Representing data

At the school library they want to visualize how many books have been lent out each day of the week. They have decided to use a pictogram where each "book" symbol represents $5$ loans.

Draw the corresponding pictogram in your notebook using a simple symbol to represent the books. Calculate how many symbols you need for each day.

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

A group of friends noted down how many pieces of fruit they eat per week. They represented the data in this pictogram, where each fruit symbol 🍎 represents $3$ pieces of fruit eaten.

1. a) How many pieces of fruit did Aoife eat in total?
2. b) Who ate exactly $6$ pieces of fruit?
3. c) How many more pieces of fruit than Seán did Niamh eat?

In an animal shelter, they counted how many bags of food they consume per month. They used this pictogram where each drawing of a "bone" $\square$ represents $15 \text{ kg}$ of food.

• January: $\square \square \square \square$

• February: $\square \square \square$

• March: $\square \square \square \square \square$

Knowing that the shelter has $30$ dogs and that each dog should eat $0.75 \text{ kg}$ per day, construct a frequency table with the kilograms consumed each month and explain whether in any month the dogs ate less than they need (assume all months have 30 days).

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

In a 1st Year class, a survey was carried out on the students' favourite video game genre. Look at the following bar chart and answer the questions:

1. a) Which video game genre do students like the most (the mode)?
2. b) How many students prefer adventure games?
3. c) If we sum all the responses, how many students took part in the survey?

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

Ximena asked her $20$ teammates what their favourite trainer colour was. The results were:

Red, Blue, Blue, White, Red, White, White, Blue, Red, White, Blue, Blue, White, Red, White, White, Blue, White, Blue, White.

1. a) Organise these data into a frequency table including the colour and the absolute frequency ($f_i$).
2. b) What is the mode of this study?
3. c) Draw a simple bar chart in your exercise book to represent these results.

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

In a 1st Year class, a vote has been held to choose the theme of the end-of-year party. The results were as follows: Superheroes ($8$ votes), Space ($6$ votes), Mystery ($10$ votes) and Sports ($4$ votes).

Calculate the relative frequency of each option (as a fraction and as a decimal number) and represent the number of votes for each theme in a bar chart. Make sure the vertical axis is properly scaled and labeled.

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

The height of the players on a youth basketball team has been measured (in $cm$). Since the data are very varied, they have been grouped into intervals:

• $[150, 160)$: $4$ players

• $[160, 170)$: $7$ players

• $[170, 180)$: $10$ players

• $[180, 190)$: $3$ players

Represent these data using a histogram. Remember that in a histogram, unlike in a bar chart, the columns are placed right next to each other because the variable is continuous.

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

A video game company has recorded how many hours its users play per day. The data have been grouped into the following intervals: $[0, 2)$, $[2, 4)$, $[4, 6)$ and $[6, 8)$. The absolute frequencies are $25, 40, 15$ and $10$ respectively.

1. a) Draw the corresponding histogram.

1. b) Next, draw the frequency polygon by joining the midpoints of the top of each bar.

1. c) If the recommended healthy limit is less than $4$ hours, what percentage of users play more than recommended? Round to the nearest tenth.

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

A regional transport consortium analyses the duration of trips on an intercity bus route during peak hours. After recording the times of $400$ trips, it groups the data into the following intervals in minutes: $[10, 20)$, $[20, 30)$, $[30, 40)$, and $[40, 50)$. The following information is available:

• $45\%$ of the trips lasted between $10$ and less than $20$ minutes.

• In the $[20, 30)$ minutes range, exactly $120$ trips were recorded.

• The relative frequency of trips in the interval $[30, 40)$ was $0.15$.

• The rest of the trips fell into the $[40, 50)$ minutes range.

1. a) Prepare the complete frequency table indicating for each interval: absolute frequency ($f_i$), relative frequency ($h_i$), and percentage.
2. b) Justify why it is mandatory to use a histogram and not a conventional bar chart to represent these data. Explain the meaning of the fact that the rectangles of the histogram share their vertical sides.
3. c) The consortium considers the route "efficient" if at least $70\%$ of the trips last less than half an hour ($< 30$ minutes). Determine whether this efficiency criterion is met and calculate how many degrees this combined group ("Under $30$ minutes") would occupy if represented in a summary pie chart against the group of "$30$ minutes or more".

The following pie chart represents how the $40$ students in a year group travel to school. Given that half of the class walks, a quarter goes by bus and the rest goes by car:

1. a) How many students walk to school?
2. b) How many students go by bus?
3. c) What angle in degrees corresponds to the sector of students who walk? (Remember that a full circle is $360^\circ$).

