Exercises: Tables

In a secondary school in Madrid, a study is to be carried out to find out the favourite sport of 1st ESO students. Classify the following statistical variables as qualitative or quantitative:

• Favourite sport (Football, Basketball, Tennis...).

• Number of hours a week they do sport.

• Brand of trainers they wear.

• Height in centimetres of each student.

• Number of medals or trophies they have at home.

The leadership team of a school wants to know if the students like the new canteen menu. So as not to ask all 800 students, they have randomly chosen 25 students from each year group to survey them. Identify the following elements of the study:

• Population: _____

• Sample: _____

• Individual: _____

During recess, we asked a group of friends what their favorite fruit is for a snack. The results were:

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

Create in your notebook a frequency table that includes a column for the Tally (using tally marks) and another for the Absolute Frequency ($f_i$).

In a video game shop, they recorded the sales from one morning:

• Action: $12$

• Adventure: $8$

• Sports: $15$

• Strategy: $5$

Calculate the value of $N$ (the total number of data values) by adding the absolute frequencies of all the categories.

A survey was conducted on the number of pets owned by 20 families. The absolute frequency of families owning "2 pets" is $4$. Calculate the relative frequency ($h_i$) of this value, expressing it as a fraction and as a decimal.

In a 1st ESO class, $20\%$ of the students walk to school. If the relative frequency in the table is $0,25$ for those who come by bus, what percentage of students does this last figure represent?

Copy and complete the missing data in this frequency table about eye colour in a group of people. Remember that the sum of the relative frequencies must be $1$.

Look at the following frequency table showing the marks on a Mathematics test for $20$ students:

Answer the following questions:

1. How many students achieved a "Good"?

2. Which mark appears most often (the most frequent)?

3. How many students passed the test (considering a pass from "Pass" up to "Excellent")?

Álvaro surveyed his 20 classmates about the number of siblings they have ($0$, $1$, $2$, or $3$). When constructing the cumulative frequency column ($F_i$), he obtained the following results written down in his notebook:

• For 0 siblings ($f_1 = 5$): he writes $F_1 = 5$.

• For 1 sibling ($f_2 = 9$): he writes $F_2 = 1 + 9 = 10$.

• For 2 siblings ($f_3 = 4$): he writes $F_3 = 2 + 4 = 6$.

• For 3 siblings ($f_4 = 2$): he writes $F_4 = 3 + 2 = 5$.

When he finishes, Álvaro says worriedly: "My last cumulative frequency gives me 5 instead of 20, so I don't know what I did wrong."

1. a) Identify Álvaro's conceptual error and explain which values he is mistakenly adding up.
2. b) Write the cumulative frequency column ($F_i$) calculated correctly.
3. c) Explain in a complete sentence what real meaning the value of $F_2$ has in the context of this class.

In a 1st year of ESO homeroom class with $N = 40$ students, a vote is held on the main means of transport they use to travel to school:

• Walking: 18 students.

• By city bus: 12 students.

• By bicycle or scooter: 6 students.

• By family car: 4 students.

1. a) Estimate without calculating:

• Observing that the total is 40, will the percentage of students who walk be greater than, less than, or equal to $50\,\%$? Justify your estimate in one sentence.

• Will the family car group reach $15\,\%$ of the total? Reason why.

1. b) Calculate: construct the complete frequency table with the columns: means of transport ($x_i$), absolute frequency ($f_i$), relative frequency ($h_i$), and percentage ($\%$).
2. c) Check: compare the exact percentages you obtained with your estimates from section a) and verify the control sums: $\sum f_i$ and $\sum h_i$.

At a secondary school in Madrid, a study is to be carried out on its students' leisure habits. To do so, the $30$ students of 1º ESO C are asked about their favorite social network. Identify, giving reasons, the following elements of the statistical study:

• Population.

• Sample.

• Individual.

• Type of variable (Qualitative or Quantitative).

A group of friends has recorded the number of times they have been to the cinema in the last six months. The data obtained are:

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

Organise these data into a frequency table including: the value of the variable ($x_i$), the absolute frequency ($f_i$) and the cumulative absolute frequency ($F_i$).

A survey was carried out among $40$ teenagers about their favourite sport to play at the municipal sports centre. Complete the missing values in the following frequency table, knowing that the sample size is $N = 40$.

In a music class, students were asked how many instruments they can play. The results for the cumulative absolute frequency ($F_i$) are:

• $0$ instruments: $F_1 = 4$

• $1$ instrument: $F_2 = 15$

• $2$ instruments: $F_3 = 22$

• $3$ instruments: $F_4 = 25$

Calculate the absolute frequency ($f_i$) for each value and determine how many students there are in total in the class. What percentage of students can play exactly $2$ instruments?

