Global Thinkers 3: Mathematics 3. Secondary (demo) by Grupo Anaya, S.A.

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INCLUDED

DIGITAL PROJECT

3 GLO BAL

THINKERS

Conntoewnledtgse in the course Basic k

How much do they spend? in proportion or in total? ............ 86 CHALLENGES THAT LEAVE THEIR MARK

CHALLENGES THAT LEAVE THEIR MARK

Arithmetic .............................................................................................10

Algebra ................................................................................................. 88

1 Fractions and decimals

5 Algebraic language

....................................................................................12

1. Fractions 2. Operations with fractions 3. Decimal numbers 4. Fractions and decimals with the calculator 5. Expressions with combined operations Exercises and problems solved Exercises and problems Maths workshop Self-assessment

2 Powers and roots

....................................................................................................... 30

1. Exponentiation 2. Scientific notation 3. Roots and radicals 4. Rational and irrational numbers Exercises and problems solved Exercises and problems Maths workshop Self-assessment

3 Arithmetic problems

.............................................................................................. 46

1. Approximations and errors 2. Calculating with percentages 3. Compound interest 4. Common problems 5. Compound proportion in arithmetic problems Exercises and problems solved Exercises and problems Maths workshop Self-assessment

4 Progressions

........................................................................................................................ 68

1. Sequences 2. Arithmetic progressions 3. Geometric progressions 4. Surprising geometric progressions Exercises and problems solved Exercises and problems Maths workshop Self-assessment

................................................................................................... 90

1. Algebraic expressions 2. Monomials 3. Polynomials 4. Identities 5. Dividing polynomials 6. Factorising polynomials 7. Algebraic fractions Exercises and problems solved Exercises and problems Maths workshop Self-assessment

6 Equations

..................................................................................................................................... 112

1. Equations. Solving an equation 2. First-degree equations 3. Second-degree equations 4. Polynomial equations with a degree greater than two 5. Solving problems with equationss Exercises and problems solved Exercises and problems Maths workshop Self-assessment

7 Systems of equations

.......................................................................................... 134

1. Lineal equations with two unknowns 2. Systems of linear equations 3. Equivalent systems 4. Types of systems by number of solutions 5. Methods for solving systems 6. Systems of non-linear equations 7. Solving problems through systems Exercises and problems solved Exercises and problems Maths workshop Self-assessment

Algebra for the water bill. ............................................................................................... 154 CHALLENGES THAT LEAVE THEIR MARK

Functions.............................................................................................156

18 Characteristics of functions

.............................................................158

1. Functions and graphs 2. Function aspects 3. Analytical expression of a function Exercises and problems solved Exercises and problems Maths workshop Self-assessment

19 Linear and quadratic functions

................................................... 176

1. Proportionality function y = mx 2. Linear function y = mx + n 3. Applying linear functions. Movement problems 4. Studying two linear functions together 5. Parabolas and quadratic functions Exercises and problems solved Exercises and problems Maths workshop Self-assessment

A graph for every container ........................................................................................ 194 CHALLENGES THAT LEAVE THEIR MARK

Geometry............................................................................................ 196

10 Metric problems in plane geometry

............................... 198

1. Angle relationships 2. Similar triangles 3. Similar shapes. Scales 4. Pythagorean theorem 5. Algebraic applications of the Pythagorean theorem 6. Areas of polygons 7. Areas of curved shapes 8. Loci 9. Conic sections as loci Exercises and problems solved Exercises and problems Maths workshop Self-assessment

11 Geometric shapes

................................................................................................. 222

1. Reguar and semiregular polyhedra 2. Truncating regular polyhedra 3. Planes of symmetry of a shape 4. Axes of rotation of a shape 5. Surface area of geometric shapes 6. Volume of geometric shapes 7. Geographic coordinates Exercises and problems solved Exercises and problems Maths workshop Self-assessment

12 Geometric transformations

..............................................................246

1. Geometric transformations 2. Motions on the plane1 3. Translations 4. Rotations. Shapes with a centre of rotation 5. Axial symmetries. Shapes with axes of symmetry 6. Composition of motions 7. Mosaics, friezes and rosettes Exercises and problems solved Exercises and problems Maths workshop Self-assessment

Builders for a day. ......................................................................................................................268 CHALLENGES THAT LEAVE THEIR MARK

Statistics and probability. ............................................................270

13 Statistical tables and graphs

......................................................... 272

1. The statistical process 2. Statistical variables 3. Population and sample 4. Making a frequancy table 5. Using the right type of graph Exercises and problems solved Exercises and problems Maths workshop Self-assessmen

14 Statistical parameters

................................................................................ 288

1. Two types of statistical parameters 2. Calculating x– and σ in frequency tables 3. Joint interpretation of x– and σ 4. Position parameters: median and quartiles 5. Using a calculator to get x– and σ 6. Statistics in the media Exercises and problems solved Exercises and problems Maths workshop Self-assessmen

15 Chance and probability

............................................................................. 306

1. Random events 2. Probability of an event 3. Probability in regular experiments. Laplace’s law 4. Probability in irregular experiments 5. Law of large numbers 6. Probabilities in compound experiments Exercises and problems solved Exercises and problems Maths workshop Self-assessmen

Let’s try to measure happiness.............................................................................. 322

Annex

Glossary. ...................................................................................................................................................324

L A C I T S I T A T S D N A TABLES GRAPHS

Language Bank 1

Censuses and the gathering of statistical information have been present in all civilizations since ancient times. However, they were limited to the collection of data and its clear and orderly presentation. Generally, the information that was collected was relative to the state, that’s where the word ‘statistics’ comes from. There are Egyptian papyrus scrolls from more than 5000 years ago showing censuses of the population and their possessions. The Egyptians were so dedicated to these concerns that they conceived a deity called Safnkit, goddess of books and accounts, who was in charge of protecting the information collected. In Babylonia, they also kept statistical counts on the clay tablets they made. They collected information especially on livestock and agriculture. They gathered 272

Have you ever seen tables and statistical graphs? Where? What information did they represent? Discuss.

so much information that, in the 8th century BCE, they built a library to keep all the documents in. Statistics became especially important with the birth of the Roman Empire, which stood out for its extraordinary organisation of State. They gathered and organised a huge amount of information from all of their territory: births, deaths, marriages, number of inhabitants per square kilometre, wealth… Since then, statistics have progressed at a staggering rate. Currently, with the large amount of data collected, especially online, and with the help of computers, the greatest governing powers possess endless information about us: hobbies, social relationships, possessions… This fact improves their ability to govern effectively, although also gives them power of manipulation.

Use what you have learnt to solve the problem 34 students in the third year of Secondary complete the following questionnaire that they themselves designed: Your hair colour (black, brown, light brown, blond, red). Number of people living in your house. Your height (in cm).

The results are collected and counted: HAIR COLOUR

NUMBER

Black

Brown

Light brown

Blond

Red

TOTAL

PEOPLE IN HOUSE

2 3 4 5 6 7 8 TOTAL

NUMBER

HEIGHT

NUMBER

3 12 10 6 2 0 1 34

150 - 155 155 - 160 160 - 165 165 - 170 170 - 175 175 - 180 180 - 185 TOTAL

3 5 6 8 5 4 3 34

And the results are represented graphically like this: 8 Black Brown Dark Blond Red

6 4

2 2

150 155 160 165 170 175 180 185

❚ Think 1 To design the questionnaire and display the results, the students (with the help of their teacher) have had to make some decisions. Think and answer. a) In relation to hair colour, why have concrete answers been required? Could they have put other options? Could they have left the options open? b) Why has the information collected on height been presented in intervals? c) Are the graphs that they have used to represent the results appropriate?

2 Alone, or in groups, do something similar in class or somewhere else if your teacher considers it appropriate. Design a questionnaire, collect the results, organise them in tables and represent them graphically. You could use the same questions or other ones that you think are more interesting. 273

THE STATISTICAL PROCESS We get statistical information from graphs or tables that have been built to make it very easy to understand the information they contain. These tables and graphs are the result of a long process. Let’s look at the steps. 1. What do we want to study? Why? For example: Suppose that a Secondary school wants to know the students’ favourite destinations for the end-of-year class trip. The head teacher of the school and his team take that information and look for the best prices, to help them decide how many options to offer. 2. Selecting the variables to analyse If each student is asked what their favourite destination is, we are likely to get too many different answers that are difficult to organise. Moreover, some of the students will say their favourite destination without thinking about travel costs, distance, length of the trip… To avoid this great amount of different answers, the survey must be very clear and include the possible options. In other words, it must be clear what the variable is and what the possible values are. For example: Which of these destinations do you prefer for your end-of-year trip? a) Paris

b) Rome

c) Marrakesh

d) Berlin

e) London

3. Collecting data The next step is to take measurements or do surveys. In the example of the end-of-year trip, we ask each student the question and write down their answer. 4. Organising and presenting data Next, we tally the answers, organise the data into tables, make the appropriate graphs and, in some cases, calculate the necessary parameters. We will spend this unit and the next one learning how to do this.

Think and practise 1 We want to do a survey to study people’s taste in music.

Explain whether or not each of these questions is reasonable:

c) Do you listen to the radio? If so, what station? d) Which of these stations do you listen to more than 2 hours a week?

a) What are your favourite musical groups?

• Cadena 100

• Los 40 principales

b) Which of these styles of music have you listened to most this month?

• Rock FM

• Kiss FM

• Radio Clásica

• Europa FM

• EDM

• M80 Radio

• Radio 3

• Cadena Dial

274

• Rock

• Pop

• Rap

• Electronic

• Hip-Hop

• Reggae

• Salsa

• Punk

• Metal

• Grunge

• Jazz

• Classical

e) What is the last concert you went to?

STATISTICAL VARIABLES Remember that statistical variables are characteristics of the population being studied. Let’s look at some examples of statistical variables:

1.48

1.77

number of fish caught

0.83

3.09

cola

backpack weight

lemon soda

orange soda

sparkling water

drink chosen

The first two variables, number of fish and weight of the backpacks, are quantitative because their values are numbers (quantities). The third variable, type of drink, is qualitative because the type of drink is a quality that cannot be described with a number. A quantitative variable is discrete when it can only take certain values (the number of fish caught can be 1 or 2, but not a number between them). A quantitative variable is continuous when it can take any value between two values (a backpack can weigh 1.245 kg, although we usually round to one decimal place). Types of statistical variables • Quantitative: Numerical.

Discrete: Can only take certain values. Continuous: Can take any value in an interval. • Qualitative: Non-numerical. Example

A health inspector writes down certain characteristics of restaurants: — Number of people who work there (1, 2, 3…): discrete quantitative. — Surface area of the restaurant (187.5 m2): continuous quantitative. — Type of food (Chinese, pasta, burgers, seafood…): qualitative.

Think and practise 1 Are these variables discrete quantitative, continuous

quantitative or qualitative?

a) In a town’s cinemas, we note the types of film shown (comedy, action…), how long the films are, and the number of people in the audience. b) In a city’s shops, we note the surface area, number of entrances and type of store (food, clothing, accessories…).

c) We note some characteristics of the mobile phones owned by the students of a school: brand, number of companies that sell it, and price. d) A scientist studies the heights of all the volcanoes in the Pacific Ocean, the number of times they have erupted in the last 100 years, and the types of volcanoes they are (cinder cone, composite volcano, shield volcano, lava dome). 275

POPULATION AND SAMPLE

Look at the graphic representation of the three distributions we saw on the previous page: N.o of fish caught by 120 fishermen

Weight of the backpacks of 30 students

30 3030

12 1212

25 2525

10 1010

20 2020

88 8

15 1515

66 6

10 1010

44 4

55 5

22 2

00 0

00 0 11 1 22 2 33 3 44 4 55 5 66 6

Drink chosen by the 40 guests at a birthday party 5% 5% 5%

Cola

20% 20% 20% 45% 45% 45% 30% 30% 30%

00 0 00 0

11 1

22 2

33 3

44 4

Lemon soda Orange soda Sparkling water

55 5

Each of them represents a group: • 120 fishermen at a lake. • 30 students in a class. • 40 guests at a birthday party. The group that is analysed in a statistical study is called the population. Sometimes, the group that we want to study is too big to be able to analyse each of its elements. When this happens, we take a sample. For example, the 120 fishermen may be a sample of a population that includes all the people who are fishing at the lake. Population or sample

Population is the group of all elements being studied.

A group can be a population or sample, depending on what we are trying to study, and can be used to infer information about a larger group.

Individual is each of the elements in the population or sample.

Sample is a subset of the population used to infer characteristics about the whole population.

Example

On the previous page, we saw an example about a health inspector who is analysing restaurants. To make a report about all the restaurants in a city, he selects some restaurants at random. The group of all the restaurants in the city is the population. The restaurants selected at random by the health inspector for the study are the sample. Each restaurant is an individual. Think and practise 1 What are the population, sample and individuals in

each of these situations?

a) 50 buildings in a city have been selected to study the number of floors, the height, and what is on the ground floor (houses, offices, stores, bars, restaurants…). 276

b) We analyse 100 books in a library: number of pages, location on the shelf, and type of book (novel, essay, manual…). c) We survey 23 of the students who bike to school and ask about the number of gears on their bicycle, the weight of the bicycle, and the brand.

UNIT 13

The role of samples As we have seen, we sometimes have to use a sample to collect data. Let’s look at when and how to do it, and what we can expect from the results.

when must we use a sample? Very often, it is useful, or even necessary, to study a sample instead of the whole population. Here are some examples: • When the population is very large. For example, if we want to know the reading habits of all the young people in a region. • When the population is difficult to count. For example, if we want to study how many times a month each customer goes to a department store. • When studying the variables requires a very expensive or destructive process. For example, if we want to know how long a new type of battery lasts. We can only measure this by using the batteries until they die.

how to select a good sample Selecting a good sample is not an easy task. Therefore, it is important to remember these guidelines: • For a sample to be valid, it must be selected at random (by chance) and all the individuals in the population must have the same probability of being selected.

Focus on English

census: offical process of counting all the people who live somewhere.

For example, to estimate the election results in a town with around 5 000 residents, we can survey 200 people. Doing the survey on the telephone, on the Internet or at a market is not a reliable method because any people who do not have a telephone, do not use the Internet, or do not go to the market would be excluded. The only reliable way to select the sample is to select 200 people at random from all the people who are registered in the census*. • The size of the sample is important. However, if we select our samples well, we can draw very valid conclusions from small samples. For example, to predict the election results of a country with millions of voters, a sample of 3 000 individuals may be a good representative. However, a much larger sample (of 50 000) taken by a newspaper is not representative because people who do not read that newspaper are excluded.

what conclusions can we draw from studying a sample? This is a very complicated question to answer. We will learn more about it in later years. Right now, we will just say that any conclusions drawn from a sample of the population will always be approximations and will have a margin of error. For example: ‘In the upcoming elections, candidate X will win with approximately 56 % of the vote’. This statement has a margin of error of 5 %. 277

MAKING A FREQUENCY TABLE Once we have collected the data, we organise them into a frequency table.

Notation n frequency tables we usually write: xi → values of the variable fi → frequency of each value

Table with individual data If there are not many values for the variable, we can use a table like the one below. It shows the number of problems that each of the 20 students solved correctly on an exam. The variable, xi, takes the values 0, 1, 2, 3, 4 and 5. frequency table

tally

Tally To do a tally, we read the results one by one and mark a line for each one. It is useful to cross out the data points as you count them so you do not count them twice. The marks are easier to count if we write them in groups of five. (The fifth mark closes the group).

Tables with data grouped into intervals If the variable is continuous, or discrete with many different values, it is useful to group the values into intervals. In this example, we see the 30 best high-jump scores this year in a region. The data is grouped into intervals using decimals so there is no question about where each score falls. frequency table tally

anayaeducacion.es Build frequency tables.

interval

195 198 201 187 192

Between 180.5 and 184.5

180.5-184.5

181 197 198 203 195

Between 184.5 and 188.5

184.5-188.5

185 187 192 196 188

Between 188.5 and 192.5

188.5-192.5

199 193 189 185 204

Between 192.5 and 196.5

192.5-196.5

198 201 184 189 202

Between 196.5 and 200.5

196.5-200.5

187 194 200 198 193

Between 200.5 and 204.5

200.5-204.5

Think and practise 1

The maths teacher of a Secondary school noted the absences for each of her 3rd of E.S.O. students during the term: 2, 3, 0, 1, 1 2, 2, 4, 3, 1 3, 0, 2, 0, 1 2, 2, 1, 2, 1

0, 3, 4, 2, 1

3, 5, 1, 1, 2

a) Make a frequency table. b) If the teacher had noted the number of problems solved correctly by each student during the year, would you make the frequency table with individual data or with data grouped into intervals? 278

2 We note the 100 m sprint time of all the members

of a Secondary school’s athletics club. These are the results: 11.62

12.03

12.15

11.54

10.95

11.56

11.08

11.38

12.08

11.73

12.11

11.52

11.72

11.23

11.66

10.87

11.32

11.58

12.01

11.06

Make a frequency table with these intervals: 10.805 - 11.075 - 11.345 - 11.615 - 11.885 - 12.155

UNIT 13

Relative frequencies and percentages It is not the same to say that 5 students solve a problem than saying that one in every four students solves a problem. There is also a difference between saying that there are 6 high jumpers in the 192.5-196.5 interval and saying that 20 % of the athletes are in this interval. The proportion or percentage corresponding to a certain value can often give us more information than the absolute frequency.

2/20 = 0.10

5/20 = 0.25

3/20 = 0.15

6/20 = 0.30

3/20 = 0.15

1/20 = 0.05

total

1.00

100

The relative frequency, fr , of a value is the proportion of times it appears. We calculate it by dividing the frequency of the value by the total number of individuals. If we multiply the relative frequency by 100, we get the percentage or the percentage frequency. The table on the left comes from the first example on the previous page. Two columns have been added: one for relative frequencies and one for percentages.

