2. 1
Despite of the crisis, chocolate business remains very dynamic and its sales continue to
grow. In November, the world chocolate show in Paris has celebrated its 20th
birthday
receiving almost 120,000 chocolate lovers in only 5 days. But if French people do love
chocolate, they are only eating an average of 6kg per person and per year which put them at
the end of the top 10 list of the biggest consuming countries. The winner is the home of the
giant confectionnaries Nestle and Lindt, Switzerland where every inhabitant ate an average
of 12 kg a year which means that each persons consumed almost 240 chocolate bars over
the course of the year.
As Christmas season was coming, the perfect moment with Easter celebrations to eat
chocolate, we were curious to discover a bit more about this small luxury that everyone falls
in love with. And after some research we found a data base made by the Candy Company
that showed the ranking of the 2014 top 100 Candy Companies. Then, as we did not want to
work on confectionnary companies, we decided to only keep the companies that produced
chocolate bars and others chocolated products.
We choose to rely on this magazine because it covered the global confectionery industry
since 1944 with in-depth analysis and research about the growth of the sector and the new
products. It offers us a good global vision over the candy industry as it reaches the biggest
candy and chocolate makers throughout the world.
Introduction
3. 2
Introduction............................................................................................................................... 1
Data base ................................................................................................................................3-4
PART I: UNIVARIATE STUDY............................................................................................... 5
A. Descriptive analysis ............................................................................................................... 5
Global analysis of the table ............................................................................................ 5
World distribution ......................................................................................................... 6
Distribution by continents.............................................................................................. 6
Distribution of the turnover........................................................................................... 9
Box & whisker plot ....................................................................................................... 10
Concentration curve..................................................................................................... 12
B. Indexes.................................................................................................................................. 13
Price ............................................................................................................................. 13
PART II: BIVARIATE STUDY ...............................................................................................16
A. Correlation between the companies sales and its number of employees ........................... 16
Scatter plot ................................................................................................................................. 16
B. Correlation between chocolate products and its consumers ........................................................... 17
Chi Square test........................................................................................................................... 17
Spearman coefficient ................................................................................................................ 18
Scatter plot................................................................................................................................... 19
C. Chocolate addiction .......................................................................................................................... 21
Star plot about clients profile .................................................................................................. 21
Star plot about the sugar ratio in chocolate.......................................................................... 23
Conclusion................................................................................................................................ 25
Sources..................................................................................................................................... 26
Summary
6. 5
A. Descriptive analysis
Global analysis of the table
Thanks to our data base, we created a global table to have an overview of the main
information we had about chocolate industry:
Net sales Number of employees Number of plants
Minimum 257 430 1
Maximum 17 640 330 000 700
Mean 2 275 20 055 35
Median 697 3 828 6
Variance 15 666 650, 9 3 718 928 061 13 723, 58
Deviation 3 958, 11 60 983, 01 117, 15
Coefficient of variation 1,74 3,04 3,34
We can notice that the average turnover of the companies is of 2,275 million dollars while
39 companies don’t reach this amount, which means that most of the sales are in the hands
of only a few companies (10 to be precise).
To look at a more reliable result, we can focus on the median value which is of 697 million
dollars, 1,578 million dollars less than the average turnover. One half of the companies has a
turnover of more than 697 million dollars and the other half has a lower turnover.
It is really interesting as well to notice that there is a real difference between the lowest
turnover and the highest. Indeed, Mars turnover is sixty eight times higher than the one of
Alfred Ritter. These figures show that the number of sales for each company fluctuates a lot.
According to the table, the number of employee for each company can change a lot. From a
general viewpoint, when a company achieves a significant turnover it is mainly thanks to its
large number of employees. For instance, Mondelez International, Nestle and Ferrero which
are among the strongest companies of the ranking have a high number of employees. But
our data base shows some exceptions. If we look at AVK Confectionary, that employs 10 000
persons, we can notice that its turnover stays lower (around 275 million dollars in 2014)
despite of this impressive number of employees. This amount is far away from the median
value so we can observe that each company has its own strategy to develop itself. We can
suppose as well that the number of employee depends on the machines and the technology
that the companies need to create their chocolate products.
