Exercises: Statistical variable

A group of researchers wants to find out the favourite music group of Spanish teenagers aged between 12 and 16. To do this, they conduct a survey among 500 students from several secondary schools in Madrid. State:

1. a) The population under study.
2. b) The chosen sample.
3. c) The individual.

Classify the following statistical variables as qualitative or quantitative:

1. a) The favourite colour of a mobile phone case.
2. b) The number of followers on a TikTok account.
3. c) The time, in minutes, it takes you to get from home to school.
4. d) The brand of trainers preferred by the students in your class.

Within quantitative variables, we must distinguish between discrete and continuous. State the type of each of the following:

1. a) The number of cousins each student has.
2. b) The capacity, in litres, of the water bottles sold in the canteen.
3. c) The number of WhatsApp messages you receive in a day.
4. d) The weight of the backpacks of 3rd ESO students.

Match each of the following situations with the nature of its statistical variable (Qualitative, Discrete Quantitative or Continuous Quantitative):

1. The sports played at the municipal sports centre.

2. The maximum daily temperature recorded in your city during the month of July.

3. The number of television channels tuned in at home.

Imagine that you want to conduct a study on the amount of money young people in your neighbourhood spend weekly on leisure.

1. a) What type of variable is "money spent"?
2. b) If $1.200$ young people live in the neighbourhood and you ask $60$, what is the sample size?
3. c) Would it be representative if you only ask young people standing outside a cinema on a Saturday afternoon? Reason your answer.

For each of the following survey questions, identify whether the response would result in a nominal qualitative, discrete quantitative or continuous quantitative variable:

1. a) "Which social media application do you use most frequently?"
2. b) "How many hours a day (with decimals) do you spend studying?"
3. c) "How many times have you been to the cinema in the last month?"
4. d) "Which is your preferred operating system (Android or iOS)?"

A study is to be conducted on fruit consumption in a secondary school. Determine whether the following samples are random (chosen at random and representative) or biased:

1. a) Asking the 25 students who arrive first at school in the morning.
2. b) Choosing 5 students at random from the class register of each class in the school.
3. c) Asking only the students who are buying food in the canteen.
4. d) Drawing 50 enrolment numbers at random among all the students in the school and asking their owners.

Look at the following table of collected data on the reading habits of a group of students and classify each column according to the type of statistical variable it represents:

Student's name	Books read per year	Favourite genre	Mean reading time (h)
Lucía	$12$	Fantasy	$0,5$
Marcos	$4$	Science Fiction	$1,2$
Elena	$0$	Does not read	$0$

1. a) Books read per year.
2. b) Favourite genre.
3. c) Mean daily reading time.

Marcos has conducted a small survey among his classmates for a Mathematics project on lifestyle and leisure habits. In his preliminary report, he classified the following four variables:

1. Home postal code: "Continuous quantitative, because it consists of a five-digit number and can take very high values."

2. Jersey number of their favorite athlete: "Discrete quantitative, because it is always an integer without decimals."

3. Daily time spent using the mobile phone: "Discrete quantitative, because when asked, everyone responds with whole numbers of minutes, such as $45$ or $60$ minutes."

4. Level of satisfaction with the neighborhood sports facilities (Low, Medium, High): "Qualitative."

Identify the conceptual errors Marcos made, explain why his justification is incorrect in each case, and write the correct and complete classification of the four variables.

The management team of your high school wants to carry out a statistical study on sustainable mobility and the transport habits of the school community. Design the study questionnaire yourself, meeting exactly the following conditions:

1. a) Write a question whose answer is a nominal qualitative variable (without any natural hierarchical order).
2. b) Write a question whose answer is a discrete quantitative variable, specifying what specific element is being counted.
3. c) Write a question whose answer is a continuous quantitative variable, indicating the physical unit of measurement to be used.
4. d) Define precisely the entire population of the study, a specific individual, and propose a reasonable sample size if a total of $750$ students are enrolled in the school.

A team of researchers wants to carry out a study on energy drink consumption among teenagers in a city with $50\,000$ inhabitants. To do so, they select $250$ students aged between $12$ and $18$ from different secondary schools and ask them how many cans they have consumed in the past week.