A recycling centre analysed the contents of a recycling bin for plastics and cans. $45\%$ of the items were plastic bottles, $30\%$ were soft drink cans, $15\%$ were milk cartons, and the rest were other metal containers. If a total of $200$ items were analysed, construct a frequency table showing the absolute frequency of each type of waste.

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

From the following bar chart representing the marks obtained by the students of a class in a resit exam, reconstruct the complete frequency table ($x_i, f_i, h_i, \%$).

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

A study was carried out on the number of siblings of the students at a school. The researcher made some mistakes when calculating the relative frequencies. State what they are and correct them so that the table is valid.

For one week, a nutritionist asked $50$ people to record how many portions of fruit they eat a day. The summarised data are:

• $20\%$ eat no fruit.

• $15$ people eat $1$ portion.

• The relative frequency of those who eat $2$ portions is $0,3$.

• The rest eat $3$ portions.

Construct the complete frequency table and answer: What number or numbers of fruit portions are the mode of this study?

In a 1st year of ESO mental arithmetic test graded with scores from 1 to 5, a group of $N = 25$ students obtained various marks. The list of individual marks has been misplaced, but these clues from the statistical report are preserved:

1. The first cumulative frequency is $F_1 = 3$.

2. The cumulative frequency up to mark 2 is $F_2 = 8$.

3. The relative frequency of mark 3 is exactly $h_3 = 0,24$.

4. The cumulative frequency up to mark 4 is $F_4 = 22$.

5. All evaluated students obtained a mark between 1 and 5.

1. a) Deduce step by step the absolute frequency ($f_i$) of each of the 5 possible marks, indicating the operation or property you use in each case.
2. b) How many students passed the test (mark 3 or higher)? Answer using two different procedures: one by adding absolute frequencies and another using cumulative frequencies.
3. c) What percentage of students obtained the top mark (5)?

Following an initiative on responsible screen use, a 1st year of ESO group surveys 50 families in their neighbourhood to record how many large-screen devices (computers, tablets, and smart TVs, not counting mobile phones) they have at home. The data obtained are:

• 1 device: 6 families.

• 2 devices: 14 families.

• 3 devices: 18 families.

• 4 devices: 8 families.

• 5 devices: 4 families.

1. a) Model these data by preparing a complete frequency table that includes: number of devices ($x_i$), absolute frequency ($f_i$), relative frequency ($h_i$), percentage ($\%$), and cumulative frequency ($F_i$).
2. b) Having at most 3 devices at home is considered "moderate equipment". Which column and value in the table directly tells you how many families meet this condition without having to add row by row? What percentage of the total do they represent?
3. c) If the municipal board offers a computer loan programme to families with 2 or fewer devices, how many families in the sample would benefit? Express the result as an irreducible fraction and as a percentage.

In a survey on the number of pets in 50 households in a neighbourhood, the frequency table got smudged and several pieces of data were lost. Your mission is to reconstruct the complete table using the remaining clues:

1. The sample total is $N = 50$.

2. The relative frequency of having 1 pet is exactly $0,32$.

3. The number of households with 3 pets is double the number of households with 4 pets.

4. There is only one household with 5 pets.

5. The percentage of households without pets (0) is $12\%$.

Complete the table by calculating the Absolute Frequency ($f_i$), Relative Frequency ($h_i$) and Percentage ($\%$) for each value.

A streaming platform analysed the number of hours that 40 teenagers spend watching series over the weekend. They have only provided us with the Cumulative Absolute Frequencies column ($F_i$):

• 1 hour: $F_1 = 6$

• 2 hours: $F_2 = 15$

• 3 hours: $h_3 = 0,25$ (Relative frequency, not cumulative)

• 4 hours: $F_4 = 34$

• 5 hours: $F_5 = 40$

From these data, obtain the complete frequency table, including the Absolute Frequency ($f_i$) of each value, the Cumulative Relative Frequency ($H_i$) and the Percentage ($\%$) of each.

In an e-sports tournament, the number of victories of 80 participants is recorded. It is known that the pie chart representing these data has the following angles for each category:

• 0 wins: $72^\circ$

• 1 win: $108^\circ$

• 2 wins: $144^\circ$

• 3 wins: The rest of the circle.

Construct the complete frequency table ($f_i$, $h_i$, $\%$) knowing that the total number of individuals is $N = 80$. How many players achieved the maximum number of wins?

