Statistics Concepts
Your city council wants to know how many teens ride the new bike lane on Main Street. It can't ask every teenager in town, so it asks a few and uses their answers to draw conclusions about everyone. Those conclusions are only worth something if the council picks the right people and records the right information. That is what statistics is about, and it's what you'll learn here.
Population, Sample, and Individual
To find out what the favorite video game of the students in a middle school of $850$ students is, a survey has been given to $40$ students chosen at random during break…
To find out what the favorite video game of the students in a middle school of $850$ students is, a survey has been given to $40$ students chosen at random during break time. Identify the following elements of the study:
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Population
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Sample
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Individual
A city council wants to find out about the digital leisure habits of teenagers between 12 and 16 years old. To do this, it randomly selects 400 students from three…
A city council wants to find out about the digital leisure habits of teenagers between 12 and 16 years old. To do this, it randomly selects 400 students from three different middle schools and asks them how many hours a day on average they spend on social media (TikTok, Instagram, etc.).
State the following elements of the statistical study:
- a) The population.
- b) The sample.
- c) The individual.
- d) The statistical variable studied and its classification.
Choosing a Good Sample
A sample is useful only if it represents the population, meaning it looks like a small copy of it. The best way to get that is a random sample: every individual has an equal chance of being chosen.
Sometimes the population is made of groups of different sizes, such as three schools. To keep the sample fair, take the same fraction from each group. This is called a proportional sample (also called a stratified sample). This is the same idea as the proportions you used in Math 6.
Part (c) of the last exercise below asks you to classify a variable. Come back to it after the next two sections.
A study is to be carried out to find out the favorite ice cream flavor of the inhabitants of a city of $50.000$ people. To do this, the first $100$ customers who enter…
A study is to be carried out to find out the favorite ice cream flavor of the inhabitants of a city of $50.000$ people. To do this, the first $100$ customers who enter an ice cream parlor in the center on a Monday morning are asked.
Do you think this sample is representative of the whole population of the city? Briefly justify your answer.
A city council wants to carry out a survey on the use of the bike lane among young people. To do this, it selects $450$ students from three different middle schools…
A city council wants to carry out a survey on the use of the bike lane among young people. To do this, it selects $450$ students from three different middle schools proportionally to the total number of students in each school. In School A there are $600$ students, in School B there are $400$ students and in School C there are $500$ students.
- a) State what the population, the sample and the individual are in this statistical study.
- b) How many students must be selected from each school so that the sample is representative and proportional?
- c) Classify the variable "daily time (in minutes) that each young person uses the bicycle" according to its nature.
Categorical Variables
A variable is the characteristic you record about each individual. Some variables are categorical (also called qualitative): their values are labels, not numbers you can do arithmetic with.
Classify the following statistical variables as qualitative or quantitative: a) The favorite color of your friends. b) The number of WhatsApp messages you receive a day.…
Classify the following statistical variables as qualitative or quantitative:
- a) The favorite color of your friends.
- b) The number of WhatsApp messages you receive a day.
- c) The brand of sneakers worn by the students in your class.
- d) The amount of money you save each month.
If in a statistical study the population is made up of all the cars in a city and the variable is the "color of the car", identify: a) The individual. b) The type of…
If in a statistical study the population is made up of all the cars in a city and the variable is the "color of the car", identify:
- a) The individual.
- b) The type of variable (Qualitative or Quantitative).
- c) Propose three possible values that this variable can take.
Numerical Variables: Discrete and Continuous
Numerical variables (also called quantitative) are counts or measurements. You can add them and find their average.
| Type | Values | Examples |
|---|---|---|
| Categorical, nominal | Labels with no order | Phone brand, movie genre |
| Categorical, ordinal | Labels with an order | Star rating, satisfaction level |
| Numerical, discrete | Numbers you count | Number of texts, siblings, goals |
| Numerical, continuous | Numbers you measure | Run time, temperature, height |
Classify these quantitative variables as discrete or continuous: a) The number of cousins each student has. b) The exact time it takes you to get from your home to…
Classify these quantitative variables as discrete or continuous:
- a) The number of cousins each student has.
