Frequency tables

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 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 ($F_i$).

In a 1st Year 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 $1$.

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 $N = 40$.

Two 1st Year 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:

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

• Aoife'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 secondary school in Dublin, a study is to be carried out to find out the favourite sport of 1st Year 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: _____

At a secondary school in Cork, a study is to be carried out on its students' leisure habits. To do so, the $30$ students of 1st Year 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).

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

1. a) Identify Seán'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 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 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.