Competitiveness of Formula 1 championship

from 2012 to 2022 as measured by Kendall

corrected evolutive coefficient
arXiv:2501.00126v1 [cs.DM] 30 Dec 2024

Francisco Pedroche1

Institut Universitari de Matemàtica Multidisciplinària,

Universitat Politècnica de València
Camı́ de Vera s/n, 46022, València, Spain
pedroche@imm.upv.es,
WWW home page: http://personales.upv.es/pedroche/

Abstract. In this paper we analyze the FIA formula one world champi-
onships from 2012 to 2022 taking into account the drivers classifications
and the constructors (teams) classifications of each Grand Prix. The
needed data consisted of 22 matrices of sizes ranging from 25 × 20 to
10 × 19 that have been elaborated from the GP classifications extracted
from the official FIA site. We have used the Kendall corrected evolutive
coefficient, recently introduced, as a measure of Competitive Balance
(CB) to study the evolution of the competitiveness along the years in
both drivers and teams championships. In addition, we have compared
the CB of F1 championships and two major European football leagues
from the seasons 2012-2013 to 2022-2023.

Keywords: Kendall’s tau, Formula One, Football, Competitive balance,

sport rankings, contest

1 Introduction
A ranking naturally appears when we sort elements, being this a key action in
more activities such as analysis of sport competitions [2], economic time series
[14], comparison of algorithms performance [25], etc. Series of rankings can be
studied from different perspectives. For example, to analyse sorting algorithms
[15], to define measures of disarray [7], to use rank transformation to develop
nonparametric methods in Statistics [5], to learn to rank in machine learning
applications [4], etc. In this paper we are interesting in characterising a series
of rankings by giving a coefficient that measures the disarray along the series in
the classic manner of [11]. Specifically, we follow the definitions of [21], [20] and
[6].

2 Kendall corrected evolutive coefficient

The Kendall corrected evolutive coefficient, denoted by τbev
•
, was introduced in
[21]. It takes as input a series of m rankings (with at most n elements) that
2 Francisco Pedroche

can be complete (that is, the n elements are ranked in all the rankings) or
incomplete. In addition, we consider the existence or not of ties between the
ranked elements. Kendall corrected evolutive coefficient can be considered as an
extension of a correlation coefficient of two rankings applied to m rankings and
therefore, as output, τbev
•
gives a real number in [−1, 1].
The coefficient τbev reduces to some particular coefficients that are well doc-
•

umented and can be found in the literature. For example, when m = 2 and
the rankings are complete and with no ties, then τbev •
reduces to the classical
Kendall’s τ coefficient of disagreement (see [11], [12], [13]) that can be written
as
2(P − Q)
τ= (1)
n(n − 1)
where P is the number of pair of elements that keep its relative order from
the first ranking to the second one and Q is the number of pairs of elements
that change its order. For example, taking n = 3, the rankings a = [1, 2, 3]
and b = [3, 2, 1] have an associated τ = −1 and the rankings a = [1, 2, 3] and
b = [1, 2, 3] have an associated τ = 1. When m = 2 and the rankings are
complete and with ties, then τbev •
is related to the Kendall distance with penalty
1
parameter p ∈ [0, 2 ] defined in [8]. When m > 2 and the rankings are complete
and with ties, then τbev •
reduces the corrected evolutive Kendall distance with
penalty parameter p introduced in [20].
In sport competitions it is most used the term Competitive Balance (CB) to
measure the balance between the teams [27], [19]. A high measure of CB means
that the competition is highly interesting since it is very difficult to predict the
result of a match (or a race, in our case), while a low measure of CB means that
the competition is very predictable, and therefore boring (see. [18], [17], [9], [10],
[2], [3]). In this regard it is more convenient to use the measure called Normalized
Strength (borrowed from complex networks terminology, see [6], [1]), and that
we define here by
1 − τbev
•
NS = (2)
2
Note that N S is a normalized index, N S ∈ [0, 1], and its value can be considered
as a measure of CB. We will use this index in our analysis. The interested reader
may find the precise definition of τbev
•
in [21] but we omit the details for the sake
of brevity.

3 Formula One World Championships

Formula One (also known as Formula 1 or F1) organised by the Fédération

Internationale de l’Automobile (FIA) is a well-known international racing for
cars [23]. The drivers championship began in the season of 1950, while the con-
structors championship began in 1958. Along the years, there has been some
modifications both in the format and in the rules that the participants must
accomplish.
 Competitiveness of Formula 1 championship from 2012 to 2022 3

A Formula One season consists of a series of races, each of them known as

Grand Prix (denoted as GP), that take place in several countries. For example,
the F1 2022 season consisted of 22 GP and participated 10 teams and 22 drivers.
A GP is held on a weekend. On friday and saturday some qualifying sessions fix
the starting order (the grid ) for the GP race that occurs on Sunday. In this
paper we are interested only in the ranking corresponding to this GP races. This
ranking is decided based on the timing of each driver and he receives a quantity
of points depending on his ranking. From 2010 to 2018 the sharing of the points
was given as shown in Table 1. The points assigned to the constructors in a GP
is the sum of the points of the two drivers of the team that participated in that
GP.

Table 1. Points scoring sharing since 2010

1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th
25 18 15 12 10 8 6 4 2 1

From 2019 one additional point is given to the pilot that occupied a position
in the top ten and furthermore has the fastest lap in the race. FIA has some
rules to break ties between the pilots and therefore the ranking of the drivers
can be considered as ranking with no ties. Note, therefore that each GP has
its own classification. The final ranking (that is, the F1 Championship) of the
season is made by accumulating the points of each GP, and, again, some rules
are applied to break the ties, if any. Our collection of rankings are precisely the
rankings of each GP in a season, both for drivers and constructors. We use these
series of rankings to compute the corresponding τbev•
of that season, and then the
corresponding N S. We precisely describe the used rankings in the next section.

4 Description of the rankings

Table 2. Number of drivers, teams and GP in each analyzed F1 Championship

Year 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022
Drivers 25 23 24 22 24 25 20 20 23 21 22
Teams 12 11 11 10 11 10 11 10 10 10 10
GP’s 20 19 19 19 21 20 21 21 17 22 22

We have selected the F1 classifications from 2012 to 2022. Our criterium to

select our dataset is based on taking the GP classifications of championships in
where 1) the regulations does not vary too much, 2) the distribution of points
4 Francisco Pedroche

(e.g. as given by Table 1) is quite stable, 3) the number of GP does not vary
too much and 4) that the standings can be easily retrieved from the official FIA
site [23]. For example, the 2012 season can be retrieved from the FIA site [24].
In Table 2 we show the number of drivers in each championship jointly with the
number of GP in that year.
To describe our rankings we use the following notation (see [26], [21]). Let
V = {v1 , v2 , · · · , vn } be the objects to be ranked, with n > 1. The ranking is
given by
a = [a1 , a2 , · · · , an ] (3)
where ai is the position of vi in the ranking. Note that if ai = aj , then vi and
vj are tied. If vi is not ranked, then it is denoted as ai = •.

Table 3. Drivers’ name, nationality, and ai vector for three of the first GP of FIA
2012 World Championship. Elaborated from [23]. Note that • means that the driver
did not start or did not finish the race. The rankings are incomplete rankings with no
ties. The order of the drivers in the first column follows the (final) classification of the
constructors championship. The drivers Raikkonen, Grosjean and D’Ambrosio belong
to the same team (Lotus F1) while the rest of teams contributed with two drivers in
the whole GP rankings of this championship.

Driver Nat GP1 GP2 GP3

Sebastien Vettel DEU 2 11 5
Fernando Alonso ESP 5 1 9
Kimi Raikkonen FIN 7 5 14
Lewis Hamilton GBR 3 3 3
Jenson Button GBR 1 14 2
Mark Webber AUS 4 4 4
Felipe Massa BRA • 15 13
Romain Grosjean FRA • • 6
Nico Rosberg DEU 12 13 1
Sergio Perez MEX 8 2 11
Nico Hulkenberg DEU • 9 15
Kamui Kobayashi JPN 6 • 10
Michael Schumacher DEU • 10 •
Paul Di Resta GBR 10 7 12
Pastor Maldonado VEN 13 19 8
Bruno Senna BRA 16 6 7
Jean-Eric Vergne FRA 11 8 16
Daniel Ricciardo AUS 9 12 17
Vitaly Petrov RUS • 16 18
Timo Glock DEU 14 17 19
Charles Pic FRA 15 20 20
Heikki Kovalainen FIN • 18 23
Jérôme D’Ambrosio BEL • • •
Narain Karthikeyan IND • 22 22
Pedro De la Rosa ESP • 21 21
 Competitiveness of Formula 1 championship from 2012 to 2022 5

