Exercises: Frequency tables

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

• Eye color of the students in a class.

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

• Time taken by runners to complete a marathon.

• Brand of cell phone of teachers in a high 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 municipality 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 $x_i = 2$.

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 $1500$ pesos, $45\%$ earn between $1500$ and $2000$ pesos, $30\%$ earn between $2000$ and $2500$ pesos, and the remaining $12$ employees earn between $2500$ and $3000$ pesos.

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.

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 analyzing a stratified sample of $300$ people drawn from a population of $15\,000$ inhabitants. The population is divided into three neighborhoods: North ($6\,000$ inhab.), Center ($4\,500$ inhab.) and South ($4\,500$ inhab.).

For the sample to be strictly representative (proportional sampling), how many people from each neighborhood should be included in the sample frequency table? If, in the actual sample used by the researcher, $100$ people from each neighborhood were surveyed, what representation error (in absolute percentage) was made in the North neighborhood 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 $24\,000$ pesos and the class midpoint of the third interval is $40\,000$ pesos. 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).