Exercises: Representing the data

In a 2nd ESO class, a survey was conducted to find out the students' favourite social network. The following results were obtained: Instagram, TikTok, TikTok, YouTube, Instagram, TikTok, YouTube, Instagram, Instagram, TikTok, YouTube, TikTok, Instagram, YouTube, Instagram.

State the population of this study, the individual, and the type of statistical variable.

Look at the following bar chart representing the favourite sport of a group of teenagers from a sports club:

Using the graph, answer the following:

1. a) How many people prefer basketball?
2. b) Which sport is the least practised in this group?
3. c) How many people were surveyed in total?

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

Students in a class were asked for the number of siblings they have, obtaining the following data:

$1, 0, 2, 1, 1, 0, 3, 2, 1, 1, 0, 2, 1, 4, 1, 0, 2, 1, 1, 2$

Construct a frequency table including the absolute frequency ($n_i$) and the relative frequency ($f_i$).

To represent the results of a survey about preferred ice cream flavours on a pie chart, we need to calculate the angles for each section. If we know that $10$ out of a total of $40$ people like chocolate, what angle size in degrees ($^\circ$) will the sector corresponding to chocolate have?

The following pictogram shows the number of trees planted by the students of a secondary school over four years. If each symbol represents $5$ trees, answer the following questions:

1. a) How many trees were planted in the year $2021$?
2. b) In which year were exactly $15$ trees planted?
3. c) What is the total number of trees planted over the four years?

The time (in minutes) taken by $25$ students to get to school from home has been recorded. The data has been grouped into the following table of intervals:

Represent these data using a histogram. (Remember to place the intervals on the horizontal axis and the frequencies on the vertical axis).

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

The following line graph shows the evolution of the maximum temperature in a city during the first week of May:

Looking at the graph:

1. a) On which day of the week was the highest maximum temperature recorded?
2. b) Between which two consecutive days was there the greatest drop in temperature?
3. c) What was the maximum temperature on Monday?

A video game company wants to find out how many hours a week young people aged $12$ to $14$ spend playing video games. They surveyed $100$ boys and girls chosen at random from different secondary schools in Spain.

In this case, identify:

1. a) What is the population?
2. b) What is the sample?
3. c) What is the individual?
4. d) Is it a qualitative or quantitative variable?

Someone else's mistake. Alba surveys the $40$ students in her class about their favourite extracurricular activity and obtains: Basketball ($10$), Theatre ($8$), Music ($14$), and Robotics ($8$). When preparing a pie chart in her notebook, she writes the following:

«For the Basketball sector: since there are $10$ people out of a total of $40$, I divide $40$ by $10$, which gives $4$, so the angle I must draw is $4^\circ$. Furthermore, to present the results, I will join the tops of the bars with a continuous line because that way you can better appreciate how the interest evolves from one activity to another».

1. a) Find and explain the two conceptual mistakes made by Alba: one in calculating the angle and another in choosing a line graph for this type of variable.
2. b) Correct her calculations and find the exact central angle that corresponds in the pie chart to «Basketball» and to «Music».
3. c) Explain which graph (separated bar chart or contiguous histogram) is appropriate to show this data and why the bars should not touch each other in this case.

Choose the method. A 2nd-year ESO work group must select the most suitable type of graph to present three statistical studies in the school newspaper:

1. The outdoor temperature recorded in the schoolyard every two hours throughout a school day (from 08:00 to 18:00).

2. The exact height of $80$ students grouped into the intervals $[140, 150)$, $[150, 160)$, $[160, 170)$, $[170, 180)$, and $[180, 190]$ (in centimetres).

3. The number of siblings ($0$, $1$, $2$, $3$ or more) in a sample of $50$ families.

1. a) For each of the three situations, determine which of the following options is the most suitable: separated bar chart, histogram with contiguous bars, or time-series line graph.
2. b) Justify each choice based on whether the variable is quantitative discrete, continuous grouped into intervals, or describes a trend over time.
3. c) If in situation 2 you were to draw separated bars with spaces between them instead of a histogram, what false message would you be conveying about intermediate height values?

