Exercises: Tables

A study is to be carried out to find out the favorite book of the $1\,200$ students of a secondary school in Madrid. To do so, $80$ students are chosen at random and asked the title of the book they have liked the most this year.

Identify the following elements of this statistical study:

• Population

• Sample

• Individual

Classify the following statistical variables as qualitative or quantitative. For those that are quantitative, state whether they are discrete or continuous:

1. a) Your classmates' favourite colour.
2. b) The number of WhatsApp messages you receive per day.
3. c) The exact height of the players on a basketball team.
4. d) The phone model used by teachers at the school.
5. e) The number of siblings of 2nd ESO students.

A group of friends recorded the number of times they went to the cinema in the last month. The results obtained are:

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

Organise these data into a frequency table including the frequency ($n_i$) for each value of the variable.

In a survey on the favourite social network of $20$ teenagers, the following frequency results were obtained:

Social Network	Frequency ($n_i$)
TikTok	$10$
Instagram	$6$
YouTube	$3$
Others	$1$
Total	$20$

Calculate the relative frequency ($f_i$) and percentage ($\%$) for each category.

The following table shows the number of pets that the families in a housing development have. Complete the column of cumulative absolute frequencies ($N_i$):

Nº of pets ($x_i$)	Absolute frequency ($n_i$)	Cumulative frequency ($N_i$)
$0$	$8$
$1$	$12$
$2$	$7$
$3$	$3$
Total	$30$	---

Hint. The cumulative frequency of each row is the sum of its absolute frequency and the cumulative frequency of the previous row. The first row carries over directly: $N_1 = n_1$. The last cumulative frequency must match the total.

Students in a class were asked how many hours on average they spend studying per day. Some data in the table have been lost. Complete it knowing that the total number of students is $25$.

Hours of study ($x_i$)	Frequency ($n_i$)	Relative frequency ($f_i$)
$1$ h	$5$	$0,20$
$2$ h		$0,40$
$3$ h	$6$
$4$ h	$4$	$0,16$
Total	$25$	$1$

Look at the following frequency table showing the shoe size of a group of $10$ friends and answer the questions:

Size ($x_i$)	Frequency ($n_i$)	Cumulative frequency ($N_i$)
$36$	$1$	$1$
$37$	$2$	$3$
$38$	$4$	$7$
$39$	$3$	$10$

1. a) How many people wear a shoe size smaller than $38$?
2. b) What percentage of people wear size $39$?
3. c) What is the most common size (the one with the highest frequency)?

For a study on eating habits, the following qualitative data were collected on the type of fruit eaten for an afternoon snack by $25$ children at a camp:

Apple, Banana, Pear, Apple, Orange, Banana, Apple, Pear, Banana, Apple, Orange, Apple, Pear, Apple, Banana, Orange, Apple, Banana, Pear, Apple, Orange, Apple, Banana, Apple, Pear.

Construct a complete frequency table including: Variable ($x_i$), Frequency ($n_i$), Relative frequency ($f_i$) and Percentage ($\%$).

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

Alejandro conducted a survey among his 20 classmates about the number of books they have read during the last term, and he presented the following results in a table:

• For 0 books: absolute frequency $n_1 = 4$; cumulative absolute frequency $N_1 = 4$; relative frequency $h_1 = 0,20$.

• For 1 book: $n_2 = 8$; $N_2 = 12$; $h_2 = 0,40$.

• For 2 books: $n_3 = 5$; $N_3 = 17$; $h_3 = 0,25$.

• For 3 books: $n_4 = 3$; $N_4 = 20$; $h_4 = 0,30$.

When adding up the relative frequency column, Alejandro gets $0,20 + 0,40 + 0,25 + 0,30 = 1,15$ and states aloud: "Since the sum of the relative frequencies is not 1, the survey is flawed and we have to ask everyone again."

1. a) Identify the specific numerical error that Alejandro made in the table calculation.
2. b) Explain why his conclusion about the survey is wrong and write the corrected values of the relative frequencies ($h_i$) and their corresponding percentages.

