Statistics Basics: Populations, Samples, and Variables
What is the most popular lunch in your school cafeteria? There are 800 students, and nobody wants to ask all of them. So who do you ask, and what exactly do you write down? Statistics starts with these two questions, and by the end of this page you will be able to answer them.
Population, Sample, and Individual
Every statistical study is about a group. The words below tell you which group, and which part of it you actually look at.
How do you read a diagram like this one? The big box is the population, and the highlighted part is the sample. Count all the dots to get the population size (24 here), and count the highlighted dots to get the sample size (4 here). When a problem gives you the numbers instead, the total number of items is the population size and the number that were actually checked is the sample size.
Notice that a population does not have to be people. It can be the screws a machine makes, the trees in a park, or the rainy days in a year. The individual is always one item of that collection.
A video game company wants to know the favorite genre of the students at your school. To avoid asking all $800$ students, they decide to survey $10$ students from each…
A video game company wants to know the favorite genre of the students at your school. To avoid asking 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 council 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 council 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:
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a) Population.
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b) Sample.
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c) Individual.
Look at the following diagram representing a study on the health of trees in a nature reserve that has $5000$ specimens. The forestry team has analyzed $150$ of them. a)…
Look at the following diagram representing a study on the health of trees in a nature reserve that has $5000$ specimens. The forestry team has analyzed $150$ of them.
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a) What is the population size?
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b) What is the sample size?
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c) If the researchers record whether the tree has a pest or not, what type of variable are they studying?
Census or Sample?
When you study every individual in the population, it is called a census. When you study only part of it, you use a sample. Why not always do a census? Sometimes it is just too big, too slow, or too expensive. Sometimes it is simply impossible.
Sometimes it is impossible to study the whole population and that is why we use a sample. Explain in which of these cases you think it is mandatory to use a sample…
Sometimes it is impossible to study the whole population and that is why we use a sample. Explain in which of these cases you think it is mandatory to use a sample instead of asking the whole population:
- a) We want to know the mean grade of the last math exam in your 6th grade class.
- b) A light bulb factory wants to know how many hours they last before burning out.
- c) The principal of the school wants to know how many students have come to class today.
- d) A hospital wants to analyze the blood group of all the inhabitants of a state.
A cell phone manufacturer wants to test the screen resistance of a new model. The factory produces $10,000$ phones a day. For the study, $50$ phones are selected and…
A cell phone manufacturer wants to test the screen resistance of a new model. The factory produces $10,000$ phones a day. For the study, $50$ phones are selected and dropped repeatedly from a height of $1,5$ feet until the screen breaks, recording how many drops each one withstands.
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a) Explain why in this case it is strictly necessary to use a sample instead of studying the whole population.
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b) Identify the studied variable and classify it.
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c) What would happen to the production if the manufacturer decided not to use a sample and analyze every individual in the population?
Representative Samples and Bias
A sample is only useful if it is representative. That means it looks like the whole population, only smaller. The best way to get one is a random sample, where every individual has the same chance of being chosen. You could number everyone and draw numbers out of a hat or use a random number generator. To be extra fair, you can also pick some students from every grade.
A sample that leans toward one kind of answer has bias. Here are the most common causes:
| Cause of bias | What goes wrong | Example |
|---|---|---|
| Convenience sample | You ask whoever is easiest to reach. | Asking only your lunch table, or only the first students who walk in. |
| Voluntary response | Only people with strong opinions answer. | An online poll anyone can click. |
| Left-out group | Part of the population has no chance to be chosen. | A phone poll that skips people without a phone. |
How do we collect the data?
Once the sample is chosen, we need a tool to get the data:
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Survey or questionnaire: you ask people (favorite sport, opinions).
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Direct observation: you watch and record (cars passing the school in an hour).
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Measuring device: you use an instrument (a ruler, a scale, a stopwatch, a thermometer).
Analyze the following situation: A city council wants to know if citizens are happy with the new bike lane. To save costs, they decide to ask only the people riding on…
Analyze the following situation: A city council wants to know if citizens are happy with the new bike lane. To save costs, they decide to ask only the people riding on the bike lane on a Saturday morning.
- a) Is the sample random?
- b) What type of bias do you think this study will present? Will the results overestimate or underestimate the general satisfaction of the city?
- c) Propose a way of choosing the sample that includes people who do not use a bicycle so that the study is objective.
- d) If the variable is "Degree of satisfaction (None, Little, A lot)", what type of variable is it? Could the arithmetic mean of this data be calculated? Justify why.
A famous influencer wants to know what American teenagers think about the new time limit on a social network. To do this, they post a poll on their Instagram stories for…
A famous influencer wants to know what American teenagers think about the new time limit on a social network. To do this, they post a poll on their Instagram stories for 24 hours. $15.400$ people participate.
- a) Identify the population under study and the sample used.
- b) Do you think the sample is representative of the population? Justify your answer by analyzing two possible factors that could invalidate the results (biases).
- c) Propose a more reliable method to choose a sample of $500$ individuals that truly represents all teenagers in the United States.
