Exercises: Frequency tables

Classify the following statistical variables, stating whether they are qualitative, discrete quantitative or continuous quantitative:

• Eye colour of the students in a class.

• Number of goals scored by a football team in a season.

• Time taken by runners to complete a marathon.

• Brand of mobile phone of teachers in a secondary school.

In a florist's shop, the number of roses sold on each of the last $15$ days has been recorded, obtaining the following raw data:

$12, 15, 12, 14, 15, 12, 13, 14, 15, 12, 14, 13, 15, 12, 14$

Construct the table of absolute frequencies ($f_i$) and relative frequencies ($h_i$) corresponding to this variable.

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

In a study on the type of transport used to get to work, the relative frequency of respondents using the underground is $0,35$. If the study was conducted with a total of $N = 400$ people, calculate the absolute frequency of this group.

Given the following absolute frequency table for the number of pets owned by $20$ families, calculate and add the column of cumulative absolute frequencies ($F_i$).

• $x_i = 0 \implies f_i = 5$

• $x_i = 1 \implies f_i = 8$

• $x_i = 2 \implies f_i = 4$

• $x_i = 3 \implies f_i = 3$

A local council wants to know the opinion of its $50\,000$ inhabitants regarding the construction of a new urban park. To do this, it conducts a telephone survey of $1\,200$ randomly selected residents. Clearly identify the population, the sample, the individual and the sample size.

The following table shows the relative frequencies ($h_i$) of the number of siblings of students at a school. Calculate the cumulative relative frequencies ($H_i$) and express them also as a cumulative percentage ($P_i$).

• $x_i = 0 \implies h_i = 0,25$

• $x_i = 1 \implies h_i = 0,45$

• $x_i = 2 \implies h_i = 0,20$

• $x_i = 3 \text{ o más} \implies h_i = 0,10$

A student has constructed a relative frequency table for exam marks and obtained the following values for the different grades: $0,15$; $0,25$; $0,35$; $0,30$. Without needing to know the total number of students, explain why it can be asserted with absolute certainty that the student made a mistake in their calculations.

Complete the following frequency table, knowing that the total number of data values is $N=50$.

• $x_i = 1 \implies f_i = 10$

• $x_i = 2 \implies f_i = x$

• $x_i = 3 \implies f_i = 15$

• $x_i = 4 \implies f_i = 5$

Calculate the value of the unknown $x$ and determine the relative frequency of the value $x_i = 2$.

Lucía has conducted a survey of the $40$ students in her year group about the number of books they have read in the last term. When presenting her frequency table in class, she makes two notable mistakes in her spreadsheet:

• In the row for the value $x_2 = 1$ book, whose absolute frequency is $f_2 = 14$ and whose cumulative absolute frequency is $F_2 = 22$, she writes as the cumulative relative frequency $H_2 = \frac{14}{40} = 0,35$.

• At the bottom of the cumulative absolute frequencies column ($F_i$), she sums all the row values ($8 + 22 + 33 + 38 + 40 = 141$) and writes below: «Cumulative total $= 141$ people surveyed».

Identify in detail the two conceptual errors made by Lucía, explain why her reasoning is incorrect, and calculate the correct numerical values that should replace her notes.

In a secondary school with $200$ Year 11 (4.º de ESO) students, a study is conducted on the main means of transport they use to get to class. The count yields the following data: Walking ($f_1 = 96$), Bus ($f_2 = 54$), Bicycle/Scooter ($f_3 = 36$), and Private car ($f_4 = 14$).

1. a) Without doing long divisions or using a calculator, make a quick mental estimate of the percentage ($p_i$) that each category represents, taking advantage of the total sample size being $N = 200$. Explain in one sentence the reasoning or mental strategy you followed to estimate them.
2. b) Next, calculate the exact values of the relative frequencies ($h_i$) and the percentages ($p_i$) for each means of transport.
3. c) Compare your estimates with the calculated values and check whether the sum of the relative frequencies and that of the percentages satisfy the fundamental properties required of any frequency table.

Group the following weights (in kg) of $20$ packages from a courier company into $4$ class intervals of equal width, starting at $2,0$ kg and ending at $10,0$ kg. Construct the frequency table indicating the intervals, class midpoints ($x_i$) and absolute frequencies ($f_i$).

Data: $2,5$; $3,1$; $4,8$; $5,2$; $2,9$; $6,5$; $7,4$; $8,8$; $9,1$; $4,5$; $5,5$; $6,2$; $3,8$; $7,1$; $8,2$; $9,5$; $4,2$; $5,9$; $6,8$; $7,7$.

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

Complete the following frequency table in which some data are missing. It is known that the sample size is $N = 40$.

• $x_i = 10 \implies f_i = 8$, $F_i = 8$, $h_i = 0,20$

• $x_i = 20 \implies f_i = ?$, $F_i = 20$, $h_i = ?$

• $x_i = 30 \implies f_i = 14$, $F_i = ?$, $h_i = ?$

• $x_i = 40 \implies f_i = ?$, $F_i = 40$, $h_i = 0,15$

The following bar chart shows the number of books read during the summer by a group of students. Using the visual information from the chart, construct the complete table indicating the absolute frequencies ($f_i$), relative frequencies ($h_i$) and percentages ($p_i$).

