# Are NASCAR Stage Lengths Arbitrary?

# Stage Racing

The 2017 NASCAR season brought us stage racing: Segmenting a race into three parts, each of which has a winner (and point reward).

If you’re me, the first thing you wondered was not how stage racing would play out. It was “how the heck did they decide on the stage lengths?

Here’s the stage lengths, in laps, as published by NASCAR for the 2017 season.

## Totally Random, AmIRight?

Sure looks that way. But you know NASCAR put some thought into these choices. The whole point is to make the racing more exciting, with more at stake, right? So I wondered how they arrived at the numbers they arrived at.

The process I used is called ‘reverse engineering’. It’s when you take something that’s been made and try to figure out ** how** it was made — often with the intent of making it yourself. I’m not going to start my own racing series and propose my own stages, so this is what we call in physics a ‘gedanken’ or ‘thought’ experiment. Einstein loved these.

## Step 1: Simplify the Data

Take a look at the table above and tell me if you see a pattern. If you do, I’m really impressed because I sure don’t.

If you want to know where the breaks are for a given race, the table is useful. If you’re trying to figure out how NASCAR arrived at those particular numbers, the table is decidedly not useful. There are too many numbers and too many columns to make sense of.

So let’s re-organize the data. We’re interested in how they decided how long each stage is, so we need to know how many laps are in each stage. That’s not what they’ve given us. So I re-did the table and wrote how long (in numbers of laps) each stage is.

**If something doesn’t jump out at you when you look at the data this way, check your pulse.**

Here’s what I noticed:

**The first two stages are always the same length**- They range from 20 laps (Watkins Glen) to 130 laps (Martinsville)
- They are all multiples of 5. There are no 83-lap first or second stages.
**The third segment is always longer than the first two**- The smallest ratio is about 1.2 times (Pocono and Indy)
- The largest ratio is 2.5 times (Watkins Glen. Sonoma is 2.4)

This makes sense, right? If we were going to split up a race into three parts, the first principle we would use is to make our scheme as simple as possible. In principle, each segment could be anywhere from 1 to 98% of the total race length.

There are some constraints. You can’t make Stage 1 and Stage 2 too short because you end up with someone getting points for leading 10 laps, which seems sort of silly.

But here are the two simples ways you could divide up the stages.

- Make all three segments the same length (i.e. split the stages into thirds)
- Make the first two segments the same and the last segment twice that amount (i.e. split the segments into quarters and use two of them for the last stage)

The first is the simplest possible way to do it — but then there’s nothing special about the last segment. Since we want to build to to crescendo at the end, the third segment must be longer than the first two.

A way forward becomes clearer.

Also note that we’ve made the problem easier because now we don’t need all three columns. We only need two of them because the first and second stages are the same.

## Step 2: What are the Actual Variables

We still have a lot of variables. If you scan our table, you’ll notice that there are just an awful lot of numbers. So I asked myself: **What changes from race to race?**

- Track Length and Type
- Race Length (i.e. number of laps run)

Track length is a number, but it’s a complicated number with decimal points and all.

Track type isn’t a number.

That leaves us with the number of laps run in each race — which varies quite a bit. That number is obviously correlated to the track length, but since we’re working with units of laps, let’s see if there’s a way of simplifying our data even further.

I’m going to write an equation. **DON’T PANIC.** I’ll talk you through it. It’s simple. Really.

The total number of laps in a race is the sum of the number of laps in each stage. So I’m going to define variables:

- S1 is the Number of Laps in Stage 1
- S2 is the Number of Laps in Stage 2
- S3 is the Number of Laps in Stage 3
- T is the total Number of Laps

The following is true:

All that equation says is that the total number of laps in a race is equal to the sum of the number of laps in each stage. That’s it. You already knew this, you just wouldn’t write it that was necessary.

BUT: There are the same number of laps in Stages 1 and 2, right? So I can simplify this equation a little. S1 = S2.

This just says twice the number of laps in Stage 1 plus the number of laps in Stage 3 equals the total number of laps in the race.

Here’s where it gets a little tricky because we are dividing. The problem we’ve got is that every race has a different total number of laps. But I can actually eliminate that from my equation by dividing the whole equation (both sides) by T

Stay with me. We’re almost there and this equation is not as scary as it looks.

What is this S1/T thing? It’s the number of laps in Stage 1 divided by the total number of laps. For example, at Martinsville last week, there were 125 laps in Stage 1 and 500 laps in the whole race. So stage 1 was 125 laps/500 laps which is 25%

So:

- S1/T is the fraction of laps in Stage 1
- S3/T is the fraction of laps in Stage 3

## See What I Just Did?

I just found a way to totally eliminate a major variable between tracks, which is the number of laps run at each track.

I also did something else: I showed that there is **only one choice to make for every track** in determining stages. A mathematician would say that there is only one independent variable.

Once you decide the first segment size, the second and third are automatically determined for you. Look at Martinsville for an example. There are 500 laps. Since the first two stages are the same, once I pick that number, there is only one possibility for the length of the third stage.

If I pick the number of laps in Stage 1 to be 70, then I have 500–70(Stage 1)-70(Stage 2) = 360 laps for Stage 3.

If I pick the number of laps in Stage 1 to be 120, then I have 500–120–120 = 260 laps left for Stage 3.

# How Did they Pick the Length of Stage 1?

This is the question.

One more time with the chart, but now let’s write the stage lengths in terms of the percentages each stage is of the total race.

For example, if Stage 1 was 125 laps out of a 500-lap race, then that would be 125 divided by 500 = 25%. The first stage contains 25% of the total race laps. All of the numbers should add up to 100% going across — the places where they look like they don’t is because I rounded everything to one decimal place to make understanding this a little easier.

