Exercises: Data representation

Classify the following statistical variables obtained from a survey conducted in a school, stating whether they are qualitative or quantitative (in the latter case, specify whether they are discrete or continuous):

1. a) The name of each student's favourite social network.
2. b) The number of hours per week they spend playing video games.
3. c) The height, measured in metres, of the basketball team players.
4. d) The number of siblings each student in a class has.

A group of $20$ teenagers has answered a question about their favourite film genre. The results are:

Action, Comedy, Horror, Action, Comedy, Science Fiction, Horror, Action, Action, Comedy, Horror, Science Fiction, Action, Comedy, Horror, Action, Action, Science Fiction, Comedy, Action.

Create a frequency table that includes the absolute frequency ($n_i$) and draw a bar chart to represent these data clearly.

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

Complete the following partial frequency table on the number of times a group of friends went to the cinema in the last month:

Films ($x_i$)	Absolute frequency ($n_i$)	Relative frequency ($h_i$)	Percentage ($\%$)
$0$	$4$	$0,20$	$20\%$
$1$	$8$
$2$	$5$	$0,25$
$3$		$0,15$	$15\%$
Total	$20$	$1,00$	$100\%$

The following pie chart shows the distribution of the $400$ students at a school according to the extracurricular language they have chosen to study. Look at the chart and answer:

1. a) What percentage of students study German?
2. b) How many students in total study English?
3. c) Which is the least popular language?

The time (in minutes) taken by $15$ students to get from home to school has been recorded. The data are as follows:

$12, 15, 8, 22, 10, 14, 18, 25, 11, 13, 20, 9, 15, 12, 17$.

Group these data in a frequency table using class intervals of width $5$, starting at $5$. That is: $[5, 10), [10, 15), \dots$

From the following interval table, draw the corresponding frequency histogram. Make sure the bars touch each other to indicate that the variable is continuous.

Interval	Frequency ($f_i$)
$[5, 10)$	2
$[10, 15)$	5
$[15, 20)$	8
$[20, 25)$	6
$[25, 30)$	3

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

An "influencer" has recorded the number of new followers gained each day over a week:

Monday: $120$, Tuesday: $150$, Wednesday: $130$, Thursday: $180$, Friday: $210$, Saturday: $250$, Sunday: $230$.

Represent this trend using a frequency polygon. To do so, place the points on a coordinate axis (Day, Followers) and join them with straight line segments.

State which type of statistical graph is the most suitable (Bar chart, Histogram or Pie chart) to represent the following situations:

1. a) We want to compare the percentage of glass, paper and plastic recycling in a neighbourhood.
2. b) We want to show the number of absences for each student in a class (from $0$ to $5$ absences).
3. c) We want to represent the weights of newborn babies in a hospital over a month (data grouped into intervals).
4. d) We want to compare the preferences for $4$ different brands of soft drinks among young people in a town.

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

In a 3rd year ESO class, you are organising the results of a project on student habits. You have collected information on two different statistical variables:

• Variable A: The means of transport each student uses to get to school (on foot, bus, bicycle, family car).

• Variable B: The outdoor average temperature (in degrees Celsius) recorded in the playground at 11:30 a.m. on each school day over three months.

1. a) For Variable A, a classmate proposes drawing a histogram with continuous bars touching each other, while you propose a bar chart with separated bars. Justify with reasoning which option is correct depending on the nature of the variable.
2. b) For Variable B, there is a debate between using a pie chart or a line graph. Compare both types of graphs and argue which is the most appropriate for analysing the trend and evolution of temperatures over time.

In a survey carried out with the $80$ students in 3rd year ESO at a school regarding their favourite type of music, $20$ of them chose "Urban music". Sofía wants to represent these data in a pie chart and writes the following calculation on the board to determine the angle of the corresponding sector:

$$\text{Angle} = \frac{20}{80} \cdot 100 = 25^\circ$$

Sofía concludes that the "Urban music" sector should be drawn with an angle of $25^\circ$.