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

In a two-dimensional study, $10$ people were asked for their shoe size ($X$) and their number of siblings ($Y$). The pairs $(X,Y)$ obtained are the following:

$(38, 1), (40, 2), (39, 1), (40, 1), (38, 2), (41, 0), (39, 2), (40, 1), (41, 1), (39, 1)$

Construct the two-way absolute frequency table for these two variables.

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

From the following two-way table, extract the marginal distribution table for the variable $X$ (shoe size), indicating its absolute frequencies ($f_i$) and its relative frequencies ($h_i$):

A company classifies the monthly salary of its employees into intervals. Given that $15\%$ earn between $1000$ and $1500$ euros, $45\%$ earn between $1500$ and $2000$ euros, $30\%$ earn between $2000$ and $2500$ euros, and the remaining $12$ employees earn between $2500$ and $3000$ euros.

Calculate the total number of employees in the company and construct the complete absolute frequency table for the salaries.

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

When analysing the marks on a Mathematics exam marked out of $10$, it is observed that the cumulative relative frequency up to mark $6$ (inclusive) is $H_6 = 0,65$. If there are $80$ students in total in the class, how many students scored a mark strictly greater than $6$? What percentage do they represent of the whole class?

A dataset of $150$ continuous data values on the duration (in hours) of certain industrial batteries is available. Use the empirical square root rule ($\sqrt{N}$) to determine the recommended approximate number of class intervals. If the minimum recorded value is $12$ hours and the maximum is $87$ hours, calculate the width that each interval should have (rounding up to the next integer) and write down what these intervals would be, starting from the minimum value.

In an evaluation survey of an educational app, $50$ users gave an integer rating from $1$ to $5$ ($x_i \in \{1, 2, 3, 4, 5\}$). The person responsible for preparing the report has misplaced the list of original responses, but retains the following notes in their notebook:

• $10\,\%$ of users gave a rating of $1$ star.

• The relative frequency of the $2$-star rating is $h_2 = 0,24$.

• The cumulative absolute frequency up to and including $4$ stars is $F_4 = 41$.

• The number of people who gave $3$ stars is exactly double the number of people who gave $4$ stars ($f_3 = 2 \cdot f_4$).

From these conditions, reconstruct the complete frequency table with justification, deducing step by step the value of each absolute frequency ($f_i$) and each cumulative frequency ($F_i$).

An AEMET weather station in an agricultural region has recorded the daily sunshine duration (direct solar radiation) in hours over $25$ spring days:

$3,2$; $5,8$; $7,1$; $4,5$; $8,4$; $6,0$; $9,2$; $5,1$; $7,8$; $10,4$; $4,0$; $6,6$; $8,1$; $7,4$; $5,5$; $9,8$; $6,9$; $8,7$; $7,0$; $11,1$; $4,8$; $6,3$; $8,9$; $7,6$; $10,0$.

1. a) Following the standard criterion for continuous data, determine the recommended number of intervals ($k \approx \sqrt{N}$) and calculate the necessary constant class width $c$ if an initial lower limit of $3,0$ hours and an exact width of $1,7$ hours per interval are set.
2. b) Construct the frequency table grouped into half-open intervals $[a, b)$, indicating the class midpoint ($x_i$), the absolute frequency ($f_i$), the relative frequency ($h_i$), and the cumulative relative frequency ($H_i$) of each class.
3. c) From the resulting table, determine what percentage of the analysed days had sunshine strictly less than $8,1$ hours, and briefly explain the usefulness of class midpoints when summarising continuous data.

An absolute frequency table has $4$ distinct values: $x_1, x_2, x_3, x_4$. It is known that the absolute frequencies depend on an unknown $x$ in the following way:

$f_1 = 2x$, $f_2 = x + 5$, $f_3 = 3x - 2$, and $f_4 = 5x$.

If the cumulative relative frequency of the third value is $H_3 = 0,60$, set up an equation, find the value of $x$, and reconstruct the complete column of absolute frequencies.

The following histogram represents the ages of visitors to a museum during a morning. Note that the intervals have different widths and that frequency density is represented on the vertical axis (height $a_i = \frac{f_i}{c_i}$, where $c_i$ is the interval width).

Reconstruct the absolute frequency table knowing that the intervals represented are $[0, 10)$, $[10, 30)$, $[30, 40)$ and $[40, 60)$.

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

A survey has been conducted on two variables, $X$ and $Y$. The two-way table is incomplete. Complete it knowing that the total number of individuals is $N=50$, the marginal distribution of $X$ has $f(X=1) = 20$ and $f(X=2) = 30$, and the marginal distribution of $Y$ has $f(Y=A) = 15$, $f(Y=B) = 25$, and $f(Y=C) = 10$.