Unless you are willfully not following, you should see a pattern: **All of the Stage 1 lengths are between 22% and 31.3% the length of the total race.**

Look back to where I mentioned what I thought the two most obvious ways to divide up the races would be: I predicted that the first stages would be between 25% and 33.3%. That first guess was just about right.

## Diving Deeper

So now, of course, I want to see if there’s a pattern as to which tracks are on the lower side and which are on the higher.

When I did this the first time, I plotted a histogram — the number of races having a Stage 1 percentage around a particular number. For example, there are 12 races that have a Stage-1 percentage between 29.5% and 30.5% — which I mark as ‘30%’.

This was my clue. Most of the races fall in one of two percentages: 25% and 30%. So I went back and re-did the table

Just to make some patterns clear, I’m going to sort the graph and include some color coding. I’m also making it one long graph, which I know is difficult to read on a phone, but putting the data side-by-side in two columns seemed to make it harder to understand the point and see the trends.

Here’s the same data from above, but:

- I’ve sorted it according to Stage-1 length in percent.
- I’ve left out the second stage because it’s the same as the first and thus only serves to confuse and not enlighten.
- I’ve left out the third stage because I’ve proven that once we know the first stage length, we know the third as well given that we know the total number of laps.
- I added the track length because it turned out to be informative.

**Less than 25% (grey)**The Stage-1 lengths for both Road Courses are 22–23% of the total race length in laps.- These races have the smallest number of laps
*in toto*(90 and 110 laps respectively). They’re long tracks, but not long races in terms of total miles. - At Watkins Glen
- 25% would be 22.5 laps
- If you round up to a multiple of 5, Watkins Glen would be 25, 25, 40. That’s not a lot of difference between Stage 3 and Stages 1 & 2
- NASCAR rounded down to 20 and the race is 20,20, 50
- 30% would be 27.5 laps
- Rounding up to 30 laps gives you 30, 30, 30
- Round down to 25 laps is the same issue as above
- You can do the same calculations for Sonoma and get the same issues, which explains why the decisions went as they did
**24–26%****(yellow)**- All tracks that are 1 mile (or slightly more) fall in this group
- Anomalies
**Dover**doesn’t appear here — and I have no explanation at this point why.- The July Daytona race does — It’s a relatively short race (400 miles at a very long track.) If you put it in the 30% group, you’d have stages of 50,50, 60, which doesn’t make for a much longer Stage 3, so it make sense to put it in the 25% group.
- There are a few 1.5 mile tracks in this group, but not all of them.
**26%-27% (green)**— Atlanta, second Charlotte and Darlington are odd tracks out. It seems like they ought to fall in the 25% or the 30%.- Each of these races is roughly 500 miles, but the laps run are 325, 334 and 367 respectively.
- Look at the numbers for Darlington.
- 25% for Stage 1 would be 92 laps. They could round to 90 and then it would be 95, 95 and 187 — which makes Stage 3 pretty long.
- 30% for Stage 1 would be 110 laps, so it would be 110, 110 and 147, which makes Stage 3 pretty short relative to Stage 1 and Stage 2.
- I think 100 laps turned out to be a nice compromise.
**29–30% (blue)**— these are mostly the larger tracks. Both Talladega races and the Daytona 500 are here- Dover is here. If you’d put it in the 25% group with the other tracks its size, you’d have 100, 100, 200. Perhaps 100 laps at Dover just goes by too quickly to make for a good stage? I don’t have a good explanation for this. Maybe NASCAR put it here just to mess up people like me trying to figure out their reasoning.
**31.3%**(peach) This is a consequence of rounding. 30% of 160 laps is 48 laps. When you round up, you get 50, which is what they went with. So these are actually consistent with the 30% group.

## But, Wait…

Yeah… something ought to be bugging you. The 1.5 mile tracks are weird, right? If my theory is correct, they should all be in the same category.

In fact, we’ve even got races at the same track that have different choices for the number of Stage 1 laps. So I charted up the races for only the 1.5 mile tracks and it became clear when I added in the total race length in miles.

You see it, right?

**the 400-mile races are all in the 30% group, the 500-mile races are in the 25% group.**

Charlotte (the 600-miler) is in-between those two. If you have 600 miles and you go with 25% for stages 1 and 2, then Stage 3 ends up being 300 miles — which is a whole race by itself at some places.

I tested out this theory by plotting all the first stages vs. the total length of the race and it sort of works.

And it’s true that, the longer the overall race length, the higher the percentage of laps contained in the first stage. If you look at the yellow box and squint, you might even argue that it’s almost linear. (But you have to squint really hard.) I didn’t say it was exact. We’re not doing quantum physics here.

# TL;DR

Here are the principles I would claim NASCAR used to develop the stages.

- The first and second stages always have the same length in laps
- The first and second stages are always be a multiple of 5 laps.
- The first and second stages are always be either 25% or 30% of the total number of laps run

- Shorter tracks and those with a small number of total laps (like road courses) are 25%;
- Longer tracks are 30%;
- 1.5 milers are 25% if the race is 500 miles and 30% if the race is 400 miles.

- In rounding so that Rule 2 is obeyed,

- Make sure that there is enough of a difference between Stages 1 & 2 and Stage 3 so that Stage 3 and thus the end of the race remain special
- Make sure that the Stage 1 and 2 aren’t too short so that drivers aren’t rewarded for leading too few laps.
- Make sure that Stage 3 isn’t too long so that it doesn’t feel like a whole race all by itself.

## PS

Everything I said up there goes for Trucks and XFINITY, too!

*Originally published at **Building Speed**.*