Imagine you want to carry out a statistical study to convince the principal of your school that classes should start 15 minutes later to improve performance. a) Clearly…
Imagine you want to carry out a statistical study to convince the principal of your school that classes should start 15 minutes later to improve performance.
- a) Clearly define the population and a suitable sample that is not biased.
- b) Design three different variables that you could measure: one qualitative, one quantitative that is a count (integers), and another quantitative that is a measurement (with decimals).
- c) Indicate what data collection instrument you would use for each one (survey, direct observation, or measuring device).
Qualitative and Quantitative Variables
A variable is the characteristic we record for each individual. Favorite lunch, number of pets, and height are all variables. There are two big types.
Some qualitative variables have an order, such as satisfaction (Very dissatisfied, Neutral, Satisfied, Very satisfied) or T-shirt size (S, M, L). They are still categories. You can sort them, but you cannot really add them.
A study usually records several variables about each individual at the same time. For example, a science class grows 100 bean plants and records the height of each plant in inches, the color of its flowers, and the number of pods it makes. That is three variables for the same individuals: height is quantitative (measured), flower color is qualitative, and the number of pods is quantitative (counted). When you are asked to design variables, pick one of each kind and say how you would get the data: a survey, observation, or a measuring device.
Classify the following statistical variables as qualitative (if they express a quality or characteristic) or quantitative (if they are expressed by a number): a) The…
Classify the following statistical variables as qualitative (if they express a quality or characteristic) or quantitative (if they are expressed by a number):
- a) The name of the most used apps on your friends' cell phones.
- b) The number of siblings each student in your class has.
- c) The brand of sneakers worn by the players of a team.
- d) The maximum temperature reached in your city each day this week.
- e) The preferred sport of the members of an athletics club.
An online clothing store wants to know the satisfaction level of its customers (Very dissatisfied, Dissatisfied, Neutral, Satisfied, Very satisfied). a) Classify this…
An online clothing store wants to know the satisfaction level of its customers (Very dissatisfied, Dissatisfied, Neutral, Satisfied, Very satisfied).
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a) Classify this variable. Is it quantitative or qualitative?
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b) Although words are used, numbers are sometimes assigned ($1$ to $5$) to calculate the mean. Do you think this makes the variable quantitative? Justify your answer.
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c) If the store sends the survey to $500$ customers who bought yesterday, who is the "individual" in this study?
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 ZIP code of the residence of the students at a school.
- b) The serial number of the computers in the library.
- c) The daily mean temperature recorded in the schoolyard during the month of January.
- d) The jersey number of the players on a basketball team.
- e) The time, expressed in seconds, that a group of swimmers takes to cross a $50$-foot pool.
Note: Think about whether the "number" acts as a label or as a real measurement.
Discrete and Continuous Variables
Quantitative variables come in two flavors.
| Type | How you get the data | Examples |
|---|---|---|
| Qualitative | Observe a category | Favorite sport, eye color |
| Quantitative discrete | Count | Siblings, goals, books in a backpack |
| Quantitative continuous | Measure | Height, weight, time |
Averages of discrete variables can be decimals. If 200 teens read a total of 450 books, the mean is $450 \div 200 = 2.25$ books per teen. No single teen read 2.25 books, but the mean describes the group.
Within quantitative variables, we distinguish between discrete (when they take isolated values, usually integers) and continuous (when they can take any value in an…
Within quantitative variables, we distinguish between discrete (when they take isolated values, usually integers) and continuous (when they can take any value in an interval, such as decimals). Classify these variables:
- a) The number of goals a team scores in each league match.
- b) The exact time it takes a swimmer to cover $50$ feet.
- c) The amount of books in the students' backpacks.
- d) The height of the members of the US national basketball team.
In a study on the reading habits of teenagers in a city, the statistical variable is defined as: "Number of books read over the summer". a) What type of variable is…
In a study on the reading habits of teenagers in a city, the statistical variable is defined as: "Number of books read over the summer".
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a) What type of variable is this?
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b) If the study is carried out on a sample of $200$ young people and the sum of all their read books is $450$, what does the value $2,25$ represent in this context?
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c) Make up three possible values that an individual could take for this variable.
Imagine you want to carry out a study on the reading habits of teenagers in your city. Give an example of: 1. A question whose answer is a qualitative variable. 2. A…
Imagine you want to carry out a study on the reading habits of teenagers in your city. Give an example of:
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A question whose answer is a qualitative variable.
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A question whose answer is a discrete quantitative variable.
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A question whose answer is a continuous quantitative variable.
Frequency
After collecting data, the first step is to count. The frequency of a value is the number of times it appears in the data. (Some problems call this the absolute frequency and write it $f$.) Adding all the frequencies gives the total number of data values, which is the sample size $n$.
A bar graph shows frequencies at a glance. The height of each bar is the frequency.
Here each gridline is one student. But graphs do not always count by ones, so always check what each step on the vertical axis stands for before you read a bar.
A group of friends is discussing which movie to go and see at the movie theater. They have noted down their preferences: Adventure, Comedy, Horror, Adventure, Adventure,…
A group of friends is discussing which movie to go and see at the movie theater. They have noted down their preferences:
Adventure, Comedy, Horror, Adventure, Adventure, Horror, Comedy, Adventure, Comedy.
Answer the following questions about the basic concepts:
- a) What is the statistical variable being studied?
- b) What type of variable is it?
- c) What is the absolute frequency (number of times it appears) of the "Adventure" category?
Look at the following bar chart representing the pets of the students in a class and answer based on the concepts learned: a) What is the sample size (total number of…
Look at the following bar chart representing the pets of the students in a class and answer based on the concepts learned:
- a) What is the sample size (total number of surveyed students)?
- b) What is the studied variable and what type is it?
- c) What is the absolute frequency of the variable "Cat"?
Relative Frequency and Circle Graphs
Is 7 students a lot? In a group of 10, yes. In a group of 700, no. To compare groups of different sizes we use the relative frequency, which is the fraction of the data that has a given value.
Why does this work? A circle graph shows the whole data set as one full circle, which is $360^\circ$. A group that is $\frac{1}{4}$ of the data gets $\frac{1}{4}$ of the circle.
You can also work backward. Use these two rules:
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From a relative frequency to a count: $f = h \times n$.
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From a slice angle to a relative frequency: $h = \text{angle} \div 360^\circ$.
When a table has missing entries, use what you know: the frequencies add up to $n$, and the relative frequencies add up to $1$. If you know the total and all but one frequency, subtract to find the last one.
In a study on fruit consumption in a class of $30$ students, some data from the frequency table has been lost. We know that the relative frequency ($h_i$) of the…
In a study on fruit consumption in a class of $30$ students, some data from the frequency table has been lost. We know that the relative frequency ($h_i$) of the students who eat "3 pieces a day" is exactly $0,2$ and that the absolute frequency ($f_i$) of those who eat "1 piece a day" is three times that of those who eat "none".
If there are $2$ students who do not eat any pieces of fruit a day, logically complete the following data:
- a) How many students eat $3$ pieces of fruit a day?
- b) How many students eat $1$ piece of fruit a day?
- c) What is the absolute frequency of those who eat "2 pieces a day", knowing that all the previous options add up to the total of the class?
A survey has been carried out on the number of children per family in a neighborhood of $200$ families. It is known that $15\%$ of the families have $3$ or more…
A survey has been carried out on the number of children per family in a neighborhood of $200$ families. It is known that $15\%$ of the families have $3$ or more children.
- a) What is the absolute frequency of the event "having 3 or more children"?
- b) If the angle in a pie chart for families with "1 child" is $144^\circ$, how many families have exactly one child?
- c) Calculate the relative frequency of the group of families that have "1 child".
Random and Deterministic Experiments
Some things can be predicted for sure. Others cannot.
For a random experiment we can still list every possible result. The list of all possible results is called the sample space. For rolling a standard die the sample space is $\{1, 2, 3, 4, 5, 6\}$. For flipping a coin it is $\{\text{heads}, \text{tails}\}$.
Try the coin yourself and notice that you can never predict the next flip, even though you know the two possible outcomes.
Real experiments in a lab or a garden are often random too. If you grow 200 plants under the same light, with the same soil and water, they will still not all end up with the same height or the same number of tomatoes. You cannot predict the exact result for each plant, so measuring them is a random experiment.
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 observing whether it lands on heads or tails.
- b) Calculating how much $3$ movie tickets will cost if each one is worth $\$8$.
- c) Asking a classmate what their favorite color is.
- d) Dropping a stone from a balcony and seeing if it falls downwards.
- e) Drawing a card from a deck of cards without looking.
We are carrying out an experiment on the growth of some tomato plants under different types of light. We have $200$ plants in total. a) If we decide to analyze only the…
We are carrying out an experiment on the growth of some tomato plants under different types of light. We have $200$ plants in total.
- a) If we decide to analyze only the plants that are in the row closest to the window, are we studying the population or a sample?
- b) Define the individual of this study.
- c) We want to measure three variables: the height of the plant in $cm$, the color of the leaves (dark green, light green, or yellowish), and the number of tomatoes each plant produces. Classify each variable.
- d) Is this a random or deterministic experiment? Justify your answer.
Identify whether the following phenomena are random or deterministic, explaining why in each case. For the random ones, describe their sample space: a) Dropping a stone…
Identify whether the following phenomena are random or deterministic, explaining why in each case. For the random ones, describe their sample space:
- a) Dropping a stone from a balcony and timing how long it takes to reach the ground (assuming there is no wind and the height is fixed).
- b) The number that will win the first prize in the lottery draw.
- c) Drawing a card from a standard deck of cards and observing whether it is a "face card" (Jack, Queen, or King).
- d) Mixing an acid with a base in a chemistry laboratory following an exact proportion to obtain a salt.
- e) The exact time it will take for the next bus to arrive at the stop.
Back to the start: To find the most popular lunch, the population is all 800 students and each individual is one student. You would choose a random sample, such as 10 students from each class, so every group is represented. The variable is "favorite lunch", which is qualitative. To report the results, you would make a frequency table and a circle graph with relative frequencies.