A group of friends wants to organise a trip and has asked $40$ classmates about their preferred destination. The results are shown in the following table:

To represent these data in a pie chart, calculate the angle corresponding to each sector. Remember that the total sum of the angles must be $360^\circ$.

The following pie chart represents how students at a school manage their free time. Given that the school has $200$ students in total, find out how many students prefer each activity based on the percentages given:

• Video games: $40\%$

• Sport: $30\%$

• Reading: $20\%$

• Music: $10\%$

Reproduce it in your exercise book by calculating the angle in degrees for each sector.

The following table shows the temperature recorded in the school playground over five hours on a winter morning:

1. a) Represent these data using a line graph (frequency polygon) in your exercise book.
2. b) In which one-hour interval did the largest temperature increase occur?
3. c) What was the temperature at ten o'clock in the morning?

In a survey conducted among the $120$ students of 1st Year about their favourite streaming platform, the following results were obtained: $40\%$ prefer Netflix, $30$ students choose Disney+, the relative frequency of HBO is $0.15$ and the rest opt for Twitch.

Calculate the absolute frequency of each category and draw a pie chart. To do this, you must indicate the exact angle in degrees for each sector using the formula: $Ángulo = \frac{f_i}{N} \times 360^\circ$.

Below is a pie chart showing the results of a vote to choose the colour of the graduation t-shirt (Blue, Red, Green). The Blue sector measures $144^\circ$ and the Red sector measures $108^\circ$.

1. Calculate the angle of the Green sector.

2. If we know that $30$ people voted for the colour Red, how many people in total took part in the vote?

3. Calculate the absolute frequencies of the other two colours and represent the final results in a bar chart.

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

Match each statistical variable with the type of chart you think is most appropriate to represent it (you may repeat options):

1. Change in a baby's height month by month.

2. Eye colour of the students in your class.

3. Distribution of a local council's budget across different areas.

A) Pie chart.

B) Bar chart.

C) Line graph.

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

A group of scientists is measuring the growth of a plant under a new LED light. Over 6 weeks, the height in centimeters has followed this progression: $2, 5, 9, 14, 20, 27$ (in week 1 the plant is $2 \text{ cm}$ tall).

Represent these data in a line graph. Then, add a second line (in a different style or color) representing the "ideal height" if the plant grew exactly $4 \text{ cm}$ per week (starting from the initial $2 \text{ cm}$ in week 1).

In which week is the difference between the actual height and the ideal height greatest?

The number of hours that $20$ teenagers spend doing sports per week has been recorded:

$2, 4, 3, 5, 2, 6, 4, 3, 3, 5, 4, 2, 6, 3, 4, 5, 2, 3, 4, 3$.

Build the absolute frequency table and represent the data using a frequency polygon. What trend do you observe in the graph?

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

Look at the following two graphs. They represent the sales of a sneaker shop in two different months.

1. a) Which of the two graphs can be misleading or give the false impression that sales have grown enormously?

1. b) Look at the vertical axis. Do both start at zero? Explain why it is important that statistical axes are not "truncated".

A YouTuber analyses the views of his latest video. In the first 4 hours, the cumulative views were:

• Hour 1: $150$ views.

• Hour 2: $400$ views (in total).

• Hour 3: $750$ views (in total).

• Hour 4: $900$ views (in total).

Build a table that distinguishes between the absolute frequency (new views each hour) and the cumulative frequency. Represent the new views of each hour in a bar chart. In which hour was there the greatest success in attracting new viewers?

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

Three 1st Year students are preparing an informative poster with data from the Central Statistics Office (CSO) on youth leisure and defend three different graphic proposals:

• Cian's proposal: To show the year-by-year evolution of the percentage of young digital readers between 2015 and 2024 ($10$ consecutive years), he proposes using a pie chart with $10$ circular sectors.

• Aoife's proposal: To represent the daily time dedicated to personal reading, grouped into the intervals $[0, 15)$, $[15, 30)$, $[30, 45)$, and $[45, 60)$ minutes, she proposes a bar chart with a visible separation between bars.

• Seán's proposal: To represent the number of books read per year per student ($0$, $1$, $2$, $3$, or more than $3$), he proposes drawing a continuous line graph connecting the top ends of each value.