- b) The exact time it takes you to get from your home to school.
- c) The temperature of a can of soda when it comes out of the machine.
- d) The number of books there are in your classroom library.
Look at the following statistical variables related to a school trip and classify them: 1. Destination country. 2. Number of days in duration. 3. Distance traveled in…
Look at the following statistical variables related to a school trip and classify them:
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Destination country.
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Number of days in duration.
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Distance traveled in miles.
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Degree of satisfaction (Very low, Low, Medium, High).
A streaming* platform analyzes the data of its teenage users to improve its algorithm. Classify the following variables collected in the study: a) Number of devices…
A streaming platform analyzes the data of its teenage users to improve its algorithm. Classify the following variables collected in the study:
- a) Number of devices connected simultaneously in the home.
- b) Favorite film genre (horror, comedy, action, etc.).
- c) Total weekly viewing time (in hours and minutes).
- d) Rating given to a series (1 star, 2 stars, 3 stars, 4 stars or 5 stars).
- e) Brand of the main device used to watch content.
Justify whether any of these quantitative variables could be treated as qualitative in a given context.
Frequency Tables
Raw data is hard to read. A frequency table counts how often each value appears.
Some problems write the frequency as $n_i$, the relative frequency as $f_i$ or $h_i$, and the cumulative columns as $N_i$ and $H_i$. They are the same ideas with different symbols.
The mode is the value with the highest frequency.
Try changing the data in this interactive and watch the summary change.
In a survey carried out in a class of $25$ students about which social network they prefer (TikTok, Instagram or Twitter), the following answers have been obtained. What…
In a survey carried out in a class of $25$ students about which social network they prefer (TikTok, Instagram or Twitter), the following answers have been obtained. What name is given to the value that is repeated the most times in this study?
An animal shelter has recorded the age (in years) of the dogs that have been adopted during the last month. The data obtained is as follows: $$2, 1, 5, 3, 2, 4, 1, 2, 3,…
An animal shelter has recorded the age (in years) of the dogs that have been adopted during the last month. The data obtained is as follows:
$$2, 1, 5, 3, 2, 4, 1, 2, 3, 2, 5, 1, 4, 2, 3$$Construct a frequency table that includes:
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Variable ($x_i$)
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Absolute frequency ($n_i$)
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Relative frequency ($f_i$)
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Cumulative absolute frequency ($N_i$)
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Percentage ($\%$)
A survey has been carried out on the number of hours that 7th grade students sleep on a weekday. We have the following incomplete data from the frequency table, where…
A survey has been carried out on the number of hours that 7th grade students sleep on a weekday. We have the following incomplete data from the frequency table, where $x_i$ is the number of hours, $n_i$ is the absolute frequency, $h_i$ is the relative frequency and $H_i$ is the cumulative relative frequency:
| $x_i$ | $n_i$ | $h_i$ | $H_i$ |
|---|---|---|---|
| 6 | 4 | 0.16 | |
| 7 | 0.24 | ||
| 8 | 10 | ||
| 9 | |||
| 10 | 2 | 1.00 |
- a) Calculate the total sample size ($N$).
- b) Fill in all the gaps in the frequency table.
- c) What percentage of students sleep 8 hours or more?
Two-Way Tables
When you record two categorical variables for each individual, a two-way table shows both at once. The rows are one variable, the columns are the other, and the totals go on the edges.
Two-way tables also help you find probabilities. When all outcomes are equally likely, the theoretical probability of an event is
$$P(\text{event}) = \dfrac{\text{number of favorable outcomes}}{\text{number of possible outcomes}}$$For example, if you pick a 7th grader at random, only the 7th grade row counts. The theoretical probability that this student plays basketball is $\dfrac{8}{20} = \dfrac{2}{5} = 0.4$.
