will a lot of the time but you shouldn't be surprised if they don't. Commented Mar 9, 2020 at 19:08

I think the main reason is because it easier to estimate in relative sizes. This is smaller or larger compared to that. If you look at different buildings from a distance, you could say that one is twice as large, but getting the exact height correct is much harder. For Agile work this means, if that took 2 days, this other thing of similair complexity will take probably also take 2 days.

The second reason is that estimating in hours has the risk of not taking slack and uncertainties into account. Saying a batch of work takes 8 hours does not mean it will be finished in a single day. Probably it will take two days, because noone works effectively for 8 hours, but the project manager already sold the estimate to the client who expected it yesterday. The larger the estimate the higher the risks.

Now why use Fibonacci? The Fibonacci sequence is a natural size, most things in nature have these relative steps. So the brain is already used to these ratios, because they are everywhere. For example, the bones in your hands follow this pattern, but also leafs, shells, etc

1. It is a convenient mathematical sequence whose growth is approximately exponential and not too steep
2. It is the only mathematical sequence whereby tasks may be split into 2 tasks whose estimates are the two previous numbers in the sequence, and such splits may be repeated until all subtasks have size 1
• @Flater counter nitpick: It is the only sequence where, as stipulated in my answer, splits may be repeated until all subtasks have size 1. Commented Mar 8, 2020 at 18:21

Mike Cohn says in his book Agile Estimating and Planning that he originally used 1, 2, 3, 5, 8, 13, 21 as his sequence until a client said to him "You must be very confident to estimate the size as exactly 21 and not 20 or 25". He realised that 21 was too precise, so he changed it to the vaguer value 20.

8 is bigger than 5 and smaller than 13. It's much bigger than 3 and much smaller than 20. Some teams use other sequences such as doubling: 1, 2, 4, 8, 16, however, in my mind that makes it too precise. It seems to be claiming that a 16 is almost exactly twice as big as an 8. I prefer the vagueness of the modified Fibonacci sequence. If a task is twice as big as an 8, it's probably a 20 because we tend to underestimate rather than overestimate. It also allows for "about 50% bigger", so 50% bigger than 8 is 13, again slightly inflated.

I find that the Fibonacci sequence has the right balance between precision and vagueness. However, the Fibonacci sequence is just one sequence for estimating the size of each task. Obviously each team should use the values that work for it. If you find that some other sequence works better for your team, by all means, use it.

II would like to ask that you genuinely try to answer/estimate every question I ask before reading on. The goal of this answer is exactly to make you understand how humans tend to estimate something when they don't know the exact answer. There's no better teacher than your own mind.

## Because it reflects how humans instinctively think about things

Which number is exactly between 1 and 9?

Most people will say 5, because 5 - 4 = 1 and 5 + 4 = 9.

Seems obvious, right? However, when you look at people who have not been "tainted" by a common education, i.e. children and remote tribesmen, you'll notice that they all agree that you're wrong.

Their reasoning is that 3 is in the middle between 1 and 9, since 3 / 3 = 1 and 3 x 3 = 9, which mirrors the reasoning for why we think it's 5, but it uses multiplication and division, instead of addition and subtraction.

Here's a much better explanation than I can give. But I will give you one example of this:

I'm going to introduce you to three people. A millionaire, a billionaire, and a trillionaire. If you think about their wealth, you'd intuitive say that the billionaire is sitting inbetween them, right?

Here's a fun fact: if you sum up the amount of money you need to give 1000 millionaires to make each of them a billionaire, and you instead decide to give this lump sum to the (original) billionaire, that would only just make them a trillionaire.
Knowing this, your interpretation of these numbers should shift, and it should become more intuitively understood that these three numbers are not equidistant from one another.

If you're still not convinced, another way to think of it is like this:

• A million seconds is 12 days
• A billion seconds is 31 years

Have a guess before you look it up. I'd put money on it that any genuine guess you make is going to be much too low.

* A trillion seconds is 316 centuries

Human are generally more capable of estimating value increases/decreases when expressed as a % of the initial value; which is effectively what a logarithmic scale does.

The billionaire who lost half a billion dollars on the stock market today is going to be as upset as the millionaire who lost half a million dollars on the stock market. We think that it's equal because both numbers represent 50% of the original value, and therefore it's proportionally equal.

But when you look at the amounts of money by themselves, they are nowhere near equal. They are off by three orders of magnitude, which is effectively the same as you not distinguishing whether I bought \$50 or \$50,000 of groceries this morning.

## Intuitive things are easier to wield than conscious things

This intuitive human behavior of thinking in logarithms lies at the root of why estimations are logarithmic (increasing exponentially) instead of linear (increasing by a constant step size).

We shouldn't forget why we call it an estimation: we spitball it because it is hard to get it exactly right. Development is not naturally intuitive, and predicting development accurately is one of the hardest parts of the development. Giving humans a tool they innately understand (logarithmic scales) means that they can focus more on the thing they should be thinking about: projected development effort.

It doesn't have to be the Fibonacci series; any series with exponential growth will do. But it's my intuitive guess that Fibonacci's exponential growth (i.e. the golden ratio) strikes a natural balance between too much and too little of exponential growth.

• I think this is an underrated comment. It nicely illustrates the reason non-linear scale is used for estimating effort. Commented Apr 25, 2023 at 13:34