Exercises: Tables

To find out opinions about the use of mobile phones in the classroom, a survey has been carried out on $45$ students of a school that has a total of $600$ students. State what the fundamental elements of this statistical study are:

1. a) The population under study.

1. b) The selected sample.

1. c) The individual.

Classify the following statistical variables as qualitative, discrete quantitative or continuous quantitative based on data from a music application:

1. a) The musical genre of the songs (Pop, Rock, Trap...).
2. b) The number of "likes" on a playlist.
3. c) The duration of a song in seconds.
4. d) The name of the artist or band.
5. e) The storage space occupied by the app on the phone (in MB).

A group of friends write down the number of times they have been to the cinema in the last month. The data obtained are:

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

Organise these data into a frequency table that includes:

• The tally (using tally marks).

• The absolute frequency ($f_i$).

In a 3rd ESO class, students were asked about their favourite sport. Complete the missing values in the following frequency table:

Sport	Absolute frequency ($f_i$)	Relative frequency ($h_i$)	Percentage (%)
Basketball	$6$	$0,24$	$24\%$
Football	$10$
Tennis	$4$	$0,16$
Swimming		$0,20$	$20\%$
Total	$25$	$1,00$	$100\%$

A study has been conducted on the number of books read by members of a book club during the summer. Look at the table and answer the questions using cumulative frequencies:

Books read ($x_i$)	Students ($f_i$)	Cumulative frequency ($F_i$)
$0$	$2$	$2$
$1$	$5$	$7$
$2$	$8$	$15$
$3$	$6$	$21$
$4$	$4$	$25$

1. a) How many students have read exactly $2$ books?
2. b) How many students have read $2$ books or fewer?
3. c) What is the total number of students participating in the study?

For a study on the weight of students' backpacks, the data have been grouped into intervals of $1$ kg. Calculate the class mark ($x_i$) for each of the intervals in the table:

Weight (kg)	Class mark ($x_i$)	Frequency ($f_i$)
$[2, 3)$		$4$
$[3, 4)$		$12$
$[4, 5)$		$9$
$[5, 6)$		$5$

A survey was conducted on the number of pets families in a neighbourhood have. Due to an error, some data in the table were erased. Knowing that a total of $40$ families were surveyed, calculate the missing values:

No. of pets	Families ($f_i$)	Relative Frequency ($h_i$)
$0$	$12$	$0,30$
$1$		$0,45$
$2$	$7$
$3$	$3$	$0,075$
Total	$40$	$1,00$

A courier company studies the weight of the parcels it sends daily. It has organised the information in the following table of cumulative relative frequencies ($H_i$):

Weight (kg)	$f_i$	$h_i$	$H_i$
$[0, 2)$	$20$	$0,40$	$0,40$
$[2, 4)$	$15$	$0,30$	$0,70$
$[4, 6)$	$10$	$0,20$	$0,90$
$[6, 8)$	$5$	$0,10$	$1,00$

Answer the following questions based on the table:

1. a) What percentage of packages weigh less than $4$ kg?
2. b) What relative frequency is represented by packages weighing $6$ kg or more?
3. c) What is the total number of packages analysed in this study?

Lucía surveyed her 25 classmates about the number of unused mobile phones they keep at home. After counting the data, she prepares the following frequency table:

• For $x_1 = 0$: $f_1 = 5$, $F_1 = 5$

• For $x_2 = 1$: $f_2 = 10$, $F_2 = 15$

• For $x_3 = 2$: $f_3 = 7$, $F_3 = 22$

• For $x_4 = 3$: $f_4 = 3$, $F_4 = 25$

When presenting her work, Lucía adds two conclusions in her notebook:

1. "The sum of the cumulative frequency column is $5 + 15 + 22 + 25 = 67$, which means that 67 phones have been analysed in total."

2. "Since the cumulative frequency for 2 phones is $F_3 = 22$, exactly 22 students have 2 unused phones."

Identify the conceptual error that Lucía makes in each of her two statements, clearly explain why her reasoning is flawed based on the mathematical definition of cumulative frequency, and write down the correct interpretation that she should have given to the numbers 25 and 22.

In a 3.º ESO class of 40 students, a survey is conducted on the main means of transport they use to get to school. The results are: 20 walk, 12 take the bus, 6 go by private car, and 2 use an electric scooter.