Cumulative frequency When the values of a variable are ordered from smallest to largest, the cumulative frequency of each value is the sum of its frequency and the frequencies of all the previous values. For example: Remember The values have to be in order if we want cumulative frequencies to make sense. We can do this with quantitative variables and with some qualitative variables. For example, the months of the year: J, F, M, A…

2+5=7

2 + 5 + 3 = 10

2 + 5 + 3 + 6 = 16

2 + 5 + 3 + 6 + 3 = 19

2 + 5 + 3 + 6 + 3 + 1 = 20

cumulative frequency

fcumulative (4) = 19 → 19 students solved 4 problems or fewer. Think and practise 3 The following table shows the sports that 40 students

like to play the most:

a) Calculate the relative and percentage frequencies of this distribution and explain why it does not make sense to find cumulative frequencies.

sport

frequency

Basketball Volleyball Football Tennis Chess

10 1 20 5 4

b) If the relative frequency of Basketball is 10/40, that means that one in every four students plays basketball. Use the same words to explain the relative frequencies of Football and Tennis and the percentage frequencies of Chess and Basketball.

4 Find the cumulative frequencies of this distribution.

What do fcumulative (3) and fcumulative (5) mean? failed subjects frequency

0 6

1 12

2 8

3 5

4 3

5 1

6 1

7 0

5 This table shows the birthday months of the 100 people

in a mountain climbing group. month freq.

J 7

F M A My Jn Jl Ag S 9 10 6 8 8 7 9 8

O N D 9 9 10

a) Find the cumulative frequencies. b) How many people were born before June? And after August? 279

USING THE RIGHT TYPE OF GRAPH

We can find examples of wonderful statistical graphs used in many different fields to transmit information. These graphs allow us to understand the information transmited in a single glance. Let’s look at them more closely.

Bar chart

Another type of graph

In bar charts, the lengths of the bars are proportional to the frequency of the values. They are used for distributions of discrete quantitative variables. This is the reason why bars are thin and appear above the individual values of the variable. They are also used to represent qualitative variables.

There are an infinite number of types of statistical graphs. This one is called a pictogram, and it shows which countries emit the most CO2 into the atmosphere.

n.° of mobile phones owned by the 400 students of a school

brand of mobile phones owned by the 300 students of a school

chonix-pera berri-back guguelne-xus

ifon-5 sanson-universe

a Jap a Ger n ma ny Iran

Histograms are used for distributions of continuous variables. They consist of rectangles with bases that are the length of the intervals. If all the intervals are all the same length, the heights are proportional to the frequencies of the values, like in a bar graph.

Rus si

Ind i

A US

ina

Frequency histogram

Pictograms are made for the general public because they are very intuitive and interesting, even if they are not very precise.

maximum temperatures during one year

laps around a track run by year 3 students in 12 minutes

5 10 15 20 25 30 35

7 8 9 10 11 12 13

n.° of cinema customers during a month

0 50 100 150 200 250 300 350

It makes sense to use a histogram instead of a bar chart when we have a continuous variable (8 laps means fewer than 9), even if the data are not given in intervals (like the laps around an athletics track). The number of audience members is a discrete quantitative variable, but because there are many different values it’s best to use intervals, and therefore, histograms. Think and practise

Make the most suitable graph for each situation:

a) Maximum temperatures in a city during one year measured every 15 days.

n.o of failed subjects

n.o of students

n.o of days

5-10

10-15

15-20

20-25

25-30

1 2 3 4 5 or more

9 3 2 1 3

temperature

280

b) Number of subjects that the students in a class have failed this term.

(oc)

UNIT 13

Frequency polygon

maximum temperatures

5 10 15 20 25 30 35

Frequency polygons are used in the same situations as histograms. They are made by drawing a line connecting the midpoints of the tops of the rectangles and extending the line to the axis at the beginning and end. They smooth out the ‘steps’ of the histogram.

Pie chart Election results At the end of an election day we often see the results represented like this:

In pie charts, the angle of each segment is proportional to the corresponding frequency of the value. They can be used for all kinds of variables, but are most often used for qualitative variables. For example, the pie chart below shows the favourite types of films of a certain population in Spain by percentages: Action

B A C Previous elections

Romance

Other

10 % 10 %

B C

Other

Prediction

Comedy

33 %

40 %

Terror

Other

A progression of pie charts can be particularly useful to represent data that changes over time. For example, we can look at how the education level of the population of a certain region changed over time, from 1950 to 2010. We see that the number of illiterate people and people with basic education decreases over time, and that the number of people with middle and higher education increases considerably. Illiterate

anayaeducacion.es GeoGebra. Pie chart.

Basic education Middle education 1950

1980

2010

Higher education

Think and practise 2 Pie charts are often used to compare the same

distribution in different countries or regions.

3 Look at how global consumption of different energy

sources has changed over time:

These charts show the distribution of the working populations of two countries: Austria and Mauritania. Which chart represents each country? Explain why. Agriculture and livestock Industry Services A

Petrol Natural Gas Coal Hydroelectric Renewable 1973

2004

a) Explain which energy sources have been consumed more and which have been consumed less over time. b) Look online for this year’s chart. 281

EXERCISES AND ED V L O S S M E L B O R P 1 Comparing population pyramids

Population pyramids are useful for studying demographics and explaining past and present events.

age (in 5-year intervals)

Men Women

The pyramids on the right represent three very different countries: Spain, Argentina and Mali. a) Which pyramid represents each country? b) Analyse the characteristics of the pyramid for each country and explain your conclusions.

a) A 8 Mali; B 8 Spain; C 8 Argentina b) The Mali pyramid shows that there is a very high percentage of people under 30 years old. The life expectancy in that country is much lower than in the other two countries (life expectancy is the average age of death). The families in this country have many children. It is an underdeveloped country. Spain’s pyramid shows that the birth rate was very high in the 1970s but that families have been having fewer and fewer children since then. There is a large percentage of older people, therefore, life expectancy is high. We can say that it is a developed country. The Argentina pyramid is similar to Spain’s. Life expectancy is a little lower, but the major difference is that the percentage of people under 40 years old gets larger (very slowly) for younger ages. This does not happen in Spain. This means that families in Argentina have more children than families in Spain, but much fewer than families in Mali.

2 Making a pie chart

The percentage of people who bought different colours of cars is shown in the table below: colour

percentage

Silver/grey

36 %

Black

22 %

Blue

18 %

Red

10 %

White

Green

Other

Make a pie chart with these data. Your turn A ferry transports different types of vehicles. Make a pie chart with these data: Cars: 53 %; Motorbikes: 24 %; Lorries: 13 %; Buses: 10 %.

282

To make a pie chart from a table, we need the column with the relative frequencies.

colour

percentage

Silver/grey

36 %

0.36

Then, we calculate the angle (in degrees) of each sector by multiplying its corresponding relative frequency by 360°.

Black

22 %

0.22

Blue

18 %

0.18

Red

10 %

0.10

In this case:

White

0.08

Silver/grey 8 0.36 · 360° = 129.6°

Green

0.04

Black 8 0.22 · 360° = 79.2°

Other

0.02

Blue 8 0.18 · 360° = 64.8° Red 8 0.10 · 360° = 36° White 8 0.08 · 360° = 28.8° Green 8 0.04 · 360° = 14.4° Other 8 0.02 · 360° = 7.2° We use the percentages to draw the pie chart and add a legend indicating what each colour means to its right.

8% 36 %

10 % 18% 22% Silver Black Blue

Red White Green

Other

folio. for your port from this unit s ce ur so re choose Remember to

S M E L B O R P D N A EXERCISES Practise

Population and sample. Variables

UNIT 13

Interpreting graphs

Indicate, in each situation:

These population pyramids show the distribution of three countries by age (in 10-year intervals) and gender (women to the right and men to the left): A

— The population and the individuals.

— The variable and what kind of variable it is. a) The weight of all newborn babies in Valencia last year. b) Amount of rain collected at a weather observatory each year in this century.

Look at problem solved 1 on the previous page, and use what you learnt to match each graph with one of these countries, then, explain your answer: I. Underdeveloped country. II. Developing country. III. Developed country with a stable system.

c) Number of pets in Spanish homes. d) Types of cars (brand and model) owned by every person in my neighbourhood. e) Number of yellow cards shown in every Premier League match this season. 2

We want to study the following: I. The gender (girl or boy) of every baby born in a hospital during one year.

average temperatures (°C)

II. What newspaper every inhabitant of a city reads. III. The heights and weights of all the students in the class. IV. The ages of the people who have seen a theatre play in a city.

b) What is the variable and what kind of variable is it? Which of the following samples are ‘reasonably’ well selected? a) We take five avocados at a greengrocers to see how hard the avocados are. b) I talk to ten of my friends about politics to find out who will win the elections this year. c) We look at ten pages of a book to see if we like the illustrations. d) I have a coffee in four bars in my neighbourhood to see how much a coffee costs in Spain.

20 J

a) What do the bars represent? b) What does the red line represent? c) What are the variables? What kind are they? d) Describe and explain the relationship between the two variables.

a) What are the population and individuals in each case?

monthly precipitation (mm)

V. Studies that the students of a school will follow after they finish Secondary school.

c) In which cases should we use a sample? Why?

We can often use the same graph to represent two data sets about a single variable. This one shows data about the weather in Badajoz during one year.

Look at the graph on the right representing the sales of some items at a small shop. a) Which of the colours represents swimsuits? And towels? And gloves?