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

A 1st ESO mathematics test has marks from 1 to 10. The teacher gives us the following information about the results of the 30 students:

1. $20\%$ of the class scored a 5.

2. The marks 1, 2, 3 and 10 have an absolute frequency of 0.

3. There is the same number of students with a 6 as with a 7.

4. The relative frequency of those who scored a 9 is $0,1$.

5. There are 4 students with a 4 and 3 students with an 8.

Work out how many students scored a 6 and a 7, and draw up the complete frequency table with all the columns studied ($f_i, F_i, h_i, H_i, \%$).

The local council has recorded the number of occupants in the vehicles that entered the city centre yesterday. Errors have been detected in the final report. Identify which data are impossible or incorrect in the following table of 200 vehicles and correct it, assuming that, in the "3 people" row, the incorrect value is the one in column $f_i$ (not those of $h_i$ or the percentage):

A study has been carried out on daily fruit consumption in a sample of people. We know that:

• $H_2$ (Cumulative relative frequency of consuming 2 portions) is $0,75$.

• $f_1 = 12$

• $f_0$ (0 portions) is half of $f_1$.

• The sample total is $N = 40$.

Calculate the absolute frequency of consuming 2 portions ($f_2$) and complete the table for the values 0, 1, 2 and 3 (knowing that 3 is the maximum value in the study).

In a quality control test at a light bulb factory, 120 units are tested to see how many hours they last before failing (rounded to thousands of hours). The collected data are:

• $35\%$ lasted 1 000 h.

• The absolute frequency of those lasting 2 000 h is 30.

• The relative frequency of those lasting 3 000 h is $0,15$.

• The rest of the bulbs lasted 4 000 h.

Organise these data into a frequency table including absolute frequencies, relative frequencies, percentages and cumulative frequencies. What percentage of light bulbs lasted 3 000 hours or more?

A group of students is asked about the number of books read in the last year. We are not given the total number of students, but we know that:

• Absolute frequency of 1 book: $f_1 = 8$.

• Relative frequency of 1 book: $h_1 = 0,16$.

• Cumulative relative frequency up to 2 books: $H_2 = 0,5$.

• Absolute frequency of 3 books: $f_3 = 15$.

• The rest of the students read 4 books (every student read at least 1 book and nobody read more than 4).

First, calculate the size of the population ($N$). Then, complete the frequency table and find out how many students read 4 books.

Two 1st year of ESO classes compete in a school container recycling campaign for one week:

• In 1st A there are 25 students, and 15 of them have recycled more than 5 containers.

• In 1st B there are 30 students, and 17 of them have recycled more than 5 containers.

To award the prize to the most committed class, two criteria are proposed:

• Sergio's criterion: "1st B should win because its absolute frequency is greater ($17 > 15$). More students have recycled."

• Laura's criterion: "It is not fair to compare absolute frequencies because the groups do not have the same size; we must compare relative frequencies or percentages."

1. a) Apply both methods: calculate the absolute frequency and the relative frequency (as a percentage) of committed students in each classroom. Which group wins according to each criterion?
2. b) Reasonably justify which of the two methods is statistically valid for comparing two samples of different sizes ($N_A \ne N_B$) and why the other can lead to misleading conclusions.
3. c) Propose a numerical example with two groups of different sizes ($N_1 = 20$ and $N_2 = 40$) where both criteria (absolute and relative) agree on pointing to the same group as the winner, and explain why this happens.

In a youth book club, the number of books read by its members during the last quarter is analysed. The variable only takes integer values from 0 to 4. Due to a computer glitch, part of the report has been deleted, but the following relationships are known:

1. The cumulative relative frequency for 2 or fewer books is $H_2 = 0,70$ (that is, $70\,\%$ of the members read at most 2 books).

2. Exactly $20\,\%$ of the members read 3 books ($h_3 = 0,20$).

3. Only 4 members read 4 books ($f_4 = 4$).

4. The number of members who read no books ($x_0 = 0$) is exactly equal to the number of members who read 2 books ($x_2 = 2$).

5. The number of members who read 1 book ($x_1 = 1$) is double that of those who read none.

1. a) From the proportions of members who read 3 and 4 books, deduce the total sample size ($N$) with reasoning.
2. b) Using the total $N$ and the conditions on 0, 1, and 2 books, calculate the absolute frequency ($f_i$) of each of those three values.
3. c) Construct the complete frequency table ($x_i$, $f_i$, $h_i$, $\%$, $F_i$) and check that all the conditions of the statement and the control sums are met.