In a study on the use of social media, in a school with $480$ teenage students, $40$ of them, chosen at random, were asked what their favorite platform is. The results obtained are as follows:

• TikTok: $18$ people.

• Instagram: $12$ people.

• YouTube: $6$ people.

• BeReal: $4$ people.

1. a) Identify the population, the sample and the type of statistical variable.

1. b) Make a frequency table that includes the absolute frequency ($n_i$), the relative frequency ($f_i$) and the percentage ($\%$) of each value.

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

The number of books read by students in a 2nd ESO class during the first term was recorded. The data are:

$1, 2, 0, 3, 1, 1, 2, 0, 1, 4, 2, 1, 1, 0, 2, 3, 1, 2, 1, 0, 5, 2, 1, 2, 1$.

1. a) Draw up the table of frequencies and cumulative frequencies ($N_i$).
2. b) What percentage of students read $2$ books or fewer?
3. c) Draw a frequency polygon over the bar chart of the frequencies.

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

A video game company wants to represent the type of console preferred by visitors at a gaming convention in a pie chart. The data are:

• PlayStation: $45\%$

• Nintendo Switch: $30\%$

• Xbox: $15\%$

• PC: $10\%$

1. a) Calculate the degrees corresponding to each circular sector using proportion or the sector angle formula: $\alpha = \frac{360^\circ \cdot n_i}{N}$.
2. b) Draw an approximate pie chart using the calculated angles.

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

The height (in cm) of the $20$ members of a youth basketball club was measured. Being a continuous variable, the data have been grouped into the following intervals:

$[160, 165), [165, 170), [170, 175), [175, 180), [180, 185]$.

The frequencies are: $2, 5, 8, 3, 2$, respectively.

1. a) State the class marks ($x_i$) for each interval.
2. b) Represent the data on a histogram. Explain how this graph differs from a conventional bar chart.
3. c) What is the modal class?

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

The local council of a town wants to raise awareness about recycling. To do so, it has published a pictogram showing the number of recycling bins installed in four different neighbourhoods.

If each ♻️ symbol is equivalent to $12$ bins:

• Neighbourhood A: ♻️♻️♻️♻️

• Neighbourhood B: ♻️♻️♻️

• Neighbourhood C: ♻️♻️♻️♻️♻️

• Neighbourhood D: ♻️♻️

1. a) Construct the frequency table for the number of bins in each neighbourhood.
2. b) What is the total number of bins installed in the town?
3. c) Represent these same data using a bar chart to compare both visual methods.

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

Look at the following two graphs showing the change in average temperature in a city during a spring morning. Both represent the same data, but have been drawn with different scales on the vertical axis.

1. a) Explain why Graph B gives the impression of much faster growth than Graph A if the data are identical.
2. b) Which of the two graphs do you think is more objective for informing the public? Reason your answer based on the scale of the vertical axis.

A climate graph (climograph) is a graph that combines a bar chart (rainfall) and a line graph (temperature). Look at the simplified data for a quarter in a city in southern Spain:

• January: $10^\circ C$, $60\text{ mm}$

• February: $12^\circ C$, $45\text{ mm}$

• March: $15^\circ C$, $30\text{ mm}$

1. a) Draw the climate graph, placing the months on the $X$ axis, rainfall on the left $Y$ axis (bars), and temperatures on the right $Y$ axis (line).
2. b) By analysing the graph, what trend do you observe moving from winter to spring?

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

A survey was conducted among $120$ people about their weekly physical activity. The results are presented in this incomplete table:

1. a) Complete all the gaps in the table knowing that the total number of individuals ($N$) is $120$.
2. b) Represent the data in a pie chart.
3. c) State the mode of this study.

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

Estimate and verify. In a school in Zaragoza, $120$ 2nd-year ESO students were surveyed about the weekly hours they spend on physical activity, obtaining the following grouped distribution: $[0, 2)$ h: $18$ students; $[2, 4)$ h: $42$ students; $[4, 6)$ h: $39$ students; $[6, 8]$ h: $21$ students. A pie chart is to be constructed to display it on the bulletin board.