A survey has been carried out in a 7th grade class on the preferred type of music according to sex. The results are shown in this two-way table: | | Pop | Rock | Trap |…
A survey has been carried out in a 7th grade class on the preferred type of music according to sex. The results are shown in this two-way table:
| Pop | Rock | Trap | Total | |
|---|---|---|---|---|
| Girls | $8$ | $3$ | $4$ | $15$ |
| Boys | $5$ | $6$ | $15$ | |
| Total | $9$ | 30 |
- a) Fill in the missing data in the table.
- b) What percentage of the whole class prefers Rock?
- c) If we choose a boy at random, what is the probability that his preferred music is Pop? Use Laplace's rule.
Bar Graphs and Circle Graphs
A bar graph shows how often each value occurs. The height of a bar is its frequency. A circle graph (pie chart) shows each value as a slice of the whole.
To draw a circle graph, remember that a full circle is $360^\circ$. Each slice has a central angle of
$$\text{angle} = \text{relative frequency} \times 360^\circ$$Look at the following graph which represents the number of books read by a group of students during the summer vacation: From the graph above, answer: a) How many…
Look at the following graph which represents the number of books read by a group of students during the summer vacation:
From the graph above, answer:
- a) How many students have been asked in total?
- b) What is the most frequent value of the variable? What is this parameter called?
- c) What percentage of students have read exactly 2 books?
In a study on the eye color of the students in a middle school, the following data has been obtained: | Color | $n_i$ | | :--- | :---: | | Brown | $120$ | | Blue | $30$…
In a study on the eye color of the students in a middle school, the following data has been obtained:
| Color | $n_i$ |
|---|---|
| Brown | $120$ |
| Blue | $30$ |
| Green | $50$ |
- a) What type of variable is "Eye color"?
- b) Can the mean of this data be calculated? Why?
- c) If we wanted to represent this data in a pie chart, calculate the size of the central angle (in degrees) that would correspond to the "Green eyes" sector.
Grouped Data and Histograms
For a continuous variable such as time, almost every value is different. So we group the values into intervals (classes). The interval $[10, 20)$ means 10 up to but not including 20.
A histogram is a bar graph for grouped data. Its bars touch, because there are no gaps between the intervals.
In an e-sports club, the reaction time (in milliseconds) of its players has been measured. As it is a continuous variable, the data has been grouped into the following…
In an e-sports club, the reaction time (in milliseconds) of its players has been measured. As it is a continuous variable, the data has been grouped into the following intervals:
| Time (ms) | Frequency ($n_i$) |
|---|---|
| $[150, 170)$ | $6$ |
| $[170, 190)$ | $12$ |
| $[190, 210)$ | $18$ |
| $[210, 230)$ | $4$ |
- a) Calculate the class midpoints of each interval.
- b) Represent the data using a histogram.
- c) In which interval is the reaction time of the majority of the players located?
The following histogram represents the time (in minutes) that students take to get from their home to the middle school. The data has been grouped into intervals of the…
The following histogram represents the time (in minutes) that students take to get from their home to the middle school. The data has been grouped into intervals of the form $[a, b)$.
- a) Construct the absolute frequency table from the graph, identifying the class midpoints of each interval.
- b) Calculate the mean time students take to get to school using the class midpoints.
- c) In which interval is the median of the data located? Explain your reasoning.
Mean, Median, and Mode
Three numbers describe the "center" of a data set.
The mean is the "fair share": if every value were replaced by the mean, the total would not change. That is why the sum equals $\bar{x} \times N$.
A student has obtained the following marks in the Mathematics tests of the second term: $$7,5; \quad 8,2; \quad 9,0; \quad 2,1; \quad 8,7$$ a) Calculate the arithmetic…
A student has obtained the following marks in the Mathematics tests of the second term:
$$7,5; \quad 8,2; \quad 9,0; \quad 2,1; \quad 8,7$$- a) Calculate the arithmetic mean ($\bar{x}$) and the median ($Me$).
- b) If the teacher decides that the term mark should be the median instead of the mean so that a "bad day" does not penalise them so much, what would their mark be?