1. a) Identify and explain with reasoning the conceptual error made by Sofía. What quantity did she actually calculate by multiplying by $100$?
2. b) Correct her procedure by applying the appropriate mathematical formula and calculate the correct central angle (in degrees) that the sector corresponding to "Urban music" must have.

A group of 30 teenagers has recorded how many hours per week they spend watching series on streaming platforms. The data obtained are:

$4, 7, 5, 8, 10, 4, 6, 7, 5, 9, 12, 6, 8, 7, 5, 4, 10, 8, 7, 6, 5, 9, 11, 4, 7, 6, 5, 8, 7, 6$.

1. a) Classify the statistical variable.

1. b) Make a frequency table that includes the absolute frequency ($f_i$) and the relative frequency ($h_i$).

1. c) Represent the data using a bar chart.

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

A study was carried out on the mode of transport used by students at a secondary school to get to school. The results are shown in the following pie chart, but some data have been rubbed out:

• The "Walking" sector has an angle of $144^\circ$.

• The "Bus" sector represents $25\%$ of the students.

• The rest of the students travel by "Bicycle".

1. a) Calculate the percentage of students who walk.
2. b) Determine the angle in degrees corresponding to the "Bus" sector and the "Bicycle" sector.
3. c) If there are 600 students in the school, how many travel by each mode of transport?

An athletics club has recorded the times (in seconds) of its members in a 400-metre sprint race. As it is a continuous variable, they have been grouped into the following intervals: $[50, 55), [55, 60), [60, 65), [65, 70)$. The absolute frequencies are $4, 12, 18$ and $6$ respectively.

1. a) Construct the frequency table including the class mark for each interval.
2. b) Draw the frequency histogram.
3. c) What percentage of athletes took less than one minute ($60$ seconds) to complete the race?

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

In the Language exam, the grades obtained by a 3rd year ESO class were as follows:

Grades: $3, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 9, 10$.

1. a) Create a frequency table with the absolute frequencies ($f_i$) and the cumulative absolute frequencies ($F_i$).

1. b) Draw the bar chart of the grades and, from it, the absolute frequency polygon (join the highest points of the bars with line segments).

1. c) Looking at the polygon, between which grades does the frequency increase, and by how much does it increase in each section?

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

An influencer analyses the growth of their followers on two different social networks over the last 5 months. The data (in thousands of new followers per month) are:

• Network A: $2, 5, 8, 12, 15$

• Network B: $6, 7, 6, 8, 7$

1. a) Represent both series of data on the same line graph (frequency polygon). Use a different colour for each network or a dashed line for one of them.
2. b) Explain, based on the shape of the graphs, which of the two social networks has more stable growth and which has a clearer upward trend.

The backpacks of 40 students were weighed at the entrance of the school for a health study. The results have been organized in the following table:

Weight (kg)	$[0, 2)$	$[2, 4)$	$[4, 6)$	$[6, 8)$	$[8, 10]$
Students ($f_i$)	$3$	$8$	$15$	$10$	$4$

1. a) Draw the frequency histogram.

1. b) On the same graph, draw the frequency polygon.

1. c) How many students carry a backpack that weighs $6$ kg or more? Calculate the relative frequency of this group (the fraction of the total that it represents), expressed as a decimal.

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

In a quality control test at a smart light bulb factory, the lifespan (in thousands of hours) of a sample of $100$ units was measured. It is known that:

• $20\%$ lasted between $10$ and $15$ thousand hours.

• $50\%$ lasted between $15$ and $20$ thousand hours.

• The rest lasted between $20$ and $25$ thousand hours.

1. a) Construct the table of absolute and relative frequencies.
2. b) Represent the relative frequency histogram. What does the area of each rectangle represent in this case?
3. c) If a light bulb is considered "excellent" if it lasts more than $20$ thousand hours, how many light bulbs in the sample are excellent?

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

The environment department of a city council wants to represent the monthly water consumption of the families in a neighbourhood. They have collected the data from the latest bill and have classified them in cubic metres ($\text{m}^3$):

Consumption ($\text{m}^3$)	$[0, 10)$	$[10, 20)$	$[20, 30)$	$[30, 40)$
Families ($f_i$)	$15$	$45$	$30$	$10$

1. a) Calculate the cumulative relative frequency ($H_i$) for each interval.