Jan Dec

Feb Mar

Nov

Apr

Oct

May

Sept Aug

June July

b) In which season are the most swimsuits sold? And the fewest? Why? c) When were the most gloves sold? d) Explain the towels graph. 283

S M E L B O R P D N A EXERCISES Making tables and graphs

A group of year three students were asked how many books they read in the last month. We got the following results: 2 1 3

1 0 2

3 2 2

1 4 1

1 1 2

5 0 3

1 2 1

2 1 2

4 2 0

Problem solving 11

3 1 2

a) Make a table of the absolute frequencies. b) Make a bar chart with these data. 8

These are the best times in 10 km races for the members of an athletics club: 42:20 40:08 47:32 49:50 43:24 48:31 51:42

The table on n.° of n.° of age men women the right describes group (millions) (millions) the population of a 0-9 2 489 2 352 country. 10-19 2 271 2 146 Bellow, you will 20-29 2 742 2 687 find the population 30-39 4 029 3 843 40-49 3 850 3 739 pyramid. However, 50-59 3 029 3 096 it shows percentages 60-69 2 307 2 504 instead of frequencies. 70-79 1 549 1 923 Be careful! There is 80-89 823 1 355 one incorrect bar in 90-99 106 279 the men’s part and one incorrect bar in the women’s part. 90-99 80-89 70-79 60-69 50-59 40-49 30-39 20-29 10-19 0-9

45:53 47:17 50:37 49:07 51:37 43:28 45:18 44:36 46:15 50:48 47:59 51:21 43:37 42:14 a) Make a table of absolute and relative frequencies with these intervals: 40 - 42 - 44 - 46 - 48 - 50 - 52

10% 8% 6% 4% 2%

Use these data to make a pie chart with the percentage of the most used browsers in the world: Chrome: 58 %

Internet Explorer: 20 %

Firefox: 12 %

Microsoft Edge: 5 %

Safari: 4 %

Others: 1 %

We did a study on how people under 26 and people between 26 and 50 years old use smartphones. The results are shown in the following table: use

under

26 to 50

Games and entertainment

35 %

12 %

Social media

33 %

26 %

News

37 %

Calls and messages

27 %

25 %

a) Make a pie chart with these data. b) Describe the similarities and differences between the two groups. c) Invent a pie chart for people over 50 years old. 284

2% 4% 6% 8% 10%

Indicate which intervals have the incorrect bars and explain the errors.

b) Draw a histogram with the data. 9

Draw the 2012 population pyramid for India using the data shown in this table: age group

men

women

0-9 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90-99

10.36 % 10.11 % 8.95 % 7.79 % 6.13 % 4.22 % 2.65 % 1.24 % 0.33 % 0.01 %

9.20 % 8.95 % 8.12 % 7.29 % 5.97 % 4.23 % 2.65 % 1.33 % 0.41 % 0.02 %

a) Look at the pyramid and classify it using the same criteria as in exercise 4. b) Build another pyramid for the same population, distributing it into these four groups: — children: 0 to 9 years old — adolescents: 10 to 24 years old — adults: 25 to 59 years old — elderly: 60 years or older

UNIT 13

These graphs show data from a study about musical tastes. We used four samples of the population taken at different locations:

These pie charts show the composition of the human body at two different ages: at 25 at 75 Body water

12 %

17% 15 %

30 %

62%

Fat tissue 53%

Bone mass

sample: At the exit to a sample: At the entrance discotheque. to a music conservatory.

a) How do the percentages of water content, bone mass, fat tissue and muscles, organs… change over these 50 years? Give your answers as a percentage increase or decrease. b) What is the water content in the body of a person who is 25 years old and weighs 80 kg? What is their content of fat tissue? c) Answer the same questions for a person who is 75 years old and weighs the same.

sample: At a party at the sample: At a graffiti Colombian Society. contest. a) Which colour represents each type of music? PopRock, Classical, Jazz, Reggaeton, Rap. b) Estimate the percentage of each type of music for each of the samples. 14

This graph describes how the age groups in a population change over time (as a percentage) in the EU for 1950-2050: 2050

13.3

9.7

2025

14.4

10.5

2000

17.1

28.2 31.1

23.7

15.5

1950

24.9

15.8 15-24

18.5 21.3

36.9

1975

0-14

18.5

16.2

12.3 32.7

25-49

11.8

35 50-64

6.5

17.2 15.4

3.4

10.7

15.2

7.9 1.2

65-79

> 80

a) Which group’s percentage will decrease the most? Which will increase the most? b) If we estimate that there will be 1 000 million inhabitants in 2050, how many will there be in each group? c) If there were around 125 200 000 people over 50 in 2000, what was the population of the EU that year? d) How much is the percentage of people under 14 estimated to decrease? How much is the percentage of people over 80 years old estimated to increase? e) Describe how each group changes over time.

Muscles, organs…

Distribution of the Spanish population by the size of the municipality where they lived, from 1900 to 2020. The data for 2020 were estimated: municipalities

1900 1930 1960 1990 2020

Up to 5 000 inhab. 51 % 40 % 5 001 to 20 000 28 % 29 % 20 001 to 100 000 12 % 16 % More than 100 000 9 % 15 %

29 % 25 % 18 % 28 %

16 % 20 % 22 % 42 %

10 % 16 % 27 % 47 %

Number of inhabitants of Spain, in millions, from 1900 to 2020: 1900

1930

1960

1990

2020

18.6

23.6

30.4

38.8

45.6

a) Calculate the number of people who lived in the smallest municipalities from 1900 to 2020. The population of these municipalities has been decreasing. But has it been decreasing constantly or decreasing less and less over time? Describe how the populations of each of the types of municipalities change over time. b) Calculate how many Spanish people lived in the largest municipalities since 1900 and describe how the number changed over time. c) Make pie charts showing the population distribution for each year shown in the table. d) Estimate from what year half of the population lived in municipalities that have more than 20 000 inhabitants. 285

P O H S K R O W S H T A M READ AND LEARN Is the sample valid? The bigger the sample used, the more reliable a statistical study will be. But more is not always better. It is essential for the sample to be representative. In the United States presidential elections of 1936, the magazine Literary Digest surveyed more than two million people, and used those data to predict that the Republican candidate Landon would win. However, a young statistician named George Horace Gallup surveyed just five thousand people and predicted that the Democratic candidate Roosevelt would win, and he was right. The magazine had surveyed its readers, who were a certain type of citizen. The sample was not representative! Could you imagine the results if we wanted to study musical tastes in Spain and we only asked around an old people’s home?

INVESTIGATE Organise the data A clever father makes a deal with his son: After the next Maths exam, he will add up the grades of all his classmates who did better than him, and add up the grades of all his classmates who did worse than him. Then: — If the sum of the low scores is 50 points or more higher than the sum of the high scores, he will buy his son a motorbike. — If the sum of the high scores is 20 points or more higher than the sum of the low scores, the son will have to stay home and study every Sunday for one month. — In any other situation, nothing will happen. The grades of his classmates were: 5-5-4-9-8-6-3-6-3-7-4-5-6-6 7 - 7 - 4 - 7 - 5 - 2 - 6 - 5 - 5 - 8 - 3 - 9 - 10 - 5 Do you think it was a good deal for the son? What happens if he gets a 5? And if he gets a 6? What grade does he need to get the motorbike?

INDEPENDENT PROBLEM SOLVING The eight rows of four numbers add up to the same sum. What is the value of a, b, c and d?

2a b

b+2 d–2

10 b

d 2a

286

UNIT 13

PRACTICE MAKES PERFECT! • A delivery woman has seven full boxes of soft drinks, seven half-full boxes and seven empty boxes in her van. If she wants to deliver them to three supermarkets, giving each one the same number of soft drinks and the same number of boxes, how should she do it?

• You have two candles. Each one takes 10 minutes to burn. But they burn at irregular speeds. In other words, half the candle does not necessarily burn in half the time. Could you measure a quarter of an hour with them? How?

Suppose she is in a hurry and does not want to walk back and forth moving bottles from one box to another. What should she do?

• Three of the vertices of a regular hexagon meet at the vertices of an equilateral triangle with area of 20 cm2. What is the area of the hexagon?

SELF-ASSESSMENT

anayaeducacion.es Answer key.

1 What are the individuals and the population in each

situation? What is the variable and what type of variable is it?

4 A study was done on fatal work accidents in a region,

distributed by job sector. These are the results:

a) Number of times a year each patient of a hospital has used their health card. b) Time each patient waits to see their doctor at a health centre throughout a day. c) Type of specialist who cares for each patient at a health centre in a month. 2 Time, in minutes, that a doctor’s patients spent in

the waiting room on one day: 28

4 12 35

26 45 22

27 16 18 32

28 37

7 39 15

6 23

8 12 34 15

25 18 17 27 15

a) Make a table using intervals ending in: 1.5 - 9.5 - 17.5 - 25.5 - 33.5 - 41.5 - 49.5 b) Use the most suitable graph to represent the results (bar chart or histogram). c) Use this classification: Short wait: 1-15 min; Medium wait: 16-30 min; Long wait: 31-50 min; to make a table and a pie chart with the data. 3 Number of days that the students of a class went to

the school library in one school year:

21 %

9% 24 %

Agriculture Industry Construction Services

a) What is the percentage of fatal accidents that occurred in the construction sector? b) If there were 135 fatal accidents in the agricultural sector, what was the total number of fatal accidents in the region? c) How many fatal accidents were there in each of the sectors? 5 Look at these population pyramids: Men

morocco 2018 age

Women

Men

france 2018 age

Women

> 60 > 60 45-59 45-59 30-44 30-44 15-29 15-29 0-14 0-14 2.5 2.0 1.5 2.0 0.5 0.0 0.0 0.5 1.0 1.5 2.0 2.5 2.5 2.0 1.5 2.0 0.5 0.0 0.0 0.5 1.0 1.5 2.0 2.5

Total population in millions

Are these statements true or false? Explain your answers: a) The population of elderly people in France is much higher than in Morocco.