- c) Explain why the median is, in this case, more representative than the mean.
In a talent show, a participant receives the following scores from 5 judges: $8, 7, 9, 8$ and a fifth unknown mark ($x$). a) If the mean of the five marks is $8,2$, what…
In a talent show, a participant receives the following scores from 5 judges: $8, 7, 9, 8$ and a fifth unknown mark ($x$).
- a) If the mean of the five marks is $8,2$, what is the score of the fifth judge?
- b) If the rules of the show state that to calculate the final mark the highest and lowest scores must be removed and the mean of the remaining three calculated, what would be the participant's final mark with the value of $x$ found previously?
- c) How would it affect the median of the five original marks if the fifth judge had given them a $10$ instead of the value $x$?
The arithmetic mean of the marks of a group of 12 students in a mathematics exam was $6,5$. After reviewing the exams, the teacher realizes that he has made a mistake…
The arithmetic mean of the marks of a group of 12 students in a mathematics exam was $6,5$. After reviewing the exams, the teacher realizes that he has made a mistake with two students: he had given one a $4$ when they actually had a $7$, and he had given another a $9$ when they actually had a $6$.
- a) What is the total sum of the marks of the 12 students before the correction?
- b) What is the new arithmetic mean of the group after correcting the mistakes?
- c) Can we know if the median has changed with this data? Justify your answer.
Outliers and Spread
The center alone doesn't tell the whole story. Two players can average the same points per game and still be very different.
Two basketball teams have scored the following points in their last 5 matches: Team A: $72, 75, 74, 73, 76$ Team B: $50, 95, 60, 110, 55$ a) Calculate the mean and the…
Two basketball teams have scored the following points in their last 5 matches:
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Team A: $72, 75, 74, 73, 76$
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Team B: $50, 95, 60, 110, 55$
- a) Calculate the mean and the range of both teams.
- b) Looking at the results, which of the two teams is more consistent in its scores? Justify your answer by relating the mean to the spread of the data.
A study on digital leisure in 13-year-old teenagers yields the following data on the number of daily hours in front of a screen: $$3, 5, 2, 4, 3, 2, 8, 3, 1, 4, 3, 2, 3,…
A study on digital leisure in 13-year-old teenagers yields the following data on the number of daily hours in front of a screen:
$$3, 5, 2, 4, 3, 2, 8, 3, 1, 4, 3, 2, 3, 5, 3, 10, 2, 3, 4, 1$$- a) Calculate the three measures of central tendency: mean, median and mode.
- b) Looking at the values of the mean and the median, would you say that the distribution is symmetrical or that there are extreme values that shift the mean? Justify your answer by naming these values.
- c) If we remove the two users with the highest consumption of hours, what would the new median be? Explain whether this parameter is sensitive to extreme values in the same way as the mean.
Misleading Graphs
The same data can look calm or dramatic depending on how a graph is drawn. The most common trick is a truncated axis: the vertical axis does not start at 0, so small differences look huge.
Both graphs show the same prices: $\$60$, $\$62$, $\$64$, $\$66$. In Store B's graph, the bars start 58 dollars up, so the April bar looks 4 times as tall as the January bar. In reality, April's price is only about 10% higher than January's.
Look at the following two graphs which represent the evolution of the price of a model of sneakers in two different stores during the same 4-month period. a) Explain why…
Look at the following two graphs which represent the evolution of the price of a model of sneakers in two different stores during the same 4-month period.
- a) Explain why the graph for Store B seems to show a much sharper price rise than that of Store A, even though the actual price data is identical in both.
- b) What is this visual effect called in statistics and why is it considered a misleading practice?
- c) Mentally redraw (or describe) what the vertical axis of Store B should look like for the comparison to be objective.
Back to the beginning: In the bike lane study, the population is all the teens in the city, the sample is the teens surveyed at random, and each teen is an individual. The variable "days per week riding the bike lane" is numerical and discrete. A fair random sample and an honest graph are what make the council's conclusions trustworthy.