1. b) Represent the data in a cumulative frequency graph (or ogive).

1. c) If the council decides to give a discount to the families that consume less than $20\ \text{m}^3$, what percentage of the neighbourhood's population does this measure benefit?

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

In a study on sustainable mobility in a Spanish municipality, data from $720$ daily student journeys are collected: $360$ are made on foot, $180$ by public transport, $120$ by bicycle or electric scooter, and $60$ by private car. These data are going to be represented in a pie chart.

1. a) Before calculating any exact angle, estimate with reasoning the fraction of the circle and the percentage that the "on foot" journey will occupy, and deduce whether its angle will be acute, right, or straight. Also estimate whether the "public transport" sector will occupy an exact quadrant of the circle ($90^\circ$).
2. b) Using the formula $\text{Angle} = h_i \cdot 360^\circ$ (where $h_i$ is the relative frequency), calculate exactly the central angle in degrees for each of the four transport modes.
3. c) Check whether your calculations confirm the estimates made in section a) and verify mathematically that the sum of all the angles obtained matches the amplitude of a full circle.

The management team of a municipal library has analysed the loans of youth books made during the last month using a pie chart. In the final report they have not included the total number of loans, but they have included the central angles of the recorded categories:

• Comic and manga: $144^\circ$

• Adventure novel: $108^\circ$

• Science fiction: $72^\circ$

• Poetry and theatre: the remaining sector.

Furthermore, the computer record indicates that exactly $54$ books of "Adventure novel" were loaned.

1. a) Calculate the measure in degrees of the central angle corresponding to the "Poetry and theatre" category.
2. b) Determine the relative frequency ($h_i$) and the percentage represented by each of the four book categories out of the total loans.
3. c) From the previous data, find out how many youth books were loaned in total during the month and reconstruct the absolute frequency ($f_i$) of each of the four categories.

Two groups of 3rd ESO students have recorded the daily time (in minutes) they spend playing video games. The results are shown in the following box plots.

Analyse the graphs and answer with reasons:

1. a) What is the maximum gaming time in Group B? And what is the value of the upper quartile ($Q_3$) in Group A?
2. b) In which group are the data more spread out? Use the range and the interquartile range to justify your answer.
3. c) If a student claims that $75\%$ of Group B students play for longer than the median of Group A, are they correct? Why?

A soft drinks brand has carried out a study on how many cans a week a group of teenagers consumes. However, the frequency table got stained and only some of the cumulative relative frequency values ($H_i$) can be read:

Cans ($x_i$)	Absolute Freq. ($f_i$)	Cum. Relative Freq. ($H_i$)
0		$0,15$
1	$12$
2		$0,65$
3		$0,85$
4	$6$	$1,00$

Given that the sample size is $40$ individuals:

1. a) Reconstruct the complete frequency table (calculating $f_i$, $F_i$, $n_i$ and $H_i$).
2. b) Represent the data using a cumulative frequency polygon.

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

A local council wants to know young people's opinions on the new skate parks. The total population is $12.000$ young people, split across three neighbourhoods: Centre ($5.000$), North ($4.000$) and South ($3.000$). They decide to take a representative sample of $600$ individuals using stratified random sampling.

1. a) How many young people from each neighbourhood should be selected for the sample to be representative?
2. b) After conducting the survey, $40\%$ in the Centre, $60\%$ in the North and $30\%$ in the South are found to be "Very satisfied". Represent these satisfaction results using a dual bar chart (side-by-side bar chart).

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

Look at the following histogram representing the resting heart rates of a group of athletes. Note that the intervals do not have the same width.

In this type of chart, the frequency is proportional to the area of the bar, not its height. If in the interval $[40, 50)$ there are $10$ athletes:

1. a) How many athletes are there in total in the study?
2. b) Construct the grouped frequency table using the intervals shown, stating the class mark for each one.

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

A social media content creator analyses the duration (in seconds) of their latest videos. The data obtained are: $15, 42, 55, 12, 18, 30, 45, 95, 110, 20, 35, 48$.