Univariate study
7. 6
Regarding the number of plants, it is usually linked with the turnover because opening new
plants is a huge risk that requires a lot of money. Never the less, we can observe that one
company is atypical compared to the others: PepsiCo and its 700 plants. This astonish result
far from the median value of 6 plants is due to the fact that some of the plants of the
company are not only dedicated to the chocolate but to other products such as beverage.
World distribution
Thanks to this pie chart, we can have a global vision of the chocolate world. This market is
dominated by Europe which almost represents half of the world sales.
North America and Asia are neck and neck at the second place. We can already feel the
power of North America in this chart because it is only represented by two countries: Mexico
and of course the USA. It is quite interesting as well to see the power of Asia over the
chocolate market as it is as well a continent represented by a few countries (6 to be precise).
South America has a good potential on the market so maybe with a few more year it will be
strong enough to be in competition with its northern neighbour. We can notice the absence
of two continents: Oceania and Africa.
It would be really interesting now to see which the leading companies are according to their
location and to understand what makes Europe so competitive.
Distribution by continents
Europe
20%
10%
48%
22%
Location of the 50 main chocolate companies
North America
South America
Europe
Asia
8. 7
Over Europe, there are 13 countries among the top 50 chocolate companies. Most of them
are part of the European Union.
First in class among the European continent, we have Germany with its 5 brands. Storck is
the most powerful of them and appears almost in the top 10 companies ranked 11th
on the
data base. It is interesting to notice that even if Europe seems to dominate the chocolate
market, European countries aren’t among the strongest in terms of turnover.
The 3 leading companies in this bar chart are Germany followed by Switzerland and Ukraine
with both 3 companies. But we can notice that Turkey, Italy and the UK are not so far behind
them with their 2 companies.
Most of the other countries have only one company so we can conclude that if Europe is so
competitive it is mainly because of the large number of countries in the territory.
Asia
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
5
Numberofcompanies Bar chart of the biggest chocolate companies in
Europe
Germany
Switzerland
Ukraine
Turkey
Italy
UK
Finland
Spain
Belgium
Norway
Denmark
France
Sweden
9. 8
When we think about chocolate, Asia is not the first continent to think about and yet it is the
second largest market over the world. What is even more surprising here is the absence of
China and India, the two most densely populated countries in the world.
In this bar chart, the two leading countries are Japan and not so far behind Korea with 4 and
3 companies respectively. It is interesting to notice the power of Japan that is the world
fourth largest chocolate company in 2014 (thanks to an impressive rising of 3 ranks
compared to 2013). Russia, Singapore, the Philippines and Israel are in a close race with only
one company in each country.
America
First of all, we can notice the large domination of the USA with 9 companies among which
Mars and Mondelez International the two leading brands in our ranking. Even all added
0
2
4
Bar chart of the biggest chocolate
companies in Asia
Japan Korea Russia Singapore Phillippines Israel
0
1
2
3
4
5
6
7
8
9
Numberofcompanies
Bar chart of the biggest chocolate
companies in America
USA
Mexico
Argentina
Brazil
Colombia
Chile
10. 9
together southern companies can’t reach the American one, which clearly shows its
superiority. For instance, Mars turnover is eight times stronger than the one of the biggest
European company Storck (Germany). It is another proof of American great power over the
world chocolate market.
Colombia is at the second place with two companies. The other countries, Argentina, Brazil,
Chile and Mexico have only one company each. It is interesting to remember the spectacular
fall of Grupo Bimbo (Mexico) on the 26th
place this year when it was among the leading
companies in 2013. By contrast, we can observe an astonish rise of Colombina (Colombia)
from 39th
to 25th
place. It reveals the nature of the market that can change from one day to
another.
To conclude about the distribution by continents, we can regret the absence of two
continents: Oceania and Africa. Finally according to our data, despite of the apparent
European dominance worldwide, the strongest companies lie in the USA. These results
reveal the instability of the market that can change from one year to another with the
creation of new plants and new companies or with the decline of some group.
Distribution of the turnover
This histogram represents the net sales of the top 25 companies between 2013 and 2014.
Thanks to this graph, we can see if there are important differences in the net sales of each
company between these two years.
We can see that for the majority of these companies, the net sales have been better in 2014
than during the previous year. Indeed, around 23 companies out of 30 had net sales bigger
in 2014.