1. a) Estimate: Without applying the formula or using a calculator, look closely at the frequency of the interval $[2, 4)$ ($42$ students out of $120$). Reason whether its sector will occupy more or less than one third of the circle ($120^\circ$) and whether the resulting angle will be acute, right, or obtuse. Write an approximate numerical estimate of the angle in degrees, justifying your reasoning.
2. b) Calculate: Find the relative frequency ($h_i$) of that interval $[2, 4)$ and calculate its central angle precisely using the formula $\text{Angle} = h_i \cdot 360^\circ$.
3. c) Verify: Compare the exact value obtained with your initial estimate. Was your approximation reasonable? What do you think caused the difference, if any?
4. d) Calculate the central angle corresponding to the interval $[0, 2)$ and check whether the sum of the four angles of the study would complete the $360^\circ$ circle exactly.

Model. An olive oil cooperative in Jaén has published in its monthly newsletter a pictogram showing the production of bottled oil during the four weeks of December. The pictogram's key states that the icon of a full jug represents $4000\text{ litres}$. The following symbols are drawn in the chart:

• Week 1: $3$ full jugs.

• Week 2: $5$ full jugs and a jug cut exactly in half.

• Week 3: $4$ full jugs and a jug of which only three quarters are drawn.

• Week 4: $2$ full jugs and a quarter of a jug.

1. a) Translate the information from the pictogram into actual litres: calculate the amount of bottled oil produced in each of the four weeks and the total for the month.
2. b) The cooperative wants to replace the pictogram with a bar chart in its official technical report so that there is no confusion with partial drawings. Describe in detail how this chart should be constructed (which variable will go on each axis, whether the bars should be touching or separated, and what vertical scale in litres you suggest so that it is clearly readable).
3. c) If in Week 2 the illustrator had mistakenly drawn the partial jug as one third instead of half a jug, how many litres of difference would have been mistakenly communicated to the public?

An e-sports club has carried out a study on the number of hours per week that its $120$ members spend training. We have the following partial data from the frequency table, where the data have been grouped into intervals of the same width:

Complete the frequency table and represent the data using a histogram with a frequency polygon superimposed on it. State what percentage of members train $10$ hours or more per week.

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

A phone company analyses the battery life of two smartphone models, A and B, after one year of use. The results are shown in the following box plot (box-and-whisker plot). Look at the graph and answer:

1. a) What is the median duration for each model?
2. b) If a user needs the battery to last at least $18$ hours, which model has a higher proportion of devices that meet this requirement? Justify your answer using the quartiles.
3. c) Which of the two models has a more spread-out distribution in the central $50\%$ of the data?

An influencer wants to show the growth of their followers on two different social networks during the last six months. To do so, they present the following graphs.

1. a) Explain why the graph for Social Network Y could be misleading to sponsors if they do not look at the vertical axis.
2. b) Knowing that on Social Network X they started with $10.000$ followers and ended with $12.000$, and on Social Network Y they started with $50.000$ and ended with $51.000$, draw a single comparative line graph using an appropriate scale that allows for comparing the actual growth of both accounts.

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

The local council of a town collected data on household organic waste recycling from $200$ families. The following pie chart shows the distribution, but the data for families who recycle "A lot" have been erased.

• None ($0$ kg/month): $15\%$

• A little ($1-5$ kg/month): $72^\circ$ (sector angle)

• Moderate ($6-10$ kg/month): $90$ families

• A lot ($>10$ kg/month): $?$

1. a) Calculate the number of families belonging to each category.
2. b) Calculate the angle in degrees of the sector corresponding to families who recycle "A lot".
3. c) Draw up a frequency and relative frequency table with all the data.

In a study on school library usage, the number of books borrowed by a sample of 2nd ESO students during a term was recorded. The results are shown in the following cumulative frequency bar chart ($N_i$):

1. a) How many students make up the sample?
2. b) Construct the frequency table ($n_i$) from the cumulative frequencies.
3. c) What are the mode and the median of the number of books borrowed?
4. d) Represent the data using a pie chart.