3 1 2 4 0

2 1 3 1 0

2 0 3 5 2

0 2 4 1 2

1 2 0 5 3

3 1 2 1 0

b) There are more elderly women than elderly men in both countries.

Make a frequency table and use the most suitable graph to represent the results.

c) The proportion of children is higher in Morocco than in France.

Commitment

Watch the video for target 13.1. Think of something you can do to contribute to achieve that goal. Make a commitment to put your idea into practice.

287

STATISTICAL PARAMETERS

Language Bank 1

A great professional Florence Nightingale (1820-1910) was a mathematician and nurse from London, who applied statistical techniques to improve conditions in hospitals and medical attention during the Crimean War. There, she confirmed that the majority of deaths were caused by the terrible conditions in the hospitals. She gathered and organised statistical information, taking into account the amount of time that had passed, the number of deaths and their causes, and expressed them in graphs that, for the first time, related three variables (before then they only represented one or two). She notably improved the organisation, interpretation and graphic presentation of statistical data. Also, she was one of the first people to use statistics with a practical objective and, as a result, managed to gradually reduce hospital deaths, improving sanitary conditions. Until then, statistics had only been used to present information.

288

Look up information about the conditions in hospitals in London during the 19th century. Write a short essay explaining what you’ve found out. Were the conditions the same they are now?

Use what you have learnt to solve the problem The marks obtained in the latest exams that a fourth-year class took in four subject areas, are presented in the bar charts below:

history

spanish language

10 5

5 0

maths

9 10

science

5 0

9 10

❚ Think 1

Look at the charts and answer: a) In one of the subjects, the average mark is quite high, more than 6. However, in another subject, it’s quite low, less than 4. Which subjects are these? b) Is it reasonable to assume that the average mark in the other subjects is close to 5, although the two corresponding charts are very different? Look closely at the marks in Spanish language. Is it fair to say that they are clearly ‘more scattered’ than the others? In this unit you’ll learn to take into account not only the average, but also the distribution of the results.

2 Alone, in small groups, or as a class, depending on what your teacher thinks best, you could follow the example described above: collect the marks of your classmates in various subjects, tabulate them and represent them in bar charts. Looking at the corresponding charts: a) Could you estimate the average of each one? b) Are any of these distributions more scattered than the others? When you finish the unit, you’ll be able to respond to these questions numerically calculating the results.

3 Consider other similar investigations. For example, with the height of the students in different classes and year groups.

289

TWO TYPES OF STATISTICAL PARAMETERS We will start by looking at just two types of statistical parameters: • Location parameters tell us the value (centre) around which the data points are distributed. • Dispersion parameters tell us how far away from the centre the values in the distribution are.

Location parameters mean Notation The ∑ sign is used to indicate a sum of several addends.

We read ∑ xi as: ‘the sum of xi’.

If x1, x2, …, xn are the values of a statistical distribution, the mean, or average, is written as x and calculated like this: x =

x1 + x2 + … + xn n

Or in its shortened form: x =

/ xi n

For example, these are the results of the physical fitness tests passed by the 10 members of a sport team: 1, 0, 3, 4, 5, 2, 3, 4, 4, 4. We calculate the mean: x = 1 + 0 + 3 + 4 + 5 + 2 + 3 + 4 + 4 + 4 = 30 = 3 10 10

median If we order the data points from smallest to largest, the median, Me, is the value in the middle. In other words, there are as many values below it as above it. If there are an even number of data points, the median is the average value of the two central terms. To find the median in the previous example, we put the data in order: Distributions with more than one mode A distribution can have more than one mode. If it has two modes it is called bimodal; if it has three modes it is called trimodal, and so on.

0, 1, 2, 3, 3, 4, 4, 4, 4, 5 The central data points are 3 and 4, so the median is 3.5.

mode The mode, Mo, is the value that appears most frequently. The mode in the previous example is 4, it is the value that appears most frequently.

Think and practise 1 Calculate the mean, the median and the mode of these

statistical distributions:

a) 4, 5, 6, 6, 6, 6, 7, 11, 12, 17 b) 10, 12, 6, 9, 10, 8, 9, 10, 14, 2 c) 2, 3, 3, 3, 4, 5, 6, 6, 6, 6, 3, 7 d) 1, 2, 3, 4, 5, 4, 3, 2, 1 290

2 Find the location parameters for the distribution shown

in this bar chart: 4 3 2 1

0 1 2 3 4 5 6 7 8 9

UNIT 14

Dispersion parameters These graphs show the ages of the players of two football teams: Liverpool Red

Chelsea Blue

The mean age is 21 on both teams. Note that, even though the mean is the same, the two distributions are very different. The data points in the second distribution are clearly more separated than the data points in the first one. The dispersion parameters described below are useful to quantify those differences.

range Remember At this level, we no longer talk about the mean deviation that we learnt in previous years. From now on, we will use the standard deviation as a dispersion measurement. Why do we use the standard deviation and not the variance? Because the variance has a major inconvenience. Imagine that we are working with a distribution of heights given in centimetres. The mean will be in cm, but the variance would be in cm2 (in other words, it would be an area instead of a length). That is why we take the square root to get the standard deviation. In our example, the standard deviation would be a length in centimetres.

The range is the difference between the largest and smallest data point. In other words, it is the length of the section containing all the data points. On the red team, the range is 23 – 20 = 3; on the blue team, it is 25 – 18 = 7.

variance The variance is the average of the squares of the distances of the data points to the mean: (x – x ) 2 + (x 2 – x ) 2 + … + (x n – x ) 2 / (x i – x ) 2 Variance: V = 1 = n n This formula is equivalent to: / x 2i x 2 + x 2 + … + x 2n Variance: V = 1 2 – x2 = – x2 n n 2 2 2 2 2 2 2 2 2 2 2 Vred = 1 + 1 + 1 + 1 + 0 + 0 + 0 + 0 + 1 + 1 + 2 = 0.91 11 2 2 2 2 2 2 2 2 2 2 2 Vblue = 3 + 3 + 3 + 2 + 2 + 1 + 1 + 2 + 3 + 4 + 4 = 7.45 11

standard deviation, σ anayaeducacion.es GeoGebra. Standard deviation.

It is the square root of the variance: σ = variance =

/ x 2i n

– x2

σred = 0.91 = 0.95; σblue = 7.45 = 2.73 Note that the dispersion of ages on Chelsea Blue is much greater than on Liverpool Red. This agrees with what we saw on their graphs. From now on, we will pay special attention to the mean ( x ) and the standard deviation (σ). Each one gives us information that complements the other one. Think and practise 3 Find the dispersion parameters for the distributions in

exercise 1 on the previous page.

4 Find the variance of this distribution in two different

ways: 8, 7, 11, 15, 9, 7, 13, 15.

291

CALCULATING x AND σ IN FREQUENCY TABLES

When the statistical data are given in frequency tables, it can be very easy to calculate the parameters.

calculating

Let’s look at an example: the frequency table on the right shows the grades that 33 students got on their last Maths exam. To calculate the mean of the grades, we have to add:

4 + (5 + 5 + … + 5) + (6 + 6 + … + 6) + (7 + 7 + … + 7) + 8 + 8 + 9 10 times

14 times

5 times

and divide the result by 1 + 10 + 14 + 5 + 2 + 1 = 33. However, we can calculate that sum more effectively like this: 4 · 1 + 5 · 10 + 6 · 14 + 7 · 5 + 8 · 2 + 9 · 1 xi

In other words, we multiply every value of the variable by its frequency, and add the results together.

fi · xi

198

∑ fi

∑ fi xi

fi · xi

f1 x1

f2 x2

… xn

… fn

… fn xn

∑ fi

∑ fi xi

To make it easier, we add a new column to the table: fi · xi . We get the total number of individuals by adding the values in the fi column: f1 + f2 + … + fn = 33 → ∑ fi = 33 We calculate the sum of all the grades by adding the values in the fi · xi column: f1x1 + f2x2 + … + fn xn = 198 → ∑ fi xi = 198 The mean is: x =

/ fi x i 198 = =6 / fi 33

To find the mean of a distribution in a frequency table, we first add the fi · xi column to the table and then: x=

/ fi x i / fi

where the sum of all the values is ∑ fi xi = f1x1 + f2x2 + … + fn xn and the number of individuals is ∑ fi = f1 + f2 + … + fn.

Think and practise 1 Calculate the mean of these distributions:

a) number of children

292

b) number of exams failed

14 15

UNIT 14

calculating σ There are two equivalent expressions for the standard deviation: v=

Rfi (x i – x )2 Rf i

where ∑fi(xi – x )2 = f1(x1 – x )2 + … + fn(xn – x )2 is the sum of the squares of the deviations from the mean.

Rfi x i2 – x2 Rf i

where ∑fi xi2 = f1x12 + … + fn xn2 is the sum of the squares of all the values.

We get the same result using either formula. However, the second formula is much more practical. Let’s see why: Since fi · xi2 is equal to (fi · xi) · xi, we add a column to the frequency table with the values obtained by multiplying the elements in columns xi and fi · xi. With the original table, the two new columns, and (xi) · (fi · xi) the sum totals, it is now easy to calculate 2 xi fi fi · xi fi · xi the mean and the standard deviation: 4

250

504

245

128

198

1 224

∑ fi

∑ fi xi

∑ fi xi2

• mean: x =

R fi x i 198 =6 = 33 R fi

• standard deviation: v= 1.04

R fi x i2 – x2 = R fi

1224 – 6 2 = 33

Tables with data grouped into intervals To make a frequency table when the data is grouped into intervals (instead of individual values), we assign a central value to each interval, called its class midpoint. For example, the class midpoint of the interval 50-58 is: 50 + 58 = 108 = 54 2 2 This allows us to turn a table with grouped data points into a table like the ones we have been working with above. From there on, we use the same methods. Think and practise 2 Find the mean and the standard deviation for this

distribution:

fi · xi

fi · xi2

1 2 3 4 5 total

12 15 24 19 10

12 30 72 76 50

12 60 216 304 250

3 Copy and complete this table in your notebook with

class midpoints, and calculate the mean and the standard deviation: weights

people

50 to 58 58 to 66 66 to 74 74 to 82 82 to 90

6 12 21 16 5

293

JOINT INTERPRETATION OF x AND σ Let’s look again at the two graphs on page 281: Liverpool Red

Chelsea Blue

Remember that the mean of both distributions is 21 and that their standard deviations are 0.95 for Liverpool Red, and 2.73 for Chelsea Blue. This agrees with what we see in the graphs, where the data points for Chelsea Blue are much farther from the mean (21) than the data points for Liverpool Red. With the values for x and σ, we can get a relatively good understanding of a distribution. The mean tells us where its centre is. The standard deviation tells us how far away from the mean, or how disperse, the data points are.

anayaeducacion.es Practise: joint interpretation of x and σ.

Problem solved I

The four graphs on the left show the number of study hours per week of the students in four classes. Look at the data in the table below and indicate which graph represents each class.

III

class

1st year A

8.2

0.8

2.5

1st

year B

1st year A and 1st year B ↔ I and III 3rd year A and 3rd year B ↔ II and IV To distinguish between classes that have similar means, we look at the dispersion of the data points. It is clear that the data points in I are much more disperse than the data points in III, so 1st year A ↔ III and 1st year B ↔ I.

3rd year A 12.5 2.3 3rd

If we look at the graphs, it is easy to see that the means of classes I and III are around 8. And the means of classes II and IV are higher, around 13. So we can say:

It is also clear that the data points in II are much more disperse than the data points in IV. So we can conclude that:

year B 13.4 1.2

3rd year A ↔ II and 3rd year B ↔ IV Think and practise 1

anayaeducacion.es GeoGebra. Joint interpretation of x and σ.

These graphs show the accuracy of the players on four basketball teams based on the number of baskets they made. Use the data in the table on the right to determine the mean and the standard deviation for each team. A 45 50 55 60 65 70 C 45 50 55 60 65 70

294

B 45 50 55 60 65 70 D 45 50 55 60 65 70

team

52.5

7.1

6.9

III

63.5

2.7

UNIT 14

Variation coefficient The distribution of the weights of the bulls at a ranch have a mean of x = 500 kg and a standard deviation of σ = 40 kg. The distribution of the weights of the dogs at a dog show have a mean of x = 20 kg and a standard deviation of σ = 10 kg.

The standard deviation of the bull weights (40 kg) is higher than the standard deviation of the dog weights (10 kg). However, 40 kg is very small compared to the enormous size of bulls. This means that the bulls are very similar in weight. However, 10 kg is much larger compared to the weight of a dog. In situations like this, the standard deviation is not a good way to compare dispersions. For that, we need to define a new statistical parameter. x

bulls

500

dogs

40 in relation to 500 is less than 10 in relation to 20.

To compare the dispersion of two different populations, the variation coefficient is: VC = v x When we divide σ by x we get a relative dispersion. The result is sometimes given as a percentage.

In the example of the bulls and the dogs, we have:

bulls

500

• For the bulls:

dogs

50 %

VC = 40 = 0.08 In other words, 8 %. 500 In other words, 50 %. • For the dogs: VC = 10 = 0.50 20 This clearly shows that the variation of the weights of the dogs (50 %) is much higher than the variation of the weights of the bulls (8 %).

Think and practise

anayaeducacion.es GeoGebra. Variation coefficient.

2 We ask about the price of certain models of pianos, flutes and harmonicas at different music

shops. The means and standard deviations are: pianos

flutes

harmonicas

mean

943

132

standard dev.

148

Compare the relative dispersion of the prices of these three products.

295

POSITION PARAMETERS: MEDIAN AND QUARTILES

We asked a group of 16 people how many times they went running this month. These are the ordered results and the graph: 0, 2, 3, 3, 5, 5, 6, 6, 7, 7, 7, 8, 8, 9, 10, 10 0

2 3 4 5 50 % of the population

7 8 9 10 Me 50 % of the population

Note that half the population is to the right of the median, Me. The other half of the population is to its left. The median divides the population in half. What if we wanted to divide the population into four parts with the same number of individuals in each part? We would have to indicate the quartiles, Q1 and Q3.

Median: position and central tendency The median and the quartiles are position parameters because they indicate a location (a position) in relation to the other values in the distribution. Among the position parameters, the median is in the central location. So it is also a parameter of central tendency.

4 Q1

7 Me

The first quartile, Q1, is the value of the variable where one-fourth of the population is below it and three-fourths are above it. The third quartile, Q 3, is the value of the variable where one-fourth of the population is above it and three-fourths are below it. They are called Q1 and Q3 because the median is the second quartile, Q 2. The difference Q3 – Q1 is called the interquartile range.

Problem solved

Since the distribution has 10 individuals, one-fourth is 10 : 4 = 2.5.

The ten guests at a birthday party played piñata, and when it broke, every one of them took as many sweets as they could. This is the ordered list of the number of sweets each guest got:

• Q1 must have ‘two and a half elements’ on its left. Therefore, the first quartile must be located on the third element, with the ‘half element’ on its left and the ‘other half ’ on its right. In other words, Q1 = 4. • Me must have 5 elements on its left and 5 elements on its right. It is therefore between the fifth (7) and sixth (8) elements. So Me = 7.5.

2, 2, 4, 6, 7, 8, 8, 9, 10, 11

• Q3 must have ‘seven and a half elements’ on its left (2.5 · 3 = 7.5). Using similar reasoning as for Q1, the third quartile is on the eighth element. So Q3 = 9.

Calculate the median and the quartiles.

Think and practise

anayaeducacion.es GeoGebra. Median and quartiles.

1 Calculate Q1, Me and Q3, and locate them in each of

these graphed distributions: a) b)

296

2 For each of these distributions:

a) Calculate Q1, Me and Q 3.

b) Graph the data points and locate Q1, Me and Q3. A: 0, 0, 2, 3, 4, 4, 4, 4, 5, 6, 7, 8, 9, 9, 10 B: 0, 1, 1, 2, 3, 4, 4, 7, 7, 7, 14, 17, 29, 35 C: 12, 13, 19, 25, 63, 85, 123, 132, 147

UNIT 14

Box-and-whisker plots The box-and-whisker plot is closely related to the parameters of position that we have learnt. Let’s look at an example of how to make these types of graphs. Example 1

22233333444666667788 Q1

The number of people in the families of a group of friends is shown on the left. We can see that: • The lowest value is 2 and the highest value is 8. • Q1 = 3; Me = 4 and Q3 = 6.

Observe

We use these results to draw the plot: 0

The length of the box is Q3 – Q1, the interquartile range.

The box describes the section between the two quartiles with the median clearly indicated, and the whiskers extend to the ends of the data range. Example 2 160 165 170 175 180 185 190 Q1 Me Q3

On the left, you can see a box-and-whisker plot showing the distribution of heights of the members of a club. Just by looking at the plot we can say that: • The shortest member is 160 cm and the tallest member is 187 cm. • The quartiles and median are: Q1 = 167.5; Me = 171 and Q3 = 175.5. • Therefore, 25 % of the members are between 160 cm and 167.5 cm; another 25 % are between 167.5 cm and 171 cm; another 25 % are between 171 cm and 175.5 cm; and the last 25 % (the tallest) are between 175.5 cm and 187 cm.

