Statistical variable
What is the favourite band in your school? How many hours a day do students in your year use their mobile phones? To answer these questions you do not need to ask everyone, but you do have to do it properly. Statistics starts by deciding who we ask and what we measure. This is the first step of a statistical investigation, like the one you will carry out for your Junior Cycle Statistical investigation (CBA 2).
Population, sample and individual
Why not always study the whole population? Because it can be very expensive, slow or even impossible (to check how long batteries last you have to run them down: you cannot test all of them). When the whole population is studied, it is called a census.
A simple random sample gives every individual the same chance of being chosen, for example by drawing names from a hat or using random numbers. Asking the first 50 people in a queue is a convenience sample, not a random one. The number 'how many in the whole population have a feature' can then be estimated by scaling up the sample proportion: if 20% of a random sample of 100 families have a dog, we estimate about 20% of all families do.
A group of researchers wants to find out the favourite music group of Irish teenagers aged between 12 and 16. To do this, they conduct a survey among 500 students from…
A group of researchers wants to find out the favourite music group of Irish teenagers aged between 12 and 16. To do this, they conduct a survey among 500 students from several secondary schools in Dublin. State:
- a) The population under study.
- b) The chosen sample.
- c) The individual.
Imagine that you want to conduct a study on the amount of money young people in your neighbourhood spend weekly on leisure. a) What type of variable is "money spent"? b)…
Imagine that you want to conduct a study on the amount of money young people in your neighbourhood spend weekly on leisure.
- a) What type of variable is "money spent"?
- b) If $1,200$ young people live in the neighbourhood and you ask $60$, what is the sample size?
- c) Would it be representative if you only ask young people standing outside a cinema on a Saturday afternoon? Reason your answer.
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…
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.
- a) Do you think the chosen sample is representative of the city's population? Reason your answer.
- b) Is there any type of bias in this choice? Explain why.
- c) Propose a sampling method that would be more suitable to obtain reliable results.
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…
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:
- a) Asking the 25 students who arrive first at school in the morning.
- b) Choosing 5 students at random from the class register of each class in the school.
- c) Asking only the students who are buying food in the canteen.
- d) Drawing 50 enrolment numbers at random among all the students in the school and asking their owners.
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…
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:
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1st Year: $180$ students.
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2nd Year: $160$ students.
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3rd Year: $140$ students.
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4th Year: $120$ students.
Calculate how many students from each year group should be chosen so that the sample maintains the proportion of the population.
Designing an investigation: how long does it take us to get to school?
We want to find out the usual journey time of the 48 students in two 3rd Year classes of 24. Before asking, we agree on what we measure: minutes from leaving home until entering the school on a normal school day, including waiting time for the bus or Luas. A question such as "Do you take too long?" collects a different thing, an opinion, not the time.
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We draw 4 students from each class at random: a sample of 8 with the same proportions as the population. We do not choose only friends or only those who arrive earliest.
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We record the class and the minutes without names. If someone does not respond, we note the absence; we do not make up their time. We agree in advance how to repeat the draw among those still to be selected.
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We check the units, impossible answers and what each person understands by "usual". We do not discard a high time just because it looks strange: we check it first.
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We organise the data, calculate measures and write down what we can conclude and what we cannot. The sample is small and does not automatically represent other year groups or schools.
In the next sections you can analyse a fictitious sample of 8 times to practise. In a real investigation we would also keep the information needed to review how each group was selected.
Types of variables
The statistical variable is the characteristic we study. There are two main types, categorical and numerical:
| Type | Values | Examples |
|---|---|---|
| Categorical | Categories, not numbers | Favourite sport, eye colour, social network |
| Discrete numerical | Numbers that are counted (isolated values) | Number of siblings, of pets, of goals |
| Continuous numerical | Numbers that are measured (any value in an interval) | Height, weight, time, temperature |
Categorical variables can be nominal (no natural order: eye colour, county) or ordinal (the categories have a natural order: 'Very dissatisfied' to 'Very satisfied', or position in a league table).
A numerical variable can be grouped into categories (for example, temperature into 'Cold', 'Optimal', 'Hot'). Once grouped, the new variable is categorical and ordinal, even though the original was continuous. How a variable is defined also matters: 'whole hours on screens' is discrete, but 'exact screen time' is continuous.
Classify the following statistical variables as qualitative or quantitative: a) The favourite colour of a mobile phone case. b) The number of followers on a TikTok…
Classify the following statistical variables as qualitative or quantitative:
- a) The favourite colour of a mobile phone case.
- b) The number of followers on a TikTok account.
- c) The time, in minutes, it takes you to get from home to school.
- 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: a) The number of cousins each student has.…
Within quantitative variables, we must distinguish between discrete and continuous. State the type of each of the following:
- a) The number of cousins each student has.
- b) The capacity, in litres, of the water bottles sold in the canteen.
- c) The number of WhatsApp messages you receive in a day.
- d) The weight of the backpacks of 3rd Year students.
For each of the following survey questions, identify whether the response would result in a nominal qualitative, discrete quantitative or continuous quantitative…
For each of the following survey questions, identify whether the response would result in a nominal qualitative, discrete quantitative or continuous quantitative variable:
- a) "Which social media application do you use most frequently?"
- b) "How many hours a day (with decimals) do you spend studying?"
- c) "How many times have you been to the cinema in the last month?"
- d) "Which is your preferred operating system (Android or iOS)?"
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…
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 |
Cian 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…
Cian 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:
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Home postal code: "Continuous quantitative, because it consists of a five-digit number and can take very high values."
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Jersey number of their favorite athlete: "Discrete quantitative, because it is always an integer without decimals."
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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."
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Level of satisfaction with the neighborhood sports facilities (Low, Medium, High): "Qualitative."
Identify the conceptual errors Cian made, explain why his justification is incorrect in each case, and write the correct and complete classification of the four variables.
Back to the beginning: To answer the questions about favourite band or mobile phone use, first define the population (students in your school or of your age), take a random and representative sample, and identify each variable: favourite band is categorical and time on the phone is continuous numerical.