In addition, the following partial joint frequencies are known:

• There are $5$ individuals with $X=1$ and $Y=A$.

• There are $10$ individuals with $X=1$ and $Y=B$.

• There are $5$ individuals with $X=2$ and $Y=C$.

Construct the two-way table by filling in the missing cells.

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

In a frequency distribution grouped into intervals, the following cumulative relative frequencies are known: $H_1 = 0,12$, $H_2 = 0,38$, $H_3 = 0,70$, $H_4 = 0,90$ and $H_5 = 1$.

If the absolute frequency of the third interval is $f_3 = 48$, determine the total sample size $N$ and calculate the absolute frequencies of all the intervals.

A researcher is analysing a stratified sample of $300$ people drawn from a population of $15\,000$ inhabitants. The population is divided into three neighbourhoods: North ($6\,000$ inhab.), Centre ($4\,500$ inhab.) and South ($4\,500$ inhab.).

For the sample to be strictly representative (proportional sampling), how many people from each neighbourhood should be included in the sample frequency table? If, in the actual sample used by the researcher, $100$ people from each neighbourhood were surveyed, what representation error (in absolute percentage) was made in the North neighbourhood compared to what was proportionally due to it?

Prove mathematically, using summation notation ($\sum$), that the sum of all relative frequencies ($h_i$) of a one-dimensional distribution of $k$ distinct values is always equal to $1$, regardless of the sample size $N$.

The salaries at a company are grouped into $3$ intervals of equal width. It is known that the class midpoint of the first interval is $1\,200$ € and the class midpoint of the third interval is $2\,000$ €. Determine the width of the intervals and write down the lower and upper limits of each of the three intervals.

There are two groups of students, A and B. For group A ($30$ students), it is known that the relative frequency of passes is $0,60$. For group B ($20$ students), it is known that the absolute frequency of fails is $5$.

If both groups are combined into a single overall frequency table, what will be the cumulative relative frequency of fails with respect to the total of $50$ students? (Assume that the variable takes only two ordered categorical values: Fail and Pass, in that order).

The school guidance department has assessed the scores obtained by $120$ Year 11 (4.º de ESO) students on a standardised numerical reasoning test, on a scale from $0$ to $100$ points with continuous decimal values. To present the report to the school leadership, three alternative methods are proposed:

• Method 1: Build a table with each individual ungrouped score, assigning each distinct numerical score its absolute frequency $f_i$.

• Method 2: Group the data into intervals of uniform width applying the rule $k \approx \sqrt{N}$, rounding $k$ to the nearest integer and calculating the constant width $c$ that covers the full range $[0, 100]$.

• Method 3: Group the data into $4$ qualitative-academic intervals predetermined by assessment criteria: Insufficient $[0, 50)$, Pass/Good $[50, 70)$, Very Good $[70, 90)$, and Outstanding $[90, 100]$.

Answer and justify with statistical rigour:

1. a) Explain why Method 1 is inappropriate and inefficient for this dataset, pointing out what practical problem the frequencies of most rows will present.
2. b) If Method 2 is adopted, determine the integer number of classes $k$ and calculate the uniform width $c$ rounded up to the most convenient integer to cover the range $[0, 100]$. What computational advantage does working with identical class widths offer compared to intervals of variable width?
3. c) Justify in what communicative or decision-making context Method 3 is more convenient and clearer than Method 2, despite its intervals having different widths ($50$, $20$, $20$, and $10$ points).

From a sample of $200$ young people aged 18 to 24 obtained in a socio-labour study in Spain (following the framework of the INE's Labour Force Survey), two variables are analysed jointly:

• Highest educational level attained ($X$): $A$ (Compulsory secondary education (ESO) or lower), $B$ (Bachillerato or Intermediate Vocational Training), and $C$ (University studies or Higher Vocational Training).

• Labour market status ($Y$): $E$ (Employed), $D$ (Unemployed), and $I$ (Inactive studying).

The following aggregate information about the sample is available:

1. $30\,\%$ of the young people have education level $A$ and $40\,\%$ have level $B$.

2. In group $C$ there is no one unemployed, and the number of employed young people is double that of inactive young people.

3. In the sample as a whole, there are $40$ unemployed people, $60\,\%$ of whom belong to education level $A$.

4. The total number of employed young people in the sample is $100$.

5. Among the young people with level $B$, exactly $24$ are in academic or training inactivity.

Carry out the following analytical tasks:

1. a) Model the joint distribution by preparing the complete two-way table (joint and marginal frequencies by rows and columns). Set up and solve the necessary equations to deduce each unknown cell, justifying your steps.
2. b) Calculate the relative sample unemployment rate within each of the three educational levels separately (that is, the percentage represented by unemployed people relative to the total number of young people at their own educational level: $A$, $B$, and $C$).
3. c) Interpret the percentages obtained in the previous section and write a reasoned conclusion: do the data support the hypothesis that a higher level of education is associated with a lower relative frequency of unemployment in this group?