1. a) Calculate the interquartile range and identify whether there are any outliers (those located at a distance greater than $1,5 \cdot \text{IQR}$ from the quartiles).
2. b) Draw the box-and-whisker plot, clearly highlighting any outliers with a prominent dot, if there are any.

A pie chart shows the distribution of the types of waste collected in a school recycling drive (paper, plastic, and glass), where the paper sector measures $120^\circ$ and the glass sector measures $60^\circ$. It is known that a total of $180$ kg of waste was collected.

1. a) Calculate the angle corresponding to the "Plastic" sector.
2. b) Calculate how many kilograms of each type of waste were collected.
3. c) If next year the amount of glass is expected to double at the expense of reducing paper, keeping the total waste constant, how would the degrees of the paper sector change?

A graph has been published in a news story about the increase in rental prices in a city. Some critics say the graph is misleading because the y-axis does not start at zero.

1. a) According to the graph, what percentage increase does there visually appear to be when comparing the height of the bars (measured from the horizontal axis)?

1. b) Calculate the actual percentage increase in the rental price between the year 2022 ($800$ €) and 2024 ($900$ €).

1. c) Mentally redraw (or describe) how the perception of the increase would change if the $Y$ axis started at $0$ €.

A weather station has recorded the maximum temperatures during the month of November. By grouping the data into intervals, the following frequency polygon was obtained:

The values on the $X$-axis are the class marks of intervals with width $4^\circ\text{C}$.

1. a) Determine the temperature intervals that have been used.
2. b) Given that the points represent the absolute frequency ($f_i$), how many days were there in November according to this record? Is the total number of days correct?
3. c) In which temperature interval is the mode?

The State Meteorological Agency (AEMET) has recorded the amount of accumulated rainfall (in litres per square metre, $\text{l/m}^2$) during $50$ stormy days in an agricultural region. Since it is a continuous variable, the data have been grouped into the following half-open intervals: $[0, 10)$, $[10, 20)$, $[20, 30)$, $[30, 40)$, and $[40, 50]$. The number of days recorded in each interval was $18$, $15$, $10$, $5$, and $2$, respectively.

1. a) Explain why it is necessary to use a histogram with adjacent (touching) rectangles and not a separated bar chart to model these data. Specify precisely which quantity is represented on the horizontal axis ($X$) and which on the vertical axis ($Y$).
2. b) To construct the associated frequency polygon, a representative point of each interval is needed. Define the mathematical concept of "class mark", calculate its value for each of the five intervals, and determine the Cartesian coordinates $(x, y)$ of the points that would be joined to draw the polygon.
3. c) Analyse the shape of the resulting frequency polygon: is the distribution symmetrical about the centre or does it show skewness? Calculate what cumulative percentage of the stormy days recorded less than $20\text{ l/m}^2$ and interpret the implication of this concentration of precipitation for the agricultural use of the area.

A municipal sports technician must present the annual statistics of a sports centre and has proposed three preliminary graphs for the management report:

• Graph 1: A histogram to represent the main sport played by users (futsal, swimming, tennis, padel, gymnastics).

• Graph 2: A separated bar chart to show the evolution of the number of daily users who have accessed the gym over the $52$ weeks of the year.

• Graph 3: A pie chart to represent the distribution of the $1\,200$ members according to their age group: children ($180$ members), youth ($360$ members), adults ($480$ members), and over 65 years old ($180$ members).

1. a) Identify and explain the methodological error made in the choice of Graph 1. Justify why it is conceptually incorrect to use a histogram with that variable and what type of graph should be used instead.
2. b) Compare the suitability of Graph 2 versus a line graph to show the data for the $52$ weeks. Justify which of the two options better reflects temporal continuity and allows identifying attendance peaks or seasonality more clearly.
3. c) In Graph 3, calculate the central angle corresponding to each of the four age groups. Argue whether it would be possible to replace that pie chart with a frequency histogram, explaining what technical difficulty the "over 65 years old" group poses when defining the width of the intervals.