Identify the following elements of the statistical study:

1. a) The target population of the study.
2. b) The chosen sample.
3. c) The individual.
4. d) The sample size.
5. e) The statistical variable and its type.

The following table shows several variables collected in a survey carried out at an e-sports club. Classify each of them as qualitative, quantitative discrete or quantitative continuous:

Variable	Type of variable
Favourite video game
Weekly training hours
Number of peripherals (keyboard, mouse...) owned
Brand of monitor used
Internet connection speed (Mbps)
World ranking position

To find out citizens' opinions on the construction of a new cycle lane in the city centre, a local newspaper carries out a survey at the entrance of a shop specialising in mountain biking.

1. a) Do you think the chosen sample is representative of the city's population? Reason your answer.
2. b) Is there any type of bias in this choice? Explain why.
3. c) Propose a sampling method that would be more suitable to obtain reliable results.

In a school there are $600$ enrolled students. We want to draw a representative sample of $60$ students using stratified sampling by year group to carry out a study on reading habits. The breakdown by year group is as follows:

• $1º$ ESO: $180$ students.

• $2º$ ESO: $160$ students.

• $3º$ ESO: $140$ students.

• $4º$ ESO: $120$ students.

Calculate how many students from each year group should be chosen so that the sample maintains the proportion of the population.

A biologist is measuring the growth of a species of fern in a greenhouse. After one month, he records the heights (in cm) of $20$ specimens. The data obtained form a continuous quantitative variable. The histogram shows the heights grouped in intervals of $5$ cm.

1. a) Explain the difference between a discrete quantitative variable and a continuous one using this example.

1. b) If the data are grouped in intervals such as $[10, 15), [15, 20), [20, 25)$..., what would be the class mark of the second interval?

1. c) What value does the class mark represent in statistical calculations?

A study on air quality in different locations of an autonomous region is carried out. To do so, measuring stations are installed to record the concentration of $PM_{10}$ particles (in $\mu g/m^3$).

State whether the following statements are true or false, justifying each response:

1. a) The variable "particle concentration" is qualitative because it indicates air quality.
2. b) If we measure the concentration at $15$ randomly chosen points, the individual is each of those measuring points.
3. c) The number of inhabitants in the areas where measurements are taken is a quantitative continuous variable.
4. d) The variable "pollution level" with values (Low, Medium, High) is a qualitative ordinal variable.

A local council wants to know the number of pets in households in a newly built neighbourhood where $1\,200$ families live. To avoid asking everyone, they decide to send a survey to the $120$ families living at the first addresses on the alphabetical list of streets.

1. a) State which is the population and which is the sample.
2. b) Is this a simple random sampling method? Why?
3. c) Identify the variable and state its type.
4. d) If $15\%$ of the families in the sample do not have a pet, how many families in the neighbourhood are estimated to have at least one pet?

Imagine that you work in the marketing department of a music streaming platform. You want to investigate the relationship between users' age and the style of music they listen to most.

To do so, you design a database with the following fields for each user:

1. Age (completed years).

2. Daily minutes of listening.

3. Preferred music genre (Pop, Rock, Trap, Others).

4. Level of satisfaction with the service (from $1$ to $5$).

1. a) Classify each of the four variables above (qualitative or quantitative; and, within each type, nominal or ordinal, discrete or continuous).

1. b) Explain which type of chart (bar chart, pie chart or histogram) you think would be the most suitable to represent the data of the variable "Preferred music genre" and why. Also state whether any of the three would not be suitable and justify your answer.

You can do it on paper and send a photo with your answer.

The State Meteorological Agency (AEMET) has automatic measuring stations distributed throughout Spain. During a heatwave episode in July, a team of analysts collects data at $16:00\text{ h}$ from $45$ selected stations in Andalusia to study the impact of extreme temperatures. For each station, the following data are recorded:

• The province in which it is located.

• The air temperature recorded in the shade (in $^{\circ}\text{C}$, with tenths-of-a-degree precision).

• The number of wind gusts exceeding $40\text{ km/h}$ recorded in the last three hours.

• The altitude above sea level at which the station is installed (in metres).

• The level of the active weather warning issued for that area (Green, Yellow, Orange, Red).