Three of these companies had an important evolution, both increase than decrease:
11. 10
Meiji and Ferrero are two companies with better net sales in 2014 than in 2013:
Meiji :
- 2014 : $11742 millions
-
- 2013 : $3415 millions
Ferrero :
- 2014 : $10900 millions
- 2013 : $5627 millions
Its represents an evolution rate increase of 243,84% for the company Meiji, and an evolution
increase of 93,71% for the company Ferrero.
However, we can clearly notice an important decrease of the net sales for the company
« Grupo Bimbo », which is passed of 14095 to 694 million dollars. It represents an evolution
decrease of 95,08%. This drop of the net sales is due to a change of strategy that did not
work.
Box & whisker plot
0
100
200
300
400
500
600
700
800
Number of plants
Q1
Maximum
Minimum
Q3
0,00
50 000,00
100 000,00
150 000,00
200 000,00
250 000,00
300 000,00
350 000,00
Number of
employees
Q1
Maximum
Minimum
Q3
12. 11
Net sales
Number of
employees
Number of
plants
Q1 457 1 417.5 3
Maximum 17 640 330 000 700
Q2 720 3 350 6
Minimum 257 430 1
Q3 1 777 8 150 9
Q3-Q1 1 320 6 732.5 6
Median 697 3 828 6
What directly draws our attention is the inequality showed by the box and whisker plot in
each category. We can notice a huge disparity between the minimum and the maximum.
This maximum can be considered as an outlier: it only represents the really of one company.
If we look at the number of plants it is really striking: the average number of plants per
company is 6 when we can observe a maximum of 700 because of one company. Indeed,
these boxes can show us how different the company’s strategies can be.
For instance, PepsiCo is well known for its beverages rather than its chocolate. It explains
why it has 700 plants: there are not all dedicated to chocolate, some of them probably works
only on beverage. This is why, despite of its worldwide reputation, PepsiCo appears only at
the 31th place on our data base: because chocolate is not its main activity. On the contrary,
some companies only work with chocolate products so it takes them more time to have a
large number of plants.
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
Net Sales
Q1
Maximum
Minimum
Q3
13. 12
Concentration curve
The concentration curve is a tool that helps us measuring inequalities according to different
variables. Here, our variables are the net sales and the number of plants.
Inequalities can be seen in this curve, because more the Lorenz curve is far from the equality
bisector, more there are inequalities. In our chart, the equality bisector represents the rank
of each company. Thanks to this curve, we can know if there is equality or not between the
companies’ rank and their number of plants or their net sales. In other words, to know if
there is a correspondence between the companies’ rank and its number of plants or its net
sales (when the company has an important rank it is mainly because it has an important net
sales or number of plants).
The result of this curve shows us inequalities between those variables. But we can notice a
more important inequality between companies’ rank and their number of plants, than
companies’ rank and their net sales. The Gini index, which was calculated, can confirm this
affirmation:
Net sales : 0,34
Plants : 0 ,48
The Gini index interpretation is the following: when the coefficient is close to 0 there are
fewer inequalities. On the contrary, when the coefficient is close to 1, there are more
inequalities.
To conclude, we can say that even if there are inequalities, coefficients are not so high.
Indeed, those two coefficients are below 0,5.
This index confirm us the fact that the companies’ rank has less correspondence with the
number of plants than with the net sales.
14. 13
B. Indexes
Price
In this part, we are going to take a look at the variation in the average selling chocolate
price.
Thanks to the INSEE’s table, we can have an idea about the chocolate price evolution over
the past twelve years. We mainly worked on the prices in January 2002 and 2014 to establish
our comparison.
January 2002 (T0) 1, 05€ (P0)
January 2014 (T1) 1, 24€ (P1)
Calculation of the rate of change:
= (P1 – P0) / P0 * 100
= (1, 24 – 1, 05) / 1, 05 * 100
= 18 %
Thanks to this formula, we can deduce that the average chocolate price between 2002 and
2014 increased by 18%. Even if it seems to be a huge rise, it was a softly move as we can see
in the scatter plot below:
Thanks to this index, we can observe that from 2002 to 2014, chocolate price continuously
increased which proves its popularity to consumers. The average price varies between 1,05€
100
105
110
115
120
125
130
2000 2002 2004 2006 2008 2010 2012 2014 2016
Prices index about chocolate consumption
between 2002 and 2014 (index base 100)
15. 14
and 1,25€. This variation that seems reasonable can explain the success of chocolate which
remains affordable despite the rise of its price.