1. a) Before carrying out any calculation or operation with decimals, estimate in a reasoned manner:

• What exact percentage does the group of those who walk represent with respect to the total class?

• Will the relative frequency of those who take the bus be greater than, less than, or equal to $0,25$? Justify it by comparing the number 12 with a quarter of 40.

• Will the group using an electric scooter exceed $10\,\%$ of the students?

1. b) Construct the frequency table, calculating for each means of transport the absolute frequency ($f_i$), the relative frequency ($h_i$) as a fraction and as a decimal, and the corresponding percentage.

1. c) Check whether your initial estimates were correct and numerically verify whether the sum of all relative frequencies satisfies the fundamental property of adding up to one.

A study is to be carried out on the most-watched type of content on a well-known streaming platform among teenagers in a city. To do this, $150$ students in $3^\text{o}$ of ESO from three different randomly chosen secondary schools are asked. Identify the following elements of the statistical study:

1. a) Population.

1. b) Sample.

1. c) Individual.

1. d) Type of statistical variable.

A group of $25$ young people recorded the number of times they went to the cinema in the last quarter, obtaining the following data:

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

Organise this information into a frequency table that includes: absolute frequency ($n_i$), relative frequency ($h_i$) and percentage frequency ($\%$).

In a survey on the number of pets that families in a neighbourhood have, the results shown in the following absolute frequency table have been obtained:

No. of pets ($x_i$)	No. of families ($n_i$)
$0$	$12$
$1$	$18$
$2$	$14$
$3$	$6$

Calculate the column of cumulative absolute frequencies ($N_i$) and the cumulative relative frequencies ($H_i$). What percentage of families have at most $2$ pets?

The time, in minutes, taken by $20$ students to get from home to school has been measured. The data are:

$5, 12, 18, 24, 7, 15, 30, 10, 22, 14, 8, 11, 28, 5, 19, 13, 16, 25, 12, 9$

Construct a frequency table by grouping the data into $5$ intervals of equal width, starting at $5$ and ending at $30$. Include in the table the class marks ($x_i$) and the absolute frequencies ($n_i$).

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

Fill in the blanks in the following frequency table for a discrete quantitative variable:

Data value ($x_i$)	Abs. frequency ($n_i$)	Rel. frequency ($h_i$)	Percentage frequency ($\%$)
$1$	$8$	$0,20$
$2$		$0,35$
$3$	$12$
$4$			$15\%$
Total	$N=$	$1,00$	$100\%$

In a Physical Education class, students were asked about their favourite sport outside school hours. The results are:

• Football: $12$ students.

• Basketball: $8$ students.

• Swimming: $6$ students.

• Tennis: $4$ students.

Create a table showing the absolute frequencies and calculate the angles required to represent these data in a pie chart (central angle size).

A nutritionist analyses the amount of sugar (in grams) per $100\text{ ml}$ in $40$ different soft drinks. The results have been grouped in the following table:

Amount of sugar ($g$)	Class mark ($x_i$)	Abs. frequency ($n_i$)
$[0, 4)$		$5$
$[4, 8)$		$12$
$[8, 12)$		$18$
$[12, 16]$		$5$

1. a) Complete the column of class marks.
2. b) Calculate the column of relative frequencies ($h_i$).
3. c) State how many soft drinks have $8$ grams of sugar or more.

A survey was conducted among office employees about the number of languages they speak. We know that $40\%$ speak only one language, $35\%$ speak two languages, $20\%$ speak three languages and the rest speak four languages.

If a total of $60$ people work in the office, construct the complete frequency table (absolute, relative and percentage). How many people speak exactly four languages?

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

In an environmental campaign organised in a municipality, a sample of $N = 50$ families were asked how many separate recycling bins or containers they have in their home ($x_i$). However, part of the report has been erased and only the following information is available:

• The possible values of the variable are $1, 2, 3$, and $4$ bins.

• The cumulative relative frequency corresponding to 2 bins or fewer is $H_2 = 0,64$.

• The percentage of families having 4 bins is exactly $12\,\%$.

• The number of families having 1 bin is three times the number of families having 2 bins.

