Exercises: Box plots

The marks of a group of $9$ students in a mathematics exam are: $3, 4, 5, 6, 7, 8, 8, 9, 10$.

Determine the minimum value, the lower quartile ($Q_1$), the median ($Me$), the upper quartile ($Q_3$) and the maximum value to construct a box plot.

The following box plot represents the ages of visitors to a museum during one morning.

From the graph, state the minimum age, lower quartile, median, upper quartile and maximum age of the visitors.

The five-number summary of the weights (in kg) of the dogs treated at a veterinary clinic is: Minimum = $5$, $Q_1 = 12$, Median = $18$, $Q_3 = 24$, Maximum = $35$.

Calculate the range of the distribution and the interquartile range (IQR).

In a boxplot representing the time (in minutes) that students take to get to school, the first quartile is $15$ and the third quartile is $30$.

What exact percentage of students take between $15$ and $30$ minutes to get to school?

A boxplot shows that the distance between the first quartile and the median is $2$ units, while the distance between the median and the third quartile is $10$ units.

What does this difference indicate about the distribution and concentration of data in the central part?

The prices (in euros) of $11$ books on a shelf are: $12, 15, 15, 18, 20, 21, 24, 25, 28, 30, 35$.

Calculate the five values needed to draw its boxplot.

State whether the following statement is true or false and briefly justify your answer:

"In any boxplot, the arithmetic mean always coincides exactly with the interior vertical line that divides the box."

In a boxplot, the left whisker (from the minimum to $Q_1$) is very short, while the right whisker (from $Q_3$ to the maximum) is extremely long.

What does this mean regarding the spread of the data at the extremes of the distribution?

Marcos is analyzing the waiting time (in minutes) at an urban bus stop and, after ordering the collected data, has obtained the following five-number summary: Minimum = $2$, $Q_1 = 5$, Median = $8$, $Q_3 = 11$, and Maximum = $22$.

When writing up his conclusions, Marcos states: "Since the interquartile range is $RIC = 11 - 5 = 6$, any time greater than $Q_3 + RIC = 11 + 6 = 17$ minutes is an outlier. Therefore, the time of $22$ minutes is an outlier because it exceeds $17$ minutes".

1. a) Identify and explain in your own words the conceptual error Marcos makes in the formula to determine the boundary for outliers.
2. b) Apply the correct box plot criterion to calculate the cutoff beyond which a data point is considered an outlier, and reason whether the value of $22$ minutes is still an outlier or if Marcos just got the conclusion right by chance.

At an AEMET station in a Spanish provincial capital, daily maximum temperatures (in $^{\circ}\text{C}$) have been recorded over a spring week. The ordered values are: $14$, $16$, $17$, $19$, $21$, $24$, and $29$.

1. a) Before calculating anything, estimate by observing the data: what percentage of the days of the week will be contained within the central box of the plot? Looking at the spacing between consecutive numbers, estimate whether the box will be symmetrical with respect to the median ($19$) or if it will be more stretched to the right or to the left. Justify your intuition.
2. b) Following the method from the notes for $N = 7$ (where $Q_1$ and $Q_3$ are the medians of the lower and upper halves, excluding the overall median), calculate the exact value of $Q_1$, $Me$, $Q_3$, and the interquartile range ($RIC$).
3. c) Compare the calculated results with your previous estimation and check whether your deduction about the shape and skewness of the box was correct.

The following graph shows the boxplots for the battery life (in hours) of two mobile phone brands, Alpha and Beta.

1. a) Which brand has a higher median battery life?
2. b) Which of the two brands shows greater variability in the central $50\%$ of its data (larger interquartile range)?

A basketball team has scored the following points in their last $15$ matches:

$65, 72, 78, 81, 82, 85, 88, 90, 92, 95, 98, 101, 105, 108, 112$.

Find the five-number summary and calculate the interquartile range.

A boxplot summarises the heights of $120$ athletes. If the third quartile ($Q_3$) is $185$ cm and the maximum value is $205$ cm, approximately how many athletes have a height greater than $185$ cm? Assume there are no repeated values coinciding exactly with the quartiles.

In a study on the resting heart rate (in beats per minute) of a group of athletes, the following summary was obtained: $Q_1 = 48$, $Me = 52$, $Q_3 = 58$.

One of the athletes recorded $75$ beats per minute. Using the mathematical rule of the interquartile range (a value is an outlier if it is greater than $Q_3 + 1,5 \cdot RIC$), determine whether this value should be represented as an outlier in the boxplot.

