Exercises: Tables

A study is to be conducted to find out the favorite book of the $1\,200$ students at a high school in Santo Domingo. To do this, $80$ students are chosen at random and asked for the title of the book they liked 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' favorite color.
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 year of secondary 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 favorite 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.

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 9th grade class, the following incomplete table was produced. Complete it in your notebook 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 favorite sport. The results are shown in this two-way table, but some data are missing. Complete it.

	Soccer	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 circle graph, 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.

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 year of secondary students are asked about their favorite 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 study on cell phone 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 municipality wants to analyze the age of young people using the new bike lane. It is known that the variable is continuous quantitative and the data has been grouped into five intervals of constant width. Complete the table knowing that:

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

• The absolute frequency of the second interval is twice that of the first.

• $N_3 = 45$

• $f_4 = 0.25$

• $n_5 = 5$

• The total sample size is $N = 60$.

Also state the class midpoints ($x_i$) for each interval.

A survey was conducted on the use of three streaming platforms (A, B, and C) among 9th grade students. An incomplete two-way table is presented relating preferred platform to main device used.

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

Given that the total number of students surveyed is $75$, complete the table and calculate what percentage of all students mainly use 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) taken by their students to run $100$ meters sprint. The results are grouped in a frequency table with intervals of width $2$ seconds.

Knowing that:

• The first interval starts at $12$ seconds.

• $f_1 = 15\%$

• $f_2 = 35\%$

• $N_3 = 21$ students.

• $F_3 = 0.7$ (cumulative relative frequency).

• The remaining students are shared equally between 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 ($x_i$)	$n_i$	$f_i$	$N_i$	$F_i$
$0$	$10$	$0.40$	$10$	$0.40$
$1$	$6$	$0.24$	$16$	$0.64$
$2$	$5$	$0.20$	$14$	$0.84$
$3$	$4$	$0.16$	$25$	$1.05$