1. a) Critically analyse each proposal: explain with mathematical arguments why each classmate's choice is incorrect or ill-advised based on the nature of the variable.
2. b) State what the correct chart is for each of the three cases and what essential graphic rule must be respected in its drawing (time orientation on the axes, continuity of the rectangles, or proportionality of angles).
3. c) In the draft of a pictogram, Cian compares those who read daily ($25\%$) against those who never read ($50\%$). For the second group, he draws an icon of a book whose height is double and whose width is also double compared to that of the first group. Reason why this representation visually distorts the data and misleads anyone viewing the poster.

At your school, the results of three statistical studies about the students and their environment are going to be published in the school magazine:

1) The number of siblings that the $80$ 1st Year students have ($0$, $1$, $2$, $3$, or more than $3$).

2) The evolution of the monthly average temperature in your municipality over the $12$ months of the year, based on Met Éireann records.

3) The height in centimetres of the $100$ participants in the school competitions, organised into the intervals $[140, 150)$, $[150, 160)$, $[160, 170)$, and $[170, 180)$.

For each of the three studies:

1. a) Identify the type of statistical variable being analysed (qualitative, discrete quantitative, or continuous quantitative grouped into intervals).
2. b) Choose with justification the most appropriate chart from: bar chart, line graph, and histogram. Justify why you rule out the other two in each case and clarify whether the bars should be drawn separated or touching.

Aoife conducted a survey among her $30$ classmates about their favourite extracurricular activity. The results were: $15$ choose Sports, $9$ choose Music, $4$ choose Drama, and $2$ choose Painting.

To represent these data in a pie chart, Aoife reasons as follows:

"Since a full circle has $360^\circ$ and there are $4$ activities, I divide $360^\circ : 4 = 90^\circ$. Therefore, I draw four equal sectors of $90^\circ$ and write the name of an activity inside each one."

1. a) Explain in detail what Aoife's mistake is and why her chart would convey a completely false conclusion to anyone who looks at it.
2. b) Correct her procedure by calculating the relative frequency ($h_i$) of each activity and the exact angle in degrees corresponding to each sector using the relationship $\text{Angle} = h_i \times 360^\circ$.

Compare the following two sets of data on the average temperature ($^\circ\text{C}$) recorded at midday over one week (from Monday to Sunday, in that order) in two different cities:

• City A: $12, 14, 15, 12, 11, 13, 14$

• City B: $18, 17, 19, 18, 20, 19, 18$

Represent both cities on the same line graph (use two different colors or line styles). Calculate the average temperature of each city and draw a dotted horizontal line representing the average of City B. Which of the two cities has more stable temperatures (smaller range)?

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

From the following bar chart about the number of pets owned by the students in a class, carry out the reverse process:

1. Create the frequency table (absolute and relative frequencies).

2. Calculate the total number of students who took part in the survey.

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

In a survey about the usual means of commuting of the $120$ employees of a company, the following data were obtained: $60$ travel by bus, $30$ travel by bicycle, $20$ walk, and $10$ use an electric scooter. You have to represent this information in a pie chart.

1. a) Estimate before calculating: without performing decimal operations or using a protractor, reason what simple fraction of the full circle ($1/2$, $1/4$, etc.) the bus sector will represent and how many degrees it will measure; anticipate whether the angle for the electric scooter will be greater or smaller than a right angle ($90^\circ$) and how it will compare to the angle of those who walk.
2. b) Calculate and verify: find the relative frequency of each mode of transport and calculate with mathematical precision the degrees of each of the four sectors. Compare the results with your previous estimates and check whether the sum of the angles equals exactly one full turn.

The head of the municipal library has prepared a report on the $80$ books borrowed during a cultural week, distributed across four literary genres: Young adult novel, Comic, Poetry, and Drama. Due to an oversight, some figures have been erased, but these data from her charts remain:

• In the pie chart, the angle for Young adult novel measures $162^\circ$, for Comic it measures $108^\circ$, and for Poetry it measures $54^\circ$.

• In the pictogram prepared for the information board, Young adult novel is represented by $9$ identical icons of an open book.

1. a) Calculate the angle in degrees corresponding to the Drama sector and determine the absolute frequency of borrowed books for each of the four genres.
2. b) Deduce how many borrowed books each icon in the pictogram represents (the pictogram's key) and calculate how many full icons will need to be drawn to represent the Comic category and the Drama category.

Look at this bar chart about the number of siblings of the students in a class and complete the frequency table below:

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

In a library, they counted how many books of each type were borrowed in a day. They have started drawing the bar chart, but the last bar is missing.

1. a) If we know that $12$ comics were borrowed, how high should the "Comic" bar reach on the vertical axis?
2. b) Which type of book has been borrowed the least so far?
3. c) Is Poetry the "mode"? Reason your answer by looking at the chart.

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