1. a) Identify the study population, the selected sample, the sample size, and the individual.
2. b) Classify each of the five recorded variables into qualitative (specifying whether it is nominal or ordinal) or quantitative (specifying whether it is discrete or continuous).
3. c) Justify whether it would make sense to replace this sample study with a complete census, and argue one advantage and one disadvantage of carrying out such a census.

In a secondary school with $800$ students ($200$ in $1.^{\circ}$ of ESO, $200$ in $2.^{\circ}$, $150$ in $3.^{\circ}$, $150$ in $4.^{\circ}$, and $100$ in Bachillerato), the School Council wants to find out the level of satisfaction with the menu and prices of the school cafeteria in order to decide whether to renew the service contract. They have the time and budget to survey $80$ students.

Three sampling methods are proposed:

• Method A: Interview the first $80$ students who line up at the cafeteria next Monday during break time.

• Method B: Leave a ballot box with slips at the reception desk and ask over the loudspeaker for $80$ people to come and vote voluntarily.

• Method C: Obtain the school's official student list and select by random draw $20$ students from $1.^{\circ}$, $20$ from $2.^{\circ}$, $15$ from $3.^{\circ}$, $15$ from $4.^{\circ}$, and $10$ from Bachillerato.

1. a) Critically analyse methods A and B: explain what biases they exhibit and why they do not guarantee a representative sample.
2. b) Prove mathematically that method C distributes the sample proportionally to the actual size of each year group, and justify why it is the most rigorous procedure of the three.
3. c) If the cafeteria were only used by students in $3.^{\circ}$, $4.^{\circ}$, and Bachillerato, would the initially defined population still be appropriate? Redefine the population for that case.

A video game developer is analysing player behaviour in a Beta stage. To optimise performance, they record the following data for each user:

1. The name of the server they connect to (North, South, East, West).

2. The unique account ID code (an 8-digit number).

3. The response time or ping measured in milliseconds.

4. The number of times the player has pressed the "pause" button in an hour.

5. The user's satisfaction level, rated as: "Very dissatisfied", "Dissatisfied", "Neutral", "Satisfied" or "Very satisfied".

Reason and classify each of the variables above. Pay special attention to variable number 2: does the fact that it is a number imply that it is a quantitative variable? Justify your answer.

A courier company wants to improve its efficiency. To do so, it defines the population as "all packages sent in Spain during one year". As it is impossible to study them all, they select a sample of 500 packages.

For each package, they record: weight in kg, volume in $m^3$, destination province and number of delivery attempts.

1. a) Identify the population, the sample and the individuals of the study.
2. b) Classify the recorded variables.
3. c) If, of the 500 packages in the sample, 450 were collected solely from a branch in an affluent area of Madrid, can we say that the sample is representative to draw conclusions about the service across the whole of Spain? Justify your answer using the concept of statistical bias.

In a biology laboratory, the growth of a bacterial colony is studied. The researcher decides to record the culture temperature every hour. The temperature can take values such as $36,42 ^\circ C$, $36,45 ^\circ C$, etc. However, to simplify the final report, the researcher groups the results into three categories: "Cold" (less than $30 ^\circ C$), "Optimal" (between $30 ^\circ C$ and $40 ^\circ C$) and "Excessive heat" (more than $40 ^\circ C$).

1. a) What type of variable is the originally measured temperature?
2. b) What type of variable is the one that finally appears in the report after grouping the data?
3. c) Has information been lost in this process? Explain what advantage and what disadvantage doing this transformation has from a statistical point of view.

Imagine that you want to carry out a statistical study on social media use among teenagers in your city. You must design a survey containing exactly:

1. A qualitative nominal variable.

2. A qualitative ordinal variable.

3. A quantitative discrete variable.

4. A quantitative continuous variable.

Write a question for each case that is realistic and useful for your research. Explain why each question corresponds to the type of variable requested.

A local council wants to know the level of acceptance of a new traffic rule. To do so, they set up an information stand at the gate of a school at 9:00 in the morning and ask the first 100 adults who pass by.

1. a) Define the target population for the study.
2. b) Describe the selected sample.
3. c) Critically analyse the sample selection method. Do you think all citizens had an equal probability of being chosen? Propose a stratified random sampling method that would be fairer for this study.

Quantitative continuous variables are often presented as discrete due to the precision of measuring instruments. A precision sensor measures the thickness of steel sheets produced in a factory. The sensor displays values such as $2,00$ mm, $2,01$ mm, $2,02$ mm...