The following frequency table shows the number of siblings of the students in a class:

• $0$ siblings: $6$ students

• $1$ sibling: $10$ students

• $2$ siblings: $8$ students

• $3$ siblings: $4$ students

• $4$ siblings: $2$ students

Determine the minimum, $Q_1$, $Me$, $Q_3$ and the maximum in order to construct the corresponding boxplot.

A boxplot has a very narrow box shifted towards the left end, with a very long right whisker. If you had to represent these same data in a histogram, what shape would the distribution have? (Symmetrical, right-skewed or left-skewed). Briefly explain why.

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

In an engineering firm, the top $25\%$ earning employees have a monthly salary greater than $3500$ €. In addition, the lowest salary in the company is $1200$ € and $50\%$ of the workforce earns less than $2100$ €.

Identify which specific parameters of the boxplot (Minimum, $Q_1$, $Me$, $Q_3$, Maximum) correspond to the values $3500$ €, $1200$ € and $2100$ €.

A boxplot has been constructed using the maximum temperatures of a city in July, obtaining $Q_1 = 28^\circ \text{C}$, $Me = 31^\circ \text{C}$ and $Q_3 = 35^\circ \text{C}$.

If, due to a heatwave, all recorded maximum temperatures the following year increase by exactly $3^\circ \text{C}$, what will the new values of $Q_1$, $Me$ and $Q_3$ be? Will the value of the interquartile range change?

The customer service of Cercanías Renfe analyzes the delay minutes on $25$ trips across two different lines ($L_1$ and $L_2$) to evaluate their reliability. The resulting statistical summaries are:

• Line $L_1$: Minimum = $1$ min, $Q_1 = 2$ min, Median = $3$ min, $Q_3 = 4$ min, Maximum = $20$ min (a single train suffered an overhead line failure).

• Line $L_2$: Minimum = $0$ min, $Q_1 = 3$ min, Median = $6$ min, $Q_3 = 8$ min, Maximum = $10$ min (no serious breakdowns, but frequent delays).

1. a) Which measure of dispersion, the range or the interquartile range ($RIC$), is more suitable for comparing the typical punctuality of both lines? Justify your choice by explaining how the isolated $20$-minute breakdown on line $L_1$ affects each indicator.
2. b) A report proposes announcing that "line $L_2$ is more reliable because its range is only $10$ minutes compared to $19$ minutes for line $L_1$". Based on the box plot (analyzing the median and the spread of the central $50\,\%$ of the trips), refute that statement and justify which line you would recommend to someone who needs to get to work on time regularly.

A Dual Vocational Training tutor summarizes in a box plot the weekly hours of workplace training completed by the $7$ students in her group. She has misplaced the original list, but she still has the following exact figures from the plot:

• Minimum value = $10$ hours

• First quartile ($Q_1$) = $14$ hours

• Median ($Me$) = $18$ hours

• Third quartile ($Q_3$) = $22$ hours

• Range = $15$ hours

1. a) Deduce the value of the maximum data point and calculate the interquartile range ($RIC$) of the distribution.
2. b) Reconstruct an ordered list of $7$ integers that strictly satisfies this five-number summary, explaining how you place each value according to the definition of median and quartiles for $N = 7$.
3. c) Is your list the only possible combination of data that produces that same box plot? Prove your answer by indicating which specific value you could change, within which integer range it could vary, and why it would not affect any of the five numbers in the plot.

A data distribution is perfectly symmetrical. It is known that its range is $40$ units, its interquartile range (IQR) is $16$ units and the minimum value is $10$.

Calculate the exact values of $Q_1$, $Me$, $Q_3$ and the maximum value.

Given the following ordered data set for the reaction time (in milliseconds) of an experiment: $120, 135, 140, 145, 150, 160, 170, x$.

Find the minimum value that the unknown $x$ must have for it to be considered an upper outlier in the boxplot. (Remember that for this calculation $N=8$).

Create a data set consisting of exactly $6$ positive integers whose boxplot has the following parameters:

Minimum = $2$, $Q_1 = 5$, Median = $8$, $Q_3 = 12$, Maximum = $15$.

(Note: Use standard interpolation rules for the median of the lower and upper halves to calculate the quartiles).

The following graph shows the execution times (in milliseconds) of three different computer algorithms processing the same amount of data.

If you need to choose an algorