Exercises: Measures of position

The number of goals scored by a handball team in their last 9 matches was: $24, 28, 21, 30, 25, 28, 22, 27, 26$. Calculate the lower quartile ($Q_1$), the median ($Me$) and the upper quartile ($Q_3$) of these data.

The battery life, in hours, of 10 mobile phones after a full charge is: $14, 18, 12, 15, 19, 11, 16, 15, 13, 17$. Determine the median and the quartiles $Q_1$ and $Q_3$ of the battery life.

The following box plot represents the prices, in euros, of tickets for various concerts in a city during the summer. Identify the minimum price, the maximum price, the median and the quartiles $Q_1$ and $Q_3$.

The number of pages read in a day by a group of 15 students is: $12, 20, 8, 15, 30, 25, 10, 18, 22, 14, 35, 28, 16, 24, 19$. Find the percentile $60$ ($P_{60}$) and interpret its meaning in the context of the problem.

In a survey carried out on 20 families about the number of pets they have at home, the following results were obtained:

• 0 pets: 6 families

• 1 pet: 8 families

• 2 pets: 4 families

• 3 pets: 2 families

Calculate the median of the number of pets.

The 25 students in a class were asked how many days a week they play sport. The results are shown in the table:

Determine the lower and upper quartiles of this distribution.

At the annual medical check-up, the paediatrician tells a child's parents that his height is at the percentile $85$ ($P_{85}$) for his age. Explain in detail what this statistical value means in relation to other children of his age.

The maximum temperatures recorded in a city over 12 consecutive spring days were, in degrees Celsius: $18, 20, 22, 19, 21, 24, 25, 23, 20, 19, 26, 22$. Calculate the interquartile range (difference between the upper and lower quartiles) of the temperatures.

Marcos has recorded the marks obtained by 8 students in an exam: $4, 8, 3, 9, 6, 5, 7, 2$. To find the quartiles, he reasons as follows: "Since there are 8 data values, for the first quartile I calculate the position $\dfrac{1 \cdot 8}{4} = 2$, so $Q_1$ is the value in second place, that is, $Q_1 = 8$. For the third quartile I calculate the position $\dfrac{3 \cdot 8}{4} = 6$, and since the sixth value in the list is a 5, then $Q_3 = 5$".

1. a) Point out the fundamental conceptual error Marcos has made when handling the data. Explain why his final conclusion ($Q_1 = 8$ and $Q_3 = 5$) is mathematically absurd according to the definition of a quartile.
2. b) Correct the procedure: order the dataset and calculate the actual values of $Q_1$, the median ($Me$), and $Q_3$, justifying your steps.

You want to design a hypothetical sample of $N = 7$ integer grades (from 1 to 10) for a small group of students that simultaneously meets these four conditions:

• The minimum grade is 2 and the maximum grade is 9.

• The sample median is exactly 6.

• The first quartile is $Q_1 = 4$.

• The third quartile is $Q_3 = 8$.

1. a) Propose an ordered list of 7 grades that satisfies all the conditions, justifying the position occupied by each measure.
2. b) Is this the only possible sample or can several valid combinations exist? Reason your answer by indicating which elements have freedom of variation and between which limits they can range.

The time, in minutes, that a group of 40 teenagers spend per day on social media has been grouped in the following table:

Identify the median class and calculate the exact value of the median using the linear interpolation formula.

The following cumulative frequency polygon shows the distribution of monthly salaries, in euros, of the 50 employees of a tech company. From the graph, estimate approximately the value of the median ($Me$), the lower quartile ($Q_1$) and the upper quartile ($Q_3$).

The shoe sizes of the 30 players in a basketball club are distributed according to the following table:

Calculate the percentiles $P_{20}$ and $P_{80}$ of the distribution and interpret the results obtained in the context of the problem.

The attached box plot represents the lifespan, in hours, of a sample of 200 LED light bulbs of a certain brand. Answer the following questions, justifying your answers:

1. a) What percentage of light bulbs last more than $1500$ hours?
2. b) Between which values is the central $50\%$ of the distribution located?
3. c) How many light bulbs exactly lasted less than $1200$ hours?

The following box plots show the marks obtained in a Science exam by two different classes, A and B.

1. a) Which class has a greater dispersion in the central $50\%$ of its marks?
2. b) In which class is the mark that marks the boundary of the lowest $75\%$ of marks higher?
3. c) Which of the two classes do you consider to have had a more consistent performance overall? Justify your answer.

The distance, in kilometres, travelled daily by 50 employees to get to their workplace is distributed as follows:

Calculate the percentile $90$ ($P_{90}$) using the linear interpolation formula and explain what information this value provides to the company for planning parking spaces or transport subsidies.

The weight, in grams, of 60 apples harvested in an orchard has been tabulated in intervals:

Calculate the interquartile range of the weights of the apples using linear interpolation to find $Q_1$ and $Q_3$ beforehand.

The ages of a group of 6 friends are: $14, 15, 15, 16, 17, 18$. A new friend joins the group. Determine what age this new member must be so that the median age of the whole group becomes exactly $16$ years.

The daily travel time (in minutes) taken by a sample of 50 4th-year ESO students to commute to their school has been tabulated as follows:

1. a) Estimate: Determine in which interval the 60th percentile ($P_{60}$) is located. Without doing any complex algebraic calculation yet, reason whether the value of $P_{60}$ should be closer to 30, to the midpoint (37,5), or to 45. Provide an intuitive estimate of its value.
2. b) Check: Accurately calculate the value of $P_{60}$ using the linear interpolation formula. Compare the value obtained with your previous estimate and explain the meaning of the result in the context of the students' commute.