1. a) Even though the sensor only displays two decimal places, why is thickness considered a quantitative continuous variable rather than discrete?
2. b) If the factory produces 10 000 sheets per day and quality control decides to measure only those sheets manufactured between 12:00 and 12:15, will the sample be representative of the total daily production? Reason your answer, mentioning possible factors that could vary throughout the day.

A study is carried out on the hearing health of young people. The volume intensity at which they listen to music with headphones is measured (in decibels, dB).

The figure shows the interval in which the measured values lie and the 85 dB threshold set by the medical team.

1. Classify the variable "Sound intensity in dB".

2. If the values obtained range between $65,4$ dB and $105,9$ dB, explain why it would not make sense to treat this variable as a nominal qualitative variable.

3. The medical team decides that, if the intensity exceeds $85$ dB, the risk is "High", and if it is lower, the risk is "Low". Classify this new variable "Hearing risk".

A streaming platform analyses its subscribers' data to improve its recommendations. Classify the following variables used by its algorithm:

1. a) Total time (in seconds) a user stays watching a video before closing it.
2. b) Type of device used (Smart TV, Smartphone, Tablet, PC).
3. c) Number of times a user has shared a link from the platform in the last month.
4. d) Average rating of watched films (from 1 to 5 stars, allowing half stars such as 3.5).
5. e) Position occupied by a series in the country's "Top 10" most watched.

Discuss whether variable (d) could be considered continuous or discrete depending on how the original data are collected (individual votes vs. calculated average).

According to data from the National Statistics Institute (INE), the age demographic structure of a municipality with $30\,000$ inhabitants is divided into: under $18$ years of age ($18\,\%$), adults from $18$ to $64$ years of age ($62\,\%$), and people aged $65$ or older ($20\,\%$). The department of mobility commissions a street survey with a sample size of $n = 300$ people to estimate the weekly frequency with which residents use the bike lane.

Before carrying out the fieldwork, the technician estimates that a sample of $300$ people will be fully representative as long as it proportionally reflects the three age brackets of the municipality. The interviewers conduct the interviews on a working Tuesday between $10:00$ and $12:30$ in the morning in the town's central square, obtaining a sample with the following actual participants: $12$ under $18$ years of age, $114$ people between $18$ and $64$ years of age, and $174$ people aged $65$ or older.

1. a) Carry out the mathematical check: calculate how many individuals from each age group should have made up the sample of $300$ people so that it faithfully represented the population of the municipality according to INE data.
2. b) Check the percentage composition of the sample actually obtained and determine which population group is severely overrepresented and which is underrepresented.
3. c) Reason in detail which factors of the data collection design (time, day, and location) caused this deviation, and, assuming that people aged $65$ or older use the bike lane less often than other residents, justify whether the variable under study ("frequency of bike lane use") will be overestimated or underestimated compared to the reality of the municipality.

A sociology of education team wants to investigate the digital habits of young people between $12$ and $16$ years of age in an autonomous community that has $120\,000$ people in that age bracket. Before starting data collection, two methodological debates arise:

Debate 1: Operational definition of the variable

• Researcher A proposes asking: "How many whole hours do you spend on screens per day? (Closed options: $1$, $2$, $3$, $4$, or more than $4$)".

• Researcher B proposes asking participants to check the 'Digital Wellbeing' app log on their phone and record the exact total screen time from the previous day, recorded in minutes and seconds (for example, $194\text{ min } 35\text{ s}$).

Rigorously classify the type of statistical variable generated by each proposal. Argue which of the two options provides greater mathematical precision and what loss of information or distortion the study suffers if proposal A is adopted.

Debate 2: Choice of sampling method

Resources are available to survey $600$ young people. Two options are considered:

• Option 1: Launch a sponsored campaign on social media with an open web form, stopping the survey as soon as the $600$ received responses are reached.

• Option 2: Select $15$ secondary schools in the community by random draw, ensuring there are public and state-subsidised private schools, from both rural and urban areas, and randomly choose $40$ students from the enrolment lists in each school.

Compare both options by analysing the three requirements of a representative sample: random selection, coverage of relevant groups, and control of non-responses. Justify which of the two options should be chosen.