Problem solved

Make a box-and-whisker plot for each of these distributions: a) 2, 2, 4, 6, 7, 8, 8, 9, 10, 11 b) 1, 1, 2, 3, 5, 5, 15, 27, 41, 43

a) On the previous page, we got: Q1 = 4; Me = 7.5: Q 3 = 9. We draw the plot and add the scale: 1

b) We first get the parameters of position: Q1 = 2; Me = 5; Q3 = 27. We look at the scale (keeping in mind that the last data points have ‘very large’ values) and draw the plot: 0

Think and practise 3 Make a box-and-whisker plot for each of the

distributions in exercise 2 on the previous page.

Use the values of Q1, Me and Q3 that you found in that exercise.

anayaeducacion.es GeoGebra. Box-and-whisker plots.

4 Make a box-and-whisker plot for the points scored on a

bullseye target: 7 6 6 8 5

5 7 9 6 8

4 7 5 8 6

7 5 6 6 7

5 6 6 5 8

6 7 5 9 3 297

5 xi

151

156

161

166

171

176

USING A CALCULATOR TO GET x AND σ Through an example, we will now study the steps that we must follow for entering data into the calculator and getting the right results. Look at the table on the left. 1

Preparation for working with statistics First, we go into the � and use the arrow ” to get to the option 6:Statistics and select it. Then, select 1:1–Variable. We see an empty table where we can enter the data points and their frequencies.

Deleting previous data If the table contains data points that you do not want to keep, make sure to delete them before entering the new data points. You may have to use the del key.

Entering the data The cursor turns black in the first box of the xi column. After entering each data point, press =, then, the cursor will move down so you can enter the next data point. Move the cursor using the ”’‘“ arrows to enter the frequencies or change incorrect data points. We start by entering all the values of the variable into the table. 151 = 156 = 161 = … 176 = Note that it automatically assigns a value of 1 to all the frequencies. Then, we enter the frequencies: 1 = 4 = 9 = 10 = 4 = 2 =

Changing incorrect data points If there is an incorrect data point, move the cursor to it, enter the correct value, and press =.

Remember After you are finished working with statistical data, you have to press � and then, select 1:Calculate to go back to doing arithmetic calculations.

Results To get a summary of the data and the values of the statistical parameters, select the calc 1-variable option. The result will look something like this: Use the cursor to move up and down to see all the results.

Think and practise 1 Use a calculator to find x and σ for the distribution a)

in exercise 1 on page 282.

298

2 Use a calculator to find x and σ for the distribution b)

in exercise 1 on page 282.

STATISTICS IN THE MEDIA Statistics are present in all areas of our lives. Most of the data we receive is prepared using statistical studies. Sometimes, private media companies do their own statistics. However, the public institution responsible for doing statistical studies about all aspects of Spanish society is the INE (National Statistics Institute). The INE and the CIS (Centre for Sociological Research) set the standard for reliable data. Fields like medicine, ecology, politics, psychology, sport, advertising and others all use statistics to do research, improve their results, diagnose problems, predict future events, and more. Let's look at some examples of news stories with statistics:

Monday, 9 November 2020

The average salary in Spain falls for the first time since 2006 The median salary has also decreased for the second year in a row, to €1 594, and the average salary has dropped for the first time in ten years, to €1 878.10. Moreover, three in ten Spanish people earn under €1 229 of gross monthly salary.

15 months, average life of a mobile phone Spanish people get new mobile phones up to 4 years before they have to. Most devices are tossed when they still have market value.

Sedentarism: A public health problem 53.5

ToTal gender

Men Women age

From 15 to 19 years old From 20 to 24 years old From 25 to 34 years old From 35 to 44 years old From 45 to 54 years old From 55 to 64 years old From 65 to 74 years old 75 years and older

Graph 2.1. People who practised sport last year by gender and age. (As a percentage of the total population studied for each age group).

One in four young people think gender violence is ‘normal’ More than 20 % of Spanish people aged 15 to 29 years old think that gender violence is ‘politicised and greatly exaggerated’, according to a report from FAD.

inTernaTional aids conference

One in every three treatments fails due to drug resistance Around 32 % of AIDS treatments fail because the virus has become resistant to antiviral drugs.

Think and practise 1

Interpret the graphs and news stories that you have seen on this page.

2 Look online for news stories that contain statistics to

underpin their arguments.

299

EXERCISES AND ED V L O S S M E L B O R P 1 Calculating statistical parameters

Twenty-five people go out on a mountain excursion. These are their ages: 8

a) Make a frequency table classifying the ages into 6 intervals that start at 7.5 and end at 67.5. Use the table to find the parameters x , σ and VC. b) Calculate x , σ and CV by entering the 25 data points into a calculator without grouping them into intervals. c) Excluding the 5 children, we get a group of 20 people. Compare their parameters with the ones you got for the original group. d) Find the position parameters of the original distribution and make the corresponding box-and-whisker plot. Your turn Make the box-andwhisker plot for the smaller group (20 adults and no children) and compare it to the one for the original group.

a) We define the 6 intervals and make the frequency table. Then, we make a table using the class midpoints as values of the variable: interval

frequency

fi · xi

fi · xi2

7.5-17.5 17.5-27.5 27.5-37.5 37.5-47.5 47.5-57.5 57.5-67.5

5 0 3 8 5 4

12.5 22.5 32.5 42.5 52.5 62.5 totals

5 0 3 8 5 4 25

62.5 0.0 97.5 340.0 262.5 250.0 1 012.5

781.25 0.00 3 168.75 14 450.00 13 781.25 15 625.00 47 806.25

• mean: x =

R fi x i 1012.5 = = 40.5 25 R fi

• standard deviation: σ =

R fi x i2 – x2 = R fi

• variation coefficient: VC = v = 16.49 = 0.407 → 40.7 % x 40.5 b) We put the calculator in stat mode, delete the memory, and enter the 25 data points. Select calc 1-var. You get: x = 40.4; σ = 17.11 Use these data to find the variation coefficient: VC = 17.11 = 0.424 → 42.4 % 40.4 c) Let’s do it with the calculator. We can delete the 5 extra data points that we entered in the previous section. We get x = 47.95; σ = 8.97; VC = 0.187 → 18.7 % When we delete the 5 children, the average age increases and the standard deviation decreases. Logically, the dispersion is much lower. We can see this with the VC, which is less than half of what it was for the full group. d) We divide the number of individuals into 4 equal parts: • 25 = 6.25. So Q1 is the individual in 7th place → Q1= 37 4 • 25 = 12.5. The median is in 13th place → Me = 43 2 • 3 · 25 = 18.75. So Q3 is in 19th place → Q3 = 52 4 The lowest value is 8 and the highest is 67. We use this to make the box-andwhisker plot: 10

40 Q1 Me

300

47 806.25 – 40.5 2 = 16.49 25

50 Q3

folio. for your port from this unit s ce ur so re choose Remember to

S M E L B O R P D N A EXERCISES

UNIT 14

Practise

Position parameters and box plots

Location and dispersion parameters

Find the median and quartiles of each distribution and make their box-and-whisker plots: a) 1, 1, 1, 2, 2, 5, 6, 6, 6, 7, 8 10, 11 b) 4, 5, 5, 6, 7, 7, 7, 8, 12, 14, 19, 22 c) 123, 125, 134, 140, 151, 173, 178, 186, 192, 198

Match each bar graph with its corresponding box-and-whisker plot:

Calculate the mean, median, mode, range, variance, standard deviation and variation coefficient for each set of numbers: a) 6, 3, 4, 2, 5, 5, 6, 4, 5, 6, 8, 9, 6, 7, 7, 6, 4, 6, 10, 6 b) 11, 12, 12, 11, 10, 13, 14, 15, 14, 12 c) 165, 167, 172, 168, 164, 158, 160, 167, 159, 162

The shoe sizes of the students in a class are: 42, 40, 43, 45, 43

44, 38, 39, 40, 43

41, 42, 38, 36, 38

45, 38, 39, 42, 40

40, 39, 37, 36, 41

46, 44, 37, 42, 39

0 1 2 3 4 5 6

n.° of boxes

51 23 11 8

58 to 62

62 to 66

66 to 70

70 to 74

a) Make a table with the class midpoints and frequencies. b) Calculate the mean, the standard deviation and the variation coefficient. c) Check your results with a calculator.

Problem solved

age

n.° of members

Find the median and the quartiles. There are 4 members who are 14 years old, 5 who are 15 years old, and so on. There are 23 people in total: 11 on one side, 11 on the other side and one in the middle. If we count them one by one, the person in 12th place is 16 years old. That is the median. We do the same thing for the quartiles: Q1 = 15 years and Q3 = 17 years.

n.° of throwers

0 1 2 3 4 5 6

The ages of the members of the theatre group of a school are shown in this table:

This table shows the javelin throws made at the Olympic trials: 54 to 58

D 0 1 2 3 4 5 6

c) Check your results with a calculator.

(m)

0 1 2 3 4 5 6

b) What is the mode?

distances

B 0 1 2 3 4 5 6

a) Calculate the mean, the standard deviation and the variation coefficient.