With 2010 data, we observed the price evolution from one month to another:
According to this index, we can notice that despite of a steady increase over the years, in
2010, there was a decrease. Indeed the main growths we can observe are in April and during
the months of September and October. These increases correspond with celebration period:
especially Easter, Halloween and of course Christmas. Indeed these times of the year are
special moments where consumers buy much more chocolate than usual. This seasonality of
the product can explain these price fluctuations.
Thus, we can conclude that chocolate has known a true success story over the year despite
its price variations.
To complete this study, we have looked at the favourite chocolate type of the consumers. To
realize our comparison table, we use data from Lindt the 8th
most popular chocolate brand in
the world.
Q0 = average chocolate consumption per person in 2010
122
122,2
122,4
122,6
122,8
123
123,2
123,4
123,6
123,8
0 2 4 6 8 10 12 14
Prices index about chocolate consumption
between January and December 2010
(index base 100)
Q0 (kg) P0 (€) for 200g Q1 (kg) P1 (€) for 200g
White chocolate 1, 32 5, 21 1, 61 5, 29
Dark chocolate 1, 85 5, 42 2, 78 5, 45
Milk chocolate 1, 97 5, 36 2, 70 5, 40
16. 15
P0 = average chocolate price in 2010
Q1 = average chocolate consumption per person in 2014
P1 = average chocolate price in 2014
LASPEYRES PRICE INDEX
= (1,32 * 5, 29 + 1, 85 * 5, 45 + 1, 97 * 5, 40) / (1,32 * 5, 21 + 1, 85 * 5, 42 + 1, 97 * 5,
36)
= 1, 008
Laspeyres price index is 100 at T0 (2010) and 8 at T1 (2014)
PAASCHES PRICE INDEX
= (1,61 * 5, 29 + 2,78 * 5, 45 + 2,70 * 5, 40) / (1,61 * 5, 21 + 2,78 * 5, 42 + 2, 70 * 5,
36)
= 1, 008
Paasches price index is 100 at T1 (2014) and 8 at T0 (2010)
Thanks to these formulas, we can deduce the general evolution of prices between 2010 and
2014: this evolution represents 100, 8 according to Laspeyres and Paasches index. This index
represents the sum needed to buy chocolate between 2010 and 2014 according to the
chocolate average price.
To make a comparison between the evolution of the price and the quantity of chocolate
eaten between 2010 and 2014 in France, we created this bar chart:
17. 16
On this bar chart, we can notice that the most popular chocolate is dark following by milk
and white chocolate. As white chocolate is the less eaten by French people, its price is the
lowest to attract the consumers.
Thanks to this chart, we can see that the gap between milk and dark chocolate is small. We
can suppose that it is due to the fact that people like the two of them. Of course, we cannot
generalise these results as they are based on only one brand but as it is among the strongest
in this industry our observations can give us a good idea of the French people’s favourite
chocolate.
A. Correlation between the companies sales and its number of employees
Scatter plot
In this chart, we can see the correlation between the net sales and the number of employees
of the companies. Moreover, thanks to those two different colors, we can dissociate the
correlation between 2013 and 2014.
Principally, we can see that in these industries when the net sales are low, the number of
employees is low too. However, we can notice some exceptions:
Some companies have a large number of employees but low net sales, which
doesn’t seem positive.
A company with a large number of employees and high net sales.
Bivariate study
18. 17
Some companies with fewer employees but important net sales, which seems
more positive.
For 2014 and 2013, the tendency is a little number of employees with lower net sales,
despite of some exceptions.
Correlation between chocolate products and its consumers
Chi Square test
During our analysis, we wondered if the chocolate aspect has an influence over the chocolate
consumption. Indeed, chocolate can be sell in various ways: bar, powder, spread, or confectioner so
we decided to see if it has an impact on the sell.
To answer this question, we investigate among 100 persons to see which chocolate forms they buy
the most. Theoretically, we were supposed to get the results below:
19. 18
But we get other answers:
As a hypothesis, we will use H0 which supposes that there is no influence on the sale of the chocolate
form.