4.1 Drivers ranking

From the FIA site, we can retrieve the drivers classification for each GP of
the considered championship. In these classifications we can see the ranking,
the points obtained by each driver, and a note indicating whether the driver
has finished the race or not. To construct our drivers ranking we consider that a
driver that has not finished the race (or has not even start it) is an absent element
in our ranking, and therefore it is indicated by •. For example, in Table 3 we
show our notation to describe the first three rankings of the 2012 championship.

4.2 Constructors ranking

From the FIA site we can retrieve the constructors classification for each GP of
the considered championship. The points given to a constructor consist of the
sum of the points of the two drivers of the corresponding team in each GP. In
this case the FIA site offers the points obtained by each constructor. This gives
us the opportunity to create two types of rankings, being the interest to see how
our measure N S is affected by these types. The two considered methods are the
following:
Method 1: We consider that the constructors that have 0 points are tied in
the last position.
Method 2: We consider that the constructors that have 0 points are absent
elements.
As an example, in Table 4 we show the constructors name, scoring and ai
vectors (by using Method 1 and Method 2) for the first three GP of FIA 2012
World Championship.

Table 4. Constructor’s name, scoring and ai vectors (by using Method 1 and Method
2) for the first three GP of FIA 2012 World Championship. The order of the teams in
the first column follows the (final) classification of the championship.

Score Method 1 Method 2

Constructors GP1 GP2 GP3 GP1 GP2 GP3 GP1 GP2 GP3
Red Bull Racing 30 12 22 2 4 3 2 4 3
Scuderia Ferrari 10 25 2 4 1 6 4 1 6
Vodafone McLaren Mercedes 40 15 33 1 3 1 1 3 1
Lotus F1 Team 6 10 8 5 5 5 5 5 5
Mercedes AMG Petronas F1 Team 0 1 25 8 8 2 • 8 2
Sauber F1 Team 12 18 1 3 2 7 3 2 7
Sahara Force India F1 Team 1 8 0 7 6 8 7 6 •
Williams F1 Team 0 8 10 8 6 4 • 6 4
Scuderia Toro Rosso 2 4 0 6 7 8 6 7 •
Caterham F1 Team 0 0 0 8 8 8 • • •
Marussia F1 Team 0 0 0 8 8 8 • • •
HRT F1 Team 0 0 0 8 8 8 • • •
6 Francisco Pedroche

5 Results

5.1 Comparison of constructors and drivers championships

In order to compare the competitivity balance of the GP of drivers and construc-

tors we have computed N S, given by (2) for the GP standings from 2012 to 2022
for drivers and for constructors (with Method 1 and Method 2). The results are
shown in Table 5.

Table 5. N S for the series of GP of the Championships from 2012 to 2022 for drivers
and constructors.

Year N S Drivers N S Constructors N S Constructors

Method 1 Method 2
2012 0.2561 0.2456 0.4052
2013 0.2136 0.1924 0.3421
2014 0.1913 0.1616 0.3106
2015 0.2270 0.2722 0.2350
2016 0.2065 0.2218 0.2143
2017 0.2140 0.2632 0.2179
2018 0.2188 0.2559 0.1886
2019 0.2157 0.2772 0.2596
2020 0.2436 0.2562 0.3652
2021 0.2270 0.2413 0.2715
2022 0.2092 0.2455 0.2376

