Exercises: Data representation

Classify the following statistical variables obtained from a survey carried out in a middle school, indicating 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 weekly hours they spend playing video games.
3. c) The height, measured in feet, of the players on the basketball team.
4. d) The number of siblings each student in a class has.

A group of $20$ teenagers has answered what their favorite movie genre is. 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 this data clearly.

Complete the following partial frequency table on the number of times a group of friends has been to the movies in the last month:

Movies ($x_i$)	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 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) it takes $15$ students to get from their home to school has been recorded. The data is as follows:

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

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

Using the interval table created in the previous exercise, draw the corresponding frequency histogram. Make sure the bars are touching each other to indicate that the variable is continuous.

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

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

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

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

1. a) We want to compare the recycling percentage of glass, paper and plastic 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 weight of newborns 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 the young people in a town.

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

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

1. a) Classify the statistical variable.
2. b) Create a frequency table that includes the absolute frequency ($f_i$) and the relative frequency ($h_i$).
3. c) Represent the data using a bar chart.

A study has been carried out on the type of transport used by students at a middle school to get to class. The results are shown in the following pie chart, but some data has been erased:

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

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

• The rest of the students come 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 come by each means of transport?

An athletics club has recorded the times (in seconds) of its members in a 400-meter sprint. 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 a minute ($60$ seconds) to complete the race?

In the Language Arts exam, the marks obtained by an 8th grade class were the following:

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

1. a) Create a cumulative absolute frequency table ($F_i$).
2. b) Represent the absolute frequency polygon from the bar chart of the grades.
3. c) Looking at the polygon, in which range of grades does the greatest increase in frequency occur?

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) is:

• Network A: $2, 5, 8, 12, 15$

• Network B: $6, 7, 6, 8, 7$

1. a) Represent both data series 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 school entrance for a health study. The results have been organized in the following table:

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

1. a) Draw the frequency histogram.
2. b) Plot the frequency polygon on the same graph.
3. c) How many students carry a backpack weighing $6$ lbs or more? Calculate the cumulative relative frequency of this group.

In a quality control check 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 absolute and relative frequency table.
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?

The environment department of a local council wants to represent the monthly water consumption of families in a neighborhood. They have collected the data from the latest bill and classified it in cubic feet ($m^3$):

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 council decides to give a bonus to families who consume less than $20 m^3$, what percentage of the neighborhood's population benefits from this measure?

Two 8th grade groups have recorded the daily time (in minutes) they spend playing video games. The results have been represented in the following box plots.

Analyze the graphs and give reasoned answers:

1. a) What is the maximum playing time in Group B? And the value of the upper quartile ($Q_3$) in Group A?
2. b) In which group is the data more spread out? Use the range and the interquartile range to justify this.
3. c) If a student claims that $75\%$ of the students in Group B 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 group of teenagers consume per week. However, the frequency table has been stained and only some of the cumulative relative frequencies ($H_i$) can be read:

Cans ($x_i$)	Absolute freq. ($f_i$)	Cumul. 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.

A local council wants to find out young people's opinions on the new skate parks. The total population is $12.000$ young people, distributed 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 must be selected for the sample to be representative?
2. b) After carrying out the survey, it is found that $40\%$ in the Center, $60\%$ in the North and $30\%$ in the South are "Very satisfied". Represent these satisfaction results using a comparative bar chart (clustered 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 graph, the frequency is proportional to the area of the bar, not its height. If there are $10$ athletes in the interval $[40, 50)$:

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

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

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

The following pie chart shows the distribution of the types of waste collected in a school recycling campaign. It is known that a total of $180$ pounds of waste was collected.

1. a) Calculate the angle corresponding to the "Plastic" sector.
2. b) Calculate how many pounds 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 degrees of the paper sector change?

A graph has been published in a news report about the increase in rent 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 appears to be there visually when comparing the height of the bars?
2. b) Calculate the actual percentage increase in the rent price between the year 2022 ($\$800$) and 2024 ($\$900$).
3. 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 has been obtained:

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

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 did the month of November have according to this record? Is the total number of days correct?
3. c) In which temperature interval is the mode found?