In a secondary school, 50 students in Year 9 (2.º de ESO) were asked how they usually commute to school: on foot, by bus, by bicycle/scooter, or by family car. The initial count indicates that 28 students go on foot, 11 by bus, and 7 by bicycle or scooter (the rest travel by family car).

1. a) Estimate: Before performing any exact calculation with pen and paper, mentally estimate what approximate percentage of students travel on foot. Will it be less than $40\,\%$, between $50\,\%$ and $60\,\%$, or over $70\,\%$? Justify your mental reasoning in one sentence.
2. b) Check: Find the absolute frequency of those who travel by family car, complete the table by calculating the relative frequency ($h_i$) and the percentage for each means of transport, and verify numerically whether your initial estimate was correct. Finally, check that the total sum of the relative frequencies is exactly 1 and that of the percentages is $100\,\%$.

A group of 20 teenagers recorded how many hours a week they spend watching series on streaming platforms. The data obtained is as follows:

$5, 8, 12, 5, 10, 15, 8, 8, 10, 12, 15, 5, 8, 10, 12, 8, 10, 12, 10, 8$.

Organise this data in a frequency table that includes:

• Absolute frequency ($n_i$).

• Relative frequency ($f_i$).

• Percentage ($\%$).

In a survey on the number of pets owned by families in a Year 9 class, the following incomplete table was produced. Complete it in your exercise book knowing that the total number of students is $30$.

Pets ($x_i$)	Abs. frequency ($n_i$)	Cum. absolute frequency ($N_i$)	Cum. relative frequency ($H_i$)
0	12
1	9
2		26
3	4

What percentage of students have at most one pet?

A study has been carried out on the distance (in km) that $40$ people travel to get to their workplace. The data have been grouped into the following intervals: $[0, 4)$, $[4, 8)$, $[8, 12)$ and $[12, 16)$.

Given that the absolute frequencies are $14$, $10$, $8$ and $8$ respectively:

1. a) Construct the frequency table including the class mark ($c_i$) of each interval.

1. b) How many people travel $8$ km or more?

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

In a local video game tournament, the ages of the participants have been recorded. Look at the relative frequency table and answer:

Age	Relative frequency ($f_i$)
12 years old	$0,15$
13 years old	$0,35$
14 years old	$0,30$
15 years old	$0,20$

If a total of $120$ people took part in the tournament:

1. a) How many participants are exactly $13$ years old?
2. b) How many participants are over $13$ years old?

A survey was carried out among secondary school students about their favourite sport. The results are shown in this two-way table, but some data are missing. Complete it.

	Football	Basketball	Tennis	TOTAL
Girls	$45$		$20$	$95$
Boys		$25$	$15$
TOTAL	$105$	$55$

Using the table, calculate the percentage of (total) students who prefer basketball.

Look at the following pie chart, which shows the preferred type of music of the $200$ students at a school:

Construct the frequency and relative frequency table corresponding to this graph.

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

A youth clothing shop wants to analyse the number of items of clothing each customer buys. It is known that the cumulative relative frequency ($H_i$) for buying $2$ items is $0,75$. If $160$ customers have been analysed:

1. a) How many customers bought $2$ items or fewer?
2. b) If the frequency of buying just $1$ item is $60$ customers, how many customers bought exactly $2$ items?

A study is to be carried out on the use of social networks. To do this, a sample of 2nd ESO students are asked about their favourite network (TikTok or Instagram) and whether they post content daily (Yes or No).

Complete the table knowing that:

• A total of $80$ students were surveyed.

• $45$ students prefer Instagram.

• Of those who prefer TikTok, $15$ do not post daily.

• In total, $50$ students in the sample post content daily.

	Posts daily (Yes)	Posts daily (No)	TOTAL
TikTok		$15$
Instagram			$45$
TOTAL	$50$		$80$

What percentage of students who prefer Instagram post content daily?