The data on Table 5 can be resumed on the box-and-whiskers plot shown on

Figure 1. In more detail, the mean values of N S on the period 2012-2022, and
the corresponding sample standard deviation, s, are as follows:
Mean value of N S for drivers: 0.2203, (s = 0.018).
Mean value of N S for constructors (Method 1): 0.2394, (s = 0.035).
Mean value of N S for constructors (Method 2): 0.2771, (s = 0.070).
Let us consider that N S is a random variable. By computing the Shapiro-
Wilk test for normality [22] we obtain the p-values 0.61, 0.08 and 0.44 for the
corresponding N S series for drivers, and constructors (Method 1 and Method 2)
respectively. Therefore we cannot reject the normality of the distribution of N S
of the corresponding samples. Regarding the mean values of N S for constructors
by using Method 1 and Method 2, since they come from the same data (as an
example, the scores in Table 4 give us the corresponding values for Method 1 and
Method 2) we can use a comparison method for means coming from paired data.
By using a t-test we obtain a p-value of 0.18 and therefore we cannot reject that
the means are equal with a confidence interval of 95%. Since the value of the
variances does not have a ratio major than 4 we can use the t-test for comparing
the mean of N S by using Method 1, and the corresponding N S for drivers. We
obtain that the p-value is 0.12 and therefore we cannot reject the null hypothesis
 Competitiveness of Formula 1 championship from 2012 to 2022 7

that the means are equal. All in all we have the statistically the three values of
N S are not different, with a confidence interval of 95%.

NS for the series of GP from 2012 to 2022

0.40

0.35

0.30

0.25

0.20

0.15
NS (Method 1) NS (Method 2) NS-drivers

Fig. 1. Box-and-whiskers diagram for N S for the series of GP of the championships

from 2012 to 2022 for drivers and constructors (by using the two methods explained
on the text).

5.2 Comparison of competitiveness between F1 championships and

two major European football leagues
A competitive balance measure like N S, based on sport ranking series, can be
used to compare the CB of two different sports. For example, by computing
the coefficient N S for two major European football leagues (Spanish League -
commercially known as Laliga Santander in the season 2022/23-, and the English
Premier league) we obtain the results shown in Table 6. We have used the series
of standings from the season 2012-2013 to the season 2022-2023 for both the
Spanish League (retrieving the data from the links on [28]) and Premier League
(retrieving the data from [29]). The summary for the football leagues in the
studied period is the following:
Mean value of N S for Spanish league: 0.059, (s = 0.0094).
Mean value of N S for Premier league: 0.056, (s = 0.0062).
As a consequence, by using the results on section 5.1 for N S of drivers and
N S of constructors by using Method 1, we obtain that the mean value of N S
8 Francisco Pedroche

for the F1 championships is about four times greater than the values of N S
corresponding to the analyzed football leagues.

Table 6. N S values for two European football leagues from season 2012/2013 to season
2022/2023.

Year NS NS
Spanish league Premier league
2012 0.0615 0.0514
2013 0.0593 0.0656
2014 0.0546 0.0597
2015 0.0613 0.0563
2016 0.0435 0.0589
2017 0.0589 0.0550
2018 0.0688 0.0489
2019 0.0600 0.0643
2020 0.0757 0.0583
2021 0.0440 0.0461
2022 0.0595 0.0515

6 Conclusions

In this communication we have shown how to apply a recently introduced metric

to calculate a measure of the competitive balance (CB) associated to Formula
1 championships, by taking into account the standings of the Grand Prix that
compose each championship. We have introduced to methods (called Method 1
and Method 2) to compute the CB values of the F1 Constructors Championship
in the period 2012-2022. We have obtained that these two methods do not offer
mean values that can be considered statistically different. We think that this
shows a good behaviour of our metric since both Method 1 and Method 2 are
obtained by computing a linear combination from the same set of data (the F1
Drivers Championship) but with different treatment of the constructors that
finish with zero points in a Grand Prix. We also have obtained that the CB of
the F1 Drivers Championship and F1 Constructors Championship show similar
values on the studied period, but with a slightly higher mean value for the
Constructors Championship. As an example of the power of our metric, we have
compared the CB of two different sports: the Formula 1 championships from
2012 to 2022 and the Spanish football league and Premier football league on
the seasons 2012-2013 to 2022-2023. Our results show that the mean value of
CB for the F1 championships is about four times greater than the values of CB
corresponding to the analyzed football leagues.
 Competitiveness of Formula 1 championship from 2012 to 2022 9

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