1. a) Reason through, step by step, the absolute frequency ($f_i$) of each of the four values of the variable.
2. b) Prepare the complete frequency table including: variable ($x_i$), absolute frequency ($f_i$), relative frequency ($h_i$), percentage, cumulative absolute frequency ($F_i$), and cumulative relative frequency ($H_i$).
3. c) Using the obtained table, determine what percentage of the families in the sample actively recycle by having at least 3 bins in their home.

From the records of the State Meteorological Agency (AEMET), the daily maximum temperatures ($^{\circ}\text{C}$) of a town in southern Spain were collected during the 31 days of July. The data have been grouped into four contiguous intervals:

• Interval $[28, 32)$: 5 days.

• Interval $[32, 36)$: 11 days.

• Interval $[36, 40)$: 12 days.

• Interval $[40, 44)$: 3 days.

1. a) Justify why in this statistical study it is convenient to group the data into half-open intervals $[a, b)$ with their corresponding class mark instead of drawing up a table with each individual recorded value.
2. b) Prepare the complete frequency table for the month, including for each row: interval, class mark ($c_i$), absolute frequency ($f_i$), relative frequency ($h_i$) rounded to three decimal places, percentage, and cumulative absolute frequency ($F_i$).
3. c) A special heat warning situation is considered when the maximum temperature reaches or exceeds $36\,^{\circ}\text{C}$. Using the frequencies from the table, calculate how many days the town was under warning and what percentage of the month this situation accounted for.

In a video game store, a study was conducted on the number of peripherals (controllers, keyboards, mice) that customers purchase in a single visit. However, part of the information in the frequency table has been lost. We know that the study was carried out on a total of $N = 80$ customers.

Complete the missing data in the table, knowing that the cumulative relative frequency of buying 3 peripherals or fewer is $0,875$.

Peripherals ($x_i$)	Absolute freq. ($f_i$)	Relative freq. ($h_i$)	Cumulative absolute freq. ($F_i$)	Cumulative relative freq. ($H_i$)
1	24
2
3	16
4
5	6

A technology laboratory is testing the lifespan in hours of a new drone battery. They have grouped the results in a continuous variable frequency table, but they have only provided us with the class marks ($x_i$) and some frequencies.

It is known that all intervals have the same width and that the lower limit of the first interval is $1,2$ hours. Reconstruct the complete table (intervals, class marks and frequencies), knowing that the cumulative absolute frequency of the third interval is $42$ and that the total number of drones tested is $N = 50$.

Class mark ($x_i$)	Abs. frequency ($f_i$)	Cumulative abs. frequency ($F_i$)
$1,45$	8
$1,95$	14
$2,45$
$2,95$
$3,45$	3

To organise an esports tournament, the organisation analyses the weekly hours that young members of a club spend training. The following cumulative relative frequency table ($H_i$) shows the distribution, but the data for the middle categories are missing.

Knowing that there is the same number of young people who train "From 5 to 10 hours", "From 10 to 15 hours", and "From 15 to 20 hours", and that the club has a total of $120$ members, calculate exactly how many members train in each of the intervals in the table.

Training hours	Cumulative relative freq. ($H_i$)
$[0, 5)$	$0,15$
$[5, 10)$
$[10, 15)$	$0,65$
$[15, 20)$
$[20, 25]$	$1,00$

In a study on water consumption per household in a housing estate, the following cumulative frequency data have been obtained, but the individual absolute frequencies have been omitted.

Consumption ($m^3$)	Cumulative abs. frequency ($F_i$)
$[0, 10)$	$12$
$[10, 20)$	$30$
$[20, 30)$	$65$
$[30, 40)$	$90$
$[40, 50]$	$100$

It has been decided that households consuming more than $30 m^3$ will pay a surcharge. However, after a review, it is discovered that $10$ households in the interval $[20, 30)$ actually consumed $32 m^3$. Calculate the new corrected percentage relative frequency table ($\%$) and determine what percentage of the total number of households will ultimately have to pay the surcharge.

A school wants to conduct a representative survey on the use of digital textbooks. The school has $400$ students in 1st ESO, $350$ in 2nd ESO, $300$ in 3rd ESO and $250$ in 4th ESO. They want to draw a sample of $130$ students using stratified random sampling with proportional allocation.