In a municipal robotics workshop, $N = 40$ students aged between 12 and 15 years old take part. When transferring the data to the computer, several cells of the frequency table were deleted, but the following information is preserved:

• The cumulative relative frequency for students aged 13 or younger is $H_2 = 0,65$.

• The cumulative absolute frequency for students aged 14 or younger is $N_3 = 36$.

• The number of 12-year-old attendees is exactly twice that of 15-year-old attendees ($n_1 = 2 \cdot n_4$).

Reasonably reconstruct the complete frequency table by determining, for each of the four ages ($12, 13, 14$, and $15$ years old), the absolute frequency ($n_i$), the cumulative absolute frequency ($N_i$), and the relative frequency ($h_i$). Clearly show the calculations and deductions you make to find each unknown value.

The Physical Education teacher timed the 60-meter sprint test for the 30 students in Year 9 (2.º de ESO). The recorded times range between $8,72\text{ s}$ and $13,25\text{ s}$, and due to the precision of the digital stopwatch, practically every student has a different time to two decimal places. To organise the data before analysing it, two students discuss which type of table to use:

• Lucía's proposal: Build a simple frequency table with a row for each of the individual times recorded, indicating how many times each one appears.

• Marcos's proposal: Group the times into four continuous intervals of width $1,2\text{ s}$ starting at $8,5\text{ s}$, and calculate the class midpoint (mark) of each interval to use as a representative value.

1. a) Compare both proposals and reason which one is suitable for this dataset. What insurmountable drawback does the discarded proposal have when it comes to summarising the information?
2. b) Write the four intervals proposed by Marcos in the form $[a; b)$ and calculate the class midpoint of each of them. Explain what the class midpoint means mathematically and what it will be used for later in the statistical study.

In a study on mobile data usage of a group of teenagers, work has started on the following frequency table. However, some data has been lost. Given that the sample size is $N = 80$ individuals and that the cumulative relative frequency of the third interval is $F_3 = 0,825$, complete the table for absolute frequency ($n_i$), relative frequency ($f_i$), cumulative absolute frequency ($N_i$) and cumulative relative frequency ($F_i$).

Data usage (GB)	$n_i$	$f_i$	$N_i$	$F_i$
$[0, 5)$	$12$
$[5, 10)$		$0,35$
$[10, 15)$
$[15, 20]$

A town hall wants to analyze the age of the young people who use the new bike lane. It is known that the variable is continuous quantitative and the data have been grouped into five intervals of constant width. Complete the table knowing that:

• The total age range is from $10$ to $30$ years old.

• The absolute frequency of the second interval is twice that of the first, and that of the third is equal to that of the first.

• $N_3 = 40$

• $f_4 = 0,25$

• $n_5 = 5$

• The total sample size is $N = 60$.

Also indicate the class marks ($x_i$) for each interval.

A survey on the use of three streaming platforms (A, B, and C) was conducted among 2nd year ESO students. This incomplete two-way table relates the preferred platform to the primary device used.

	Tablet	Smartphone	PC	TOTAL
Platform A	$8$		$5$
Platform B		$14$		$22$
Platform C	$6$		$7$	$25$
TOTAL	$18$	$35$

Knowing that the total number of surveyed students is $69$, complete the table and calculate what percentage of the total students mainly uses a Smartphone to watch Platform B.

In an esports club, the weekly training hours of its 40 members have been recorded. They have been organised into four class intervals: $[0, 4), [4, 8), [8, 12)$ and $[12, 16]$.

The following data are known:

1. The relative frequencies of the first two intervals are identical: $f_1 = f_2$.

2. The cumulative relative frequency of the second interval is $F_2 = 0,5$.

3. The last interval has an absolute frequency of $n_4 = 4$.

Deduce the complete table of absolute and relative frequencies. How many players train for at least $8$ hours a week?

