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

A group of $20$ teenagers answered what their favorite movie genre is. The results are:

Action, Comedy, Horror, Action, Comedy, Sci-Fi, Horror, Action, Action, Comedy, Horror, Sci-Fi, Action, Comedy, Horror, Action, Action, Sci-Fi, Comedy, Action.

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

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 circle graph shows the distribution of the $400$ students at a school according to the extracurricular language they have chosen to study. Look at the graph 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 graph, Histogram, or Circle graph) to represent the following situations:

1. a) We want to compare the percentage of glass, paper, and plastic recycling in a neighborhood.
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.

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 transportation used by students at a high school to get to school. The results are shown in the following circle graph, but some data have been erased:

• 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 transportation?

An athletics club has recorded the times (in seconds) of its members in a 400-meter 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?

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 analyzes 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 color 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 environmental department of a local municipality wants to represent the monthly water consumption of families in a neighborhood. They have collected the data from the latest bill and classified them in cubic meters ($m^3$):

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

1. a) Calculate the cumulative relative frequency ($H_i$) for each interval.
2. b) Represent the data on a cumulative frequency graph (or ogive).
3. c) If the municipality decides to give a discount to families who consume less than $20 m^3$, what percentage of the neighborhood's population will benefit from this measure?

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 municipality wants to know young people's opinions on the new skate parks. The total population is $12,000$ young people, split across three neighborhoods: Center ($5,000$), 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 neighborhood should be selected for the sample to be representative?
2. b) After conducting the survey, $40\%$ in the Center, $60\%$ 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).

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 $1.5 \times \text{IQR}$ from the quartiles).
2. b) Draw the box-and-whisker plot, clearly highlighting any outliers with a prominent dot, if there are any.

The following circle graph shows the distribution of the types of waste collected during a school recycling campaign. 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 while keeping the total waste constant, how would the angle 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 ($800$ pesos) and 2024 ($900$ pesos).

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

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?