Statistical concepts
What is your class's favourite sport: hurling, camogie, Gaelic football or soccer? How many brothers and sisters does each classmate have? To answer these questions you don't need to ask the whole school, but you do have to choose well who you ask and what you write down. That is the first thing statistics teaches.
Population, sample and individual
Why not always ask everybody? Because it can take a lot of time or cost a lot of money, or because testing destroys the item (a company testing how long its light bulbs last cannot burn out every bulb). When a whole population is studied, it is called a census. In Ireland the CSO runs a census of every household.
A video game company wants to know what the favourite genre of students at your school is. In order not to ask all $800$ students, they decide to survey $10$ students…
A video game company wants to know what the favourite genre of students at your school is. In order not to ask all $800$ students, they decide to survey $10$ students from each class in the school.
Identify the following elements of the statistical study:
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Population:
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Sample:
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Individual:
To improve the quality of the food in the cafeteria of a large school with $1200$ students, the school board decides to ask the opinion of the first $5$ students from…
To improve the quality of the food in the cafeteria of a large school with $1200$ students, the school board decides to ask the opinion of the first $5$ students from each class who enter the school on Monday morning. Identify the following elements of this statistical study:
- a) Population.
- b) Sample.
- c) Individual.
A famous influencer wants to know what Irish teenagers think about the new time limit on a social media platform. To do this, they post a poll on their Instagram…
A famous influencer wants to know what Irish teenagers think about the new time limit on a social media platform. To do this, they post a poll on their Instagram "stories" for 24 hours. $15,400$ people take part.
- a) Identify the population under study and the sample used.
- b) Do you think the sample is representative of the population? Reason your answer by analysing two possible factors that could invalidate the results (biases).
- c) Propose a more reliable method for choosing a sample of $500$ individuals that truly represents all teenagers in Ireland.
Types of variable
A statistical question is a question that can be answered by collecting data that varies, for example: "Which sport do students in our school prefer?" or "How many brothers and sisters do 1st Year students have?" ("What is Aoife's favourite sport?" is not statistical, as it has only one answer.)
Data you collect yourself, for example with a survey, is primary data. Data that someone else has already collected and published, for example by the CSO, is secondary data.
The statistical variable is what we measure about each individual. In Junior Cycle we call the two main types categorical (qualitative) and numerical (quantitative):
| Type | Values | Examples |
|---|---|---|
| Categorical | Qualities, not numbers | Favourite sport, eye colour, favourite band |
| Numerical, discrete | Numbers that are counted | Number of brothers and sisters, pets, goals |
| Numerical, continuous | Numbers that are measured | Height, weight, time |
Frequency
The frequency of a value is the number of times it appears in the data. The relative frequency is the frequency divided by the total number of data values (it can be written as a fraction, a decimal or a percentage).
Random or deterministic?
An experiment is random if we cannot predict the result with certainty, even though we know the possible outcomes (the list of them is the sample space). It is deterministic if the result is certain from the start.
Many things happen in our daily lives, but not all of them depend on chance. State which of the following situations are random experiments and which are deterministic:…
Many things happen in our daily lives, but not all of them depend on chance. State which of the following situations are random experiments and which are deterministic:
- a) Tossing a coin and seeing whether it lands on heads or tails.
- b) Calculating how much $3$ cinema tickets will cost if each one is worth $8$ euros.
- c) Asking a classmate what their favourite colour is.
- d) Dropping a stone from a balcony and seeing if it falls downwards.
- e) Drawing a card from a standard deck of cards without looking.
Among quantitative variables, we distinguish between discrete ones (when they take isolated values, usually whole numbers) and continuous ones (when they can take any…
Among quantitative variables, we distinguish between discrete ones (when they take isolated values, usually whole numbers) and continuous ones (when they can take any value in an interval, such as decimals). The figure shows the difference: above, isolated values; below, a stretch of the line in which any value fits, including decimals.
Classify these variables as discrete or continuous:
- a) The number of goals a team scores in each league match.
- b) The exact time a swimmer takes to cover $50$ meters.
- c) The number of books in the students' backpacks.
- d) The height of the members of the Irish national basketball team.
A well-known streamer wants to know what their followers' favourite video game in Ireland is. To do this, they conduct a quick poll during a livestream in which $1500$…
A well-known streamer wants to know what their followers' favourite video game in Ireland is. To do this, they conduct a quick poll during a livestream in which $1500$ people participate. Classify the following variables collected in the study as qualitative or quantitative:
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a) Title of the favourite video game.
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b) Number of hours they play per week.
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c) Console or platform used (PC, PS5, Switch, etc.).
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d) Amount of money spent on "skins" or battle passes per month.
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e) Rating of the livestream from 1 to 10.
Classify the following statistical variables as qualitative or quantitative. In the case of quantitative variables, reflect on whether it makes sense to perform…
Classify the following statistical variables as qualitative or quantitative. In the case of quantitative variables, reflect on whether it makes sense to perform arithmetic operations (such as the mean) with their values:
- a) The postcode of the residence of students at a secondary school.
- b) The serial number of the computers in the library.
- c) The daily mean temperature recorded in the school playground during the month of January.
- d) The shirt number of the players in a basketball team.
- e) The time, expressed in seconds, taken by a group of swimmers to cross a $50$-metre swimming pool.
Note: Think about whether the "number" acts as a label or as a real measurement.
Back to the start: To answer those questions, the population is your class; favourite sport is a categorical variable and number of brothers and sisters is a discrete numerical variable.