²exp = (31 – 25)² / 25 + (22 – 25)² / 25 + (5 – 25)² / 25 + (42 – 25)² / 25
²exp = 1,44 + 0,36 + 16 + 11,56
²exp = 29,36
With this result and a degree of freedom around 3, we can use the ² law. According to this table,
with a risk of 1%, ² needs to be inferior to H0 to validate the hypothesis. Here it is not the case so
we can definitely say that the chocolate forms matter in the consumers’ spirit. So we can conclude
that H0 is false.
Spearman coefficient
To understand what people think about chocolate around the world, we decided to ask 2
people and to create a table to classify chocolate by country, from the best to the worst (1
corresponds to the best and 5 is the worst).
We ask our 2 judges to classify chocolate into the 5 most represented countries in our
ranking.
USA UK GERMANY SWITZERLAND KOREA
JUGE 1 1 3 4 2 5
JUGE 2 1 2 5 4 3
Di 0 1 -1 -2 2
Di² 0 1 1 4 4
Thanks to this table, we were able to calculate the Spearman coefficient:
Chocolate
form
Bar Spread Powder Confectionery TOTAL
Theoretical
basis
25 25 25 25 100
Chocolate
form
Bar Spread Powder Confectionery TOTAL
Real basis 31 22 5 42 100
20. 19
Rs = 1 – (6 * 10)/ [5 * (5² - 1)]
Rs = 1 – 60/ 120
Rs = 1 – 0,5
Rs = 0, 5
Spearman coefficient is here equal to 0.5 which means that there is a link between the 2
judges ranking but this link is weak. The opinions of the 2 people are different but they are
able to order their preference. The USA is at the 1st place for both judges: indeed Mars
company is well known for its chocolate products.
We can conclude that even if it is possible to make a ranking it will always depend on the
taste and the subjectivity of the people that answer.
Scatter plot
In this part, we are about to see if the age and the chocolate consumption are linked
together. Do we eat more chocolate when we are young or are the old people the most
voracious? We ask 4 people about their average consumption per year
X represents the age and Y the average chocolate consumption in kg.
X Y X - X Y - Y (X - X) *(Y - Y)
8 4,5 -52 -0,39 20,33
12 4,2 -48 -0,69 33,16
16 3,4 -44 -1,49 65,60
23 3,4 -37 -1,49 55,16
29 2,6 -31 -2,29 71,02
34 2,2 -26 -2,69 69,96
39 3,1 -21 -1,79 37,61
46 1,6 -14 -3,29 46,07
55 1,4 -5 -3,49 17,45
68 0,5 8 -4,39 -35,13
Total 330 26,9 -270 -22,01 381,5
Average 60 4,89 -49,09 -4,00 69,32
Thanks to the above data, we realized a scatter plot that linked the age of the consumer to
its consumption per year:
21. 20
Thanks to these data, we were able to calculate the covariance between X and Y :
COVARIANCE -23,67
As the result is negative, we can conclude that the more we grow the less we eat chocolate.
Of course, this is a global vision: some children don’t like chocolate and some elderly people
are mad about chocolate, exceptions always exist.
This graph is not revelant enough at the moment as it concerns only 10 persons but we tried
to question people from age as different as possible to make our results as close as possible
to the reality. We calculated as well the correlation coefficient.
COEFFICIENT DE CORRELATION -0,96
As this coefficient is close to -1, we can say that X and Y are linked in a negative way. This is
why the dots are close to the line and show a strong relation between age and chocolate
consumption.
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
5
0 10 20 30 40 50 60 70 80
Link between the age and the chocolate consumption
22. 21
To conclude over this part, we can say that the chocolate is much more popular to the
children than to the old people that eat it with moderation. This conclusion makes us think
about sugar addiction and its source. What in chocolate makes us so addict?
B. Chocolate addiction
Star plot about the sugar ratio in chocolate
To study the components of chocolate and to understand better why it makes us so
addicted, we decided to take a look at Mars, the most popular chocolate in the world.
In this part we are going to answer 3 questions:
Why do we love Mars?
From what it is made?
What is the main ingredient of this bar?