Draw the table for the total population and, from it, create the frequency table for the sample indicating how many students from each year group should be surveyed. What is the relative frequency of the 3rd ESO students in the sample compared to that in the total population?

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

A survey has been conducted on the number of pets in $200$ families. The results are presented in the following incomplete relative frequency table ($h_i$).

No. of pets	$h_i$
0	$0,35$
1	$x$
2	$y$
3	$0,12$
4 or more	$0,08$

Calculate the values of $x$ and $y$, knowing that the mean number of pets is exactly $1,25$ (for the interval "4 or more", use the value $4$ as the class mark). Once found, construct the complete absolute frequency table.

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

A local council analyses the age of the users of a sports centre. The data for the first $160$ users are summarised in this cumulative frequency table:

Age	$F_i$
$[0, 18)$	$40$
$[18, 35)$	$95$
$[35, 60)$	$140$
$[60, 85]$	$160$

If during the afternoon $40$ new users register, all of them belonging to the age group $[18, 35)$, how does the relative frequency table change? Calculate the resulting percentage relative frequency table ($\%$) after the new registrations and comment on how this change has affected the relative importance of the group over $60$ years old.

You are given the following frequency table for a discrete quantitative variable $X$ representing the number of times a group of students went to the cinema this month. However, the absolute frequencies $f_2$ and $f_3$ have been erased.

$x_i$	$f_i$	$F_i$	$H_i$
0	5	5	$0,125$
1			$0,400$
2		31
3	9		$1,000$

Calculate the total number of data values ($N$) and the missing frequencies to complete the table. Justify each step using the properties of cumulative and relative frequencies.

Two secondary schools in the same district want to compare the reading habits of their 3rd year ESO students by looking at the weekly hours dedicated to recreational reading. School A has 120 students in 3rd year ESO and School B has 200 students in that same year. To present the results, the teaching staff debate two key methodological decisions:

1. Decision on the type of frequency: The teacher from School A proposes directly comparing the absolute frequencies ($f_i$) of both schools (for example, how many students read more than 4 hours a week). The teacher from School B states that this comparison lacks statistical validity and that it is essential to use relative frequencies ($h_i$) or percentages.

2. Decision on grouping: Part of the team wants to show each integer value separately ($0, 1, 2, \dots, 10$ hours per week), while another part proposes grouping the data into intervals of width 2: $[0, 2), [2, 4), [4, 6), \dots$

1. a) Justify who is right in the debate about the type of frequency. To do this, invent a realistic numerical case with the totals of both schools (120 versus 200 students) where one school has a higher absolute frequency in a bracket but a lower real proportion of readers, demonstrating why absolute frequency leads to error.
2. b) Evaluate the two methods of organising the variable: what exact mathematical information is lost when transforming individual data into grouped intervals represented by their class mark, and what analytical advantage is obtained in return when interpreting the overall sample?

A branch of the National Statistics Institute (INE) conducts a survey on monthly spending on digital services and connectivity from a sample of $N = 200$ families in a district. The amounts in euros have been grouped into four intervals of equal width: $[10, 30)$, $[30, 50)$, $[50, 70)$, and $[70, 90)$. Due to a computer error, only the following statistical clues appear in the final report:

• The cumulative relative frequency of the third interval is $H_3 = 0,90$.

• The percentage of families belonging to the first spending bracket ($[10, 30)$) exceeds the percentage of families in the highest bracket ($[70, 90)$) by 10 percentage points.

• The absolute frequency of the second interval ($[30, 50)$) exactly matches the arithmetic mean of the absolute frequencies of the first and third intervals.

1. a) Formulate with reasoning the mathematical relationships and equations needed to deduce the absolute frequency ($f_i$) of each of the four intervals.
2. b) With the values obtained, construct the complete frequency distribution table: intervals, class marks ($c_i$), absolute frequencies ($f_i$), relative frequencies ($h_i$), cumulative absolute frequencies ($F_i$), and percentages.
3. c) A neighbourhood spokesperson asserts: "In light of the table, exactly 60 families spend €40 per month on connectivity". Critically analyse whether this statement is necessarily true, false, or indeterminable based solely on the information in the table, explaining rigorously what the class mark represents and what limitation it poses when working with grouped data.