From the cumulative frequency polygon of the grades of a mathematics exam, reconstruct the absolute frequency table. Knowing that the total number of students is $30$, determine how many students obtained a grade greater than or equal to $5$ but strictly less than $9$.

Note: The points on the graph are $(1, 3), (3, 7), (5, 15), (7, 24), (9, 28), (10, 30)$, where the first coordinate is the grade and the second is the cumulative absolute frequency $N_i$.

A survey on the number of books read in the past year was carried out among a group of people. It is known that the mode is $3$ books and that the arithmetic mean is exactly $2,5$ books. Using the data in the incomplete table, find the values of $x$ and $y$ (absolute frequencies).

Books read ($x_i$)	$n_i$
$1$	$5$
$2$	$x$
$3$	$y$
$4$	$3$
TOTAL	$20$

A Physical Education teacher records the time (in seconds) it takes his students to run the $100$-meter sprint. The results are grouped into a frequency table with intervals of width $2$ seconds.

Knowing that:

• The first interval starts at $12$ seconds.

• $f_1 = 15\%$

• $f_2 = 35\%$

• $N_3 = 28$ students.

• $F_3 = 0,7$ (cumulative relative frequency).

• The rest of the students are evenly distributed across the last two intervals up to a maximum time of $22$ seconds.

Calculate the total number of students ($N$) and complete the absolute frequency table.

Analyze the following frequency table corresponding to the number of absences of a class of $25$ students during one month. Find the two logical consistency errors in the cumulative frequency columns ($N_i$ and $F_i$) and justify why they are errors based on the properties of frequency tables.

Absences ($x_i$)	$n_i$	$f_i$	$N_i$	$F_i$
$0$	$10$	$0,40$	$10$	$0,40$
$1$	$6$	$0,24$	$16$	$0,64$
$2$	$5$	$0,20$	$14$	$0,84$
$3$	$4$	$0,16$	$25$	$1,05$

Based on a study inspired by data from the National Statistics Institute (INE) on adolescent lifestyles, information was collected from a sample of 120 secondary school students. Two variables are cross-tabulated in a contingency table: "Weekly sports practice" (Yes / No) and "Daily screen time for leisure" (Low: less than 2 hours; Medium: 2 to 4 hours; High: more than 4 hours). The following data are known:

• $60\,\%$ of the respondents state that they do practice sports weekly.

• Of the students who practice sports, 24 have low screen time and 36 have medium screen time.

• Among those who do not practice sports, exactly 12 have low screen time, and the number of those with high screen time is three times that of those with medium screen time.

1. a) Model this data by building a complete two-way table with the absolute frequencies, clearly showing the internal cells, row totals, and column totals.
2. b) Calculate what percentage of sports players have high screen time and what percentage of non-sports players have high screen time.
3. c) In light of the two calculated percentages, can it be deduced that in this sample there is a relationship between not doing sports and high screen time? Justify your conclusion by comparing both proportions.

In a study on reading habits in a municipality, the weekly time (in hours) spent reading by public library users was recorded. The data have been grouped into four intervals of equal width: $[0; 2)$, $[2; 4)$, $[4; 6)$, and $[6; 8)$. Following a computer glitch, the total number of respondents ($N$) was erased from the report, but the following information is preserved:

• The cumulative absolute frequency of the third interval is $N_3 = 54$.

• In the last interval ($[6; 8)$), exactly 6 people are recorded.

• The relative frequency of the second interval is $h_2 = 0,35$.

• In the first interval, there are twice as many people as in the third ($n_1 = 2 \cdot n_3$).

1. a) Reasonably deduce the total sample size ($N$) and calculate the absolute frequency ($n_i$) of each of the four intervals.
2. b) A councillor reads the report's results and states: "Since the class midpoint of the interval $[2; 4)$ is $3$ hours and there are $n_2$ people in it, if we add up the actual reading times of all the people in that group, we will get exactly $3 \cdot n_2$ hours." Explain in detail the conceptual error the councillor is making and build a specific numerical example showing that their statement does not necessarily hold.