0 1 2 3 4 5 6

A factory counted the number of glasses in each box that broke on the way to the shop. These are the results: 0

4 0 1 2 3 4 5 6

b) Find the mean, the standard deviation and the VC.

n.° of broken glasses

0 1 2 3 4 5 6

a) Make a frequency table with these intervals: 35.5 - 38.5 - 40.5 - 42.5 - 44.5 - 46.5. 3

This table shows the distribution of the number of subjects failed by the students of a class: n.° of subjects failed

n.° of students

Make a box-and-whisker plot for this distribution. 301

S M E L B O R P D N A EXERCISES Problem solving 9

Two classes, A and B, with 30 students each, took the same exam. Their means and standard deviations are: x A = 6, σA = 1, x B = 6, σB = 3. a) Assign one of these graphs to A and the other one to B.

a) Compare these distributions of the grades of three classes. Indicate what are the medians and quartiles of each class. 0

9 10

I II

III

b) In one of the classes, there were 11 fails and 4 distinctions. In the other class, there were 5 fails and 1 distinction. Which is A and which is B? c) If Laura needs to get a distinction and Michael just needs to pass, which class is better for each one? 10

These four graphs show the heights of the players of four basketball teams: A, B, C and D. We can see their parameters in the table. Which graph represents each team? I

180

195

210 180

III

195

210

198.5 9.7

198.1 3.9

D 180

195

210 180

195

team

193

4.6

193.4 8.1

210

Find the VC of each team and order them from less regular to more regular. 11

A basketball player named Ellen makes an average of 17 points per game with a standard deviation of 9 points. Her team-mate, Marta, makes an average of 20 points per game with a standard deviation of 3. For the next game, the coach needs a player who will try to make 30 or more points. Which of the two should he choose? Why?

Lydia and Mark play a game where they try to guess the most words, given their definitions, in one minute. These are the results: lydia

mark

a) Find the mean and the standard deviation for each one. b) Calculate their VC and say who is more regular. 302

b) These comments were made about the three groups: i. 50 % of the class passed. ii. The grades are very similar. iii. A quarter of the class has grades above 7. iv. This is the best class, but with the most dispersion. Which class is each comment about? 14

These are the box plots for the maths grades of four classes of 20 students: 0

9 10

I II III IV

a) What are the highest and lowest values, and Q1, Me and Q3 for each class? b) The parameters, not in order, are: a

2.3

3.1

2.5

1.3

Match each set of parameters with the correct class. c) The 20 grades for class I are: 2 2 2 2 3 3 4 4 4 5 5 5 5 6 6 7 8 8 10 10 Check that these agree with its box plot. Invent 20 values for the box plots of the classes ii, iii and iv. d) Calculate x and σ for the distributions you invented in the previous section and compare them with what appears in the table in section b). e) Find the variation coefficient of every distribution in section b). Which is the most regular?

UNIT 14

Ralph works as a travelling salesman six days a week. Yesterday was Friday, and he calculated that he had earned an average of €48 per day this week. When he did the same calculation today, Saturday, his average daily earnings were now €60. How much did he earn today? To calculate a student’s final grade for a subject, the second exam is worth twice the first exam, and the third exam is worth triple the first exam. a) What is the final grade of a student who got a 5, a 6 and a 4 on the exams? b) And what would her final grade be if the exams were worth 10 %, 40 % and 50 %?

We know that a class’s average score on an exam was 5, with a standard deviation of 1.5. The same class got an average of 5 on a different exam, with a standard deviation of 1. If a student gets an 8 on the first exam and a 7.5 on the second, which grade is better? Why?

We know the number of days per month it rained this year in a region. The quartile values are 6, 9 and 14. It rained the most in March, with 21 days, and we know that the range of the distribution is 18. a) Make the box-and-whisker plot. b) Do you think that it is a rainy region? Justify your answer.

Advanced problem solving 19

height (m) (class midpoints) 1.52 1.56 1.60 1.64 1.68 1.72 1.76 1.80 1.84 1.88 n.° of soldiers

(kg)

n.° of animals

These are the number of hours a group of students studies each week: 14 9 9 20 18 12 14 6 14 8 15 10 18 20 2 7 18 8 12 10 20 16 18 15 24 10 12 25 24 17 10 4 8 20 10 12 16 5 4 13 a) Make a frequency table with these intervals: 1.5 - 6.5 - 11.5 - 16.5 - 21.5 - 26.5. b) Calculate the mean and the standard deviation.

These are the grades a class got on an exam: grades n.° of students

1 2 3 4 5 6 7 8 9 10 4 3 2 1 7 3 2 8 3

Calculate the average grades for: the class ( x ), the students who passed ( x A ) and the students who failed ( x B ). Can we find x by taking the mean of x A and x B ? 23

In my class, there are 16 students and we have a mean grade of 7.1. If the mean of the girls is 8 and the mean of the boys is 6.4, what is the proportion of girls and boys in the class?

The mean of three numbers is 7 units more than the smallest number, and 10 units less than the largest number. If we know that the median is 7, which are the numbers?

3.5 - 4.5

4.5 - 5.5

5.5 - 6.5

6.5 - 7.5

Let’s think!

7.5 - 8.5

8.5 - 9.5

a) Find the mean and the standard deviation. b) What percentage of animals weighed between x – σ and x + σ? And more than x + σ? And less than x – σ? Make a reasonable estimation.

62 186 530 812 953 860 507 285 126 29

Let’s say that soldiers with heights between x + σ and x + 2σ are tall, soldiers with heights between x – σ and x – 2σ are short, and soldiers with heights between x – σ and x + σ are normal. Estimate what percentage are tall, short and normal. What percentage are very tall and very short?

These data show the birth weights of a particular animal species: weight

These are the heights of 4 350 soldiers:

What happens to the x and σ of a distribution if we add the same number to all its data points? And if we multiply all its data points by the same number? Test your theories using these data points: 4, 3, 6, 7, 5, 4, 5, 3, 2, 6, 5 303

P O H S K R O W S H T A M READ AND LEARN The Gaussian bell curve Many statistical variables related to real-world phenomena, with random components, have very low frequencies at values that are very far from the central value, and the frequencies get higher as they get closer to the central values. c This is true, for example, for the distribution of heights of a group of people: very few are really short (under 1.55 m); very few are really tall (over 1.95 m) and most people are somewhere in between (around 1.75 m). It is also true for distributions of weights, clothing sizes, temperature data, river b flows, energy costs, income… This type of distribution, in its ideal form, is what is called a normal a distribution. Its graph (values-frequencies) is known as the Gaussian bell curve due to its shape and because Gauss was the first mathematician to apply these concepts in practical studies for other sciences. x

• What is the mean of the distribution in the red graph? And the median? And the mode? • What are the values for those parameters in the green graph? And in the purple graph?

PARADOX A statistician is a scientist who can drown in a lake averaging 30 cm in depth.

THINK AND GENERALISE

This die has two hidden faces and four visible faces. How many dots are on its hidden faces?

Here, there are four hidden faces. How many dots are on those four faces?

And here?

And if there were x dice? The number of dots on the hidden faces is a function of the number of dice. Write and graph a function relating the number of dice, x, to the number of dots on the hidden faces, y . 304

And here?

UNIT 14

PRACTICE MAKES PERFECT! • A group of 17 boys and girls of the same age are organising a big trip. All their parents come to the first meeting. The average age of the parents is 45 years old. However, if we take the group formed by both parents and children, the average age is 35 years old. How old are the boys and girls? • Place 10 toy soldiers on a table so that there are 5 rows of 4 soldiers each.

• A cook wants to fry three steaks. Each steak has to cook for five minutes on each side in the frying pan. But only two steaks fit in each frying pan. How should she do it to take the shortest time possible? • Copy the image on the right and draw a broken line of five segments that passes through all thirteen dots.

SELF-ASSESSMENT

anayaeducacion.es Answer key.

1 Find the mean, the median, the standard deviation

and the variation coefficient for each of these distributions. Which is most disperse?

3 The grades of a class on a 5-question exam are:

a) 6, 9, 1, 4, 8, 2, 3, 4, 4, 9 b) 120, 95, 87, 111, 116, 82, 121, 92, 76 c) 987, 1 010, 1 004, 995, 998, 1 001, 999, 982 2 Calculate x , σ and the VC for these distributions:

a) Number of days the students in a class went to the library: n.° of days

frequency

b) Time, in minutes, that a doctor’s patients spent in the waiting room on a particular day: time

(min)

frequency

From 1 to 9

From 9 to 17

From 17 to 25

From 25 to 33

From 33 to 41

From 41 to 49

Commitment

3 3 2 4 5

4 1 3 3 2

3 2 4 4 3

1 2 0 5 3

2 0 3 5 3

3 5 2 1 4

a) Calculate the median and the quartiles. b) Draw the box plot. 4 The height distributions of the members of three

school basketball teams, A, B and C, are shown in these graphs: I

III

170 175 180 185 165 170 175 180 185 165 170 175 180 185 190

Their parameters are: a

177.8

176.8

174.6

6.4

3.2

4.5

Which graph represents each team? 5 This week, I studied: 3 h on Monday, 2 h on Tuesday,

2.5 h on Wednesday, 5 h on Thursday, 2 h on Friday, and 3.5 h on Saturday. a) How long do I have to study on Sunday for the mean to stay the same? And for the median to stay the same? b) How long do I have to study for the mean to be 5 h?

Watch the video for target 16.9. Think of something you can do to contribute to achieve that goal. Make a commitment to put your idea into practice.

305

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