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

A survey of $400$ teenagers was carried out about their favourite weekend activity, distinguishing between boys and girls. The data are shown in the following contingency table, but some values are missing:

1. a) Fill in all the missing values in the table.
2. b) Represent the data using a clustered bar chart (one bar for boys and another for girls for each activity).
3. c) What percentage of girls prefers sports? Is it greater or smaller than the percentage of boys who prefer sports?

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

The reaction time (in seconds) of a group of athletes to the starting signal is represented in the following histogram. The intervals are of the form $[a, b)$.

1. a) Calculate the total number of athletes who took part in the race, knowing that there were $12$ athletes in the interval $[0,15, 0,20)$.
2. b) Calculate the class mark for each interval.
3. c) Estimate the mean reaction time from the histogram data.
4. d) What percentage of athletes have a reaction time of less than $0,20$ seconds?

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

An NGO has carried out a survey on the number of children in $50$ families of a neighborhood. The results have been lost, but we have kept the pictogram, where each house symbol represents $4$ families. In the neighborhood no family has more than $3$ children.

• $0$ children: 🏠 🏠 🏠

• $1$ child: 🏠 🏠 🏠 🏠 🏠

• $2$ children: 🏠 🏠 🏠 🏠

• $3$ children: $?$

1. a) Knowing that the total number of families is $50$, how many families have $3$ children? How would you represent that quantity in the pictogram?

1. b) Calculate the mean, the median and the mode of the distribution.

1. c) If a family is chosen at random, what is the probability that it has fewer than $2$ children? Represent this situation with a pie chart.

Model and analyse scales. An electricity company has sent its customers in Valladolid a leaflet showing the average monthly consumption (in $\text{kWh}$) of a household during the last four months of the year: September: $320\text{ kWh}$; October: $310\text{ kWh}$; November: $295\text{ kWh}$; December: $280\text{ kWh}$.

To illustrate the decrease in consumption, the company designs two bar charts with the same outer dimensions but different vertical axes:

• Chart A: The vertical axis is truncated; it starts at $270\text{ kWh}$ and ends at $330\text{ kWh}$.

• Chart B: The vertical axis is full; it starts strictly at $0\text{ kWh}$ and ends at $350\text{ kWh}$.

1. a) Calculate the visual height (measured from the baseline of the horizontal axis) of the September bar and the December bar in Chart A (where the base marks $270$) and in Chart B (where the base marks $0$).
2. b) In Chart A, how many times taller does the September bar appear compared to the December one? Then calculate the actual mathematical ratio between both consumptions ($\frac{320}{280}$) and compare it with the visual impact of the truncated chart.
3. c) Model the actual percentage reduction experienced between September and December. Next, write a short critical report explaining why Chart A manipulates the user's perception and what erroneous conclusions it encourages in someone who does not read the numbers on the vertical axis.

Inverse problem. In the emergency department of a hospital in Asturias, the waiting time (in minutes) of $120$ patients treated during one day has been recorded. The data have been grouped into four intervals of equal width: $[0, 15)$, $[15, 30)$, $[30, 45)$, and $[45, 60]$. The original data in the table have been lost, but we have the following clues from the graphs that were produced:

1. In the pie chart, the central angle corresponding to the interval $[0, 15)$ measures exactly $54^\circ$.

2. In the histogram, the height of the bar for the interval $[15, 30)$ is exactly three times the height of the bar for the interval $[45, 60]$.

3. $70\,\%$ of the patients waited less than $30$ minutes in total.

1. a) Deduce step by step the absolute frequency ($n_i$) of each of the four waiting time intervals.
2. b) Determine the central angle in degrees corresponding to each interval in the pie chart so that it can be reconstructed accurately.
3. c) Justify why, in the graphical representation of this continuous data, the bars must be drawn contiguous (touching each other) and what the base of each rectangle on the horizontal axis represents.