According to Mars website, here are the components of the delicious bar:
Components Quantity (in gram)
Protein 1,9
Carbohydrate 35,4
Lipid 9
Sodium 0,08
Roughage 0,6
Thanks to this star plot we can notice a high domination of carbohydrate in the chocolate
which corresponds to sugar. We can suppose that this passion for Mars chocolate bar is due
Protein
Carbohydrate
LipidSodium
Roughage
STAR PLOT OF THE DIFFERENT COMPONENTS OF THE MARS CHOCOLATE BAR
23. 22
to the sugar it contents. Indeed sugar can create addiction as we can see with the world
“chocoholic” that appears more and more in the Medias. With almost 40g of carbohydrate,
we can easily say that sugar is the main component of the bar: 30g separated the sugar from
any of the other components.
This large amount of sugar explains some troubles linked to the chocolate consumption as
diabetes, obesity or liver trouble.
If we look at the other components they are almost non-existent: around 0.8g of each on
average in one bar. Thus, sodium, protein or roughage even added don’t reach the quantity
of sugar.
To create a comparison, we took another example with Loacker chocolate (brand among the
last places in the ranking). As for Mars, we found our information on the brand website.
Components Quantity (in gram)
Protein 8
Carbohydrate 19
Lipid 2
In this star plot, we didn’t count the sodium as its presence was close to zero. Here again, we
can notice a huge presence of carbohydrate.
Those two analyses show us how important it was to use sugar in chocolate conception and
we can suppose that it is this magic component that creates our dependency to chocolate
products.
Star plot about clients’ profiles
Protein
CarbohydrateLipid
STAR PLOT OF THE DIFFERENT COMPONENTS OF THE LOACKER CHOCOLATE
BAR
24. 23
In this part, we tried to show the different chocolate consumer profiles according to their
age. To do so, we used the table below from INSEE :
Consumers age
Average quantity of chocolate eaten (g/day)
Chocolate
Chocolate
Bars
Chocolate
Confectionaries
Chocolate
Spreads TOTAL
3-6 years old 8.2 1.6 2.5 4 16.3
7-11 years old 10.6 2 2.9 5.6 21.1
12-14 years old 11.7 2.3 2.5 6.9 23.4
15-24 years old 9.4 1.5 3 4.9 18.8
25-49 years old 3.7 1.5 1 1.3 7.5
50 years old
and beyond
2.2 1.7 0.2 0.3 4.4
First of all, we can classify these profiles in two distinct categories: 25 years old and more
and 24 years old and under. According to the star plot, we can see that the older you are,
the less you eat chocolate. This decrease of chocolate consumption seems to start at 25
years old. The biggest consumers are the teenagers especially between 12 and 14 years old.
This is probably due to the fact that at this age you eat a big more sugar because you spend
more energy. We can suppose that at this age, children don’t care about their weight
because they can lose their extra pounds more easily than adults. Indeed the old people are
0
2
4
6
8
10
12
Chocolate
Chocolate Bars
Chocolated
Confectioneries
Chocolate Spreads
Chocolate consumers profile according to their
age and daily consumption
3-6 years old
7-11 years old
12-14 years old
15-24 years old
25-49 years old
50 years old and +
25. 24
the one that eat the less chocolate maybe because they prefer salty meals or simply because
they cannot go to the supermarket anymore to buy their favourite confectionery.
The star plot reveal as well that the favourite product of everyone is chocolate itself even if
the teenagers seem to like as well the chocolate spread. These different profiles are a good
help to understand the marketing strategy of chocolate company because their ads don’t
need to target all categories. For example, Ferrero a famous Italian brand has developed
different products adapted to each person according to its age. There is Kinder for the
youngest, Nutella for teenagers and Ferrero Rocher or Mon Chéri for the adults.
26. 25
Thanks to our analysis, we were able to understand better the chocolate industry by using
our theoretical knowledge in a real case. We have discovered that even if we directly think
about America or Europe in term of chocolate, the Asian market represents as well a true
potential. The companies in our data base were not all similar: we had indeed some outliers’
results in our study that make us think beyond the data, about the environment, the supply
and the demand.
It was interesting to see that companies have different strategies to develop themselves:
some has a large amount of employee, other has numerous plants because chocolate isn’t
their only activity.
By creating correlations, we understood better what makes us so fond of chocolate and
what our profiles about this product were. We discovered that the age was determinant in
terms of chocolate consumption for example.
Finally, it was amazing to see all the possibilities that a data base offer and in how many way we can
use one formula or another to prove phenomenon from a mathematical viewpoint.
Conclusion