In a quality survey conducted with 200 users about a mobile app, the rating is measured using discrete stars from 1 to 5. The results were: 20 people gave 1 star; 45 gave 2; 75 gave 3; 40 gave 4; and 20 gave 5.

Three students debate how to calculate the quartiles of this dataset:

• Sergio proposes listing the 200 data points individually on a sheet of paper and crossing them out from the extremes toward the center.

• Lucía proposes treating the data as if they were continuous by creating intervals $[0,5; 1,5), [1,5; 2,5), \dots$ and applying the continuous linear interpolation formula.

• Alba proposes calculating the cumulative frequencies ($F_i$) directly on the discrete table and finding the first value where $F_i \ge \dfrac{k \cdot N}{4}$.

1. a) Compare the three alternatives: explain what evident practical problem Sergio's approach presents and what conceptual error Lucía makes by applying continuous interpolation to a discrete variable with isolated integer values.
2. b) Apply Alba's method to calculate $Q_1$, the median ($Me$), and $Q_3$, justifying your steps. Interpret the users' satisfaction level based on the median obtained.

On a music app, the ratings (from 1 to 5 stars) given by a group of users have the following distribution:

It is known that the median of the ratings is $3,5$ stars. Determine the value of the unknown frequency $x$, reasoning your answer, and then calculate the lower quartile ($Q_1$) of the distribution.

The monthly rainfall, in mm, recorded in a region over several years is grouped in the following table:

Find the value of $k$ such that the percentile $P_k$ is exactly $65$ mm. What percentage of months had less rainfall than that amount?

The following graph represents the cumulative frequency polygon of the charging time, in minutes, of the battery of 40 electric vehicles. From the graph, reconstruct the absolute frequency table by intervals and calculate analytically the upper quartile ($Q_3$). Compare the analytical result with the value observed directly on the graph.

An electronics shop has a price list for 20 models of headphones. It is known that the lower quartile of the prices is $30$ €, the median is $45$ € and the upper quartile is $60$ €. For Black Friday week, the shop applies a $20\%$ discount to all headphones and, in addition, deducts an extra $5$ € at the till. Determine what the new values of $Q_1$, $Me$ and $Q_3$ will be after applying the promotion, justifying your answer mathematically.

The reaction times, in milliseconds, of 15 professional gamers in a video game are: $210, 245, 215, 230, 220, 250, 215, 225, 240, 235, 218, 222, 260, 228, 232$.

Calculate the five-number summary (minimum, $Q_1$, median, $Q_3$, maximum) for this group. If an amateur player has a box plot whose quartiles are $Q_1 = 240$ and $Q_3 = 280$, reason statistically whether it is possible to state that more than $25\%$ of professional gamers have worse (higher) reaction times than the fastest $25\%$ of amateur players.

The test scores of 6 students are integers from 1 to 10. It is known that the arithmetic mean is $6,5$, the median is $6,5$, the mode is $7$ (and is unique), and the first quartile ($Q_1$) is $5$. Reconstruct the exact scores of the 6 students, explaining step by step the reasoning followed.

The lifespan, in thousands of kilometres, of two brands of tyres, X and Y, has been studied.

For brand X:

For brand Y:

Calculate the interquartile range of both brands and determine which of the two offers a more consistent lifespan (less spread out in the central $50\%$).

The distribution of monthly income, in hundreds of euros, of a residential neighbourhood is grouped in the following table:

Calculate the first decile ($D_1$) and the ninth decile ($D_9$). Subsequently, find the ratio $\frac{D_9}{D_1}$ and explain what this quotient indicates about economic inequality in the neighbourhood.

From a sample of 120 workers in an industrial sector in Spain (with data inspired by the INE's Wage Structure Survey), the gross monthly salary (in euros) has been recorded in the following table:

1. a) Calculate the median ($Me$) and the 80th percentile ($P_{80}$) of the wage distribution using linear interpolation.
2. b) The local administration claims: "Since the arithmetic mean salary of this group is around €1850, the vast majority of the staff earns more than €1800." Use the calculated position measures and the analysis of the table to rigorously refute this claim, explaining why the median represents reality better than the mean in skewed economic variables.
3. c) A transport subsidy is proposed for the $20\,\%$ of workers with the lowest wages. Deduce the maximum salary threshold to receive the subsidy and indicate which position measure it corresponds to exactly.

In a residential community of 80 homes, the monthly electricity consumption in kilowatt-hours (kWh) is analyzed. The consumptions were grouped into 4 intervals of identical width ($c = 50$): $[100, 150)$, $[150, 200)$, $[200, 250)$, and $[250, 300)$.

Due to a computer glitch, the data in the table were deleted, but the administrator clearly remembers three key pieces of information:

1. In the first interval, $[100, 150)$, there are exactly 12 homes registered.

2. The first quartile, obtained using the linear interpolation formula, was exactly $Q_1 = 170$ kWh.

3. The 90th percentile of the community was $P_{90} = 275$ kWh.

1. a) Set up the corresponding equations from the interpolation formulas and determine, with reasoning, the absolute frequencies ($f_1, f_2, f_3, f_4$) of the four intervals in the table.
2. b) With the table reconstructed, calculate the median consumption ($Me$) of the community.
3. c) What exact percentage of the homes in this community has a consumption between $Q_1$ and the median? Reason whether your answer depends on the specific numerical data of this problem or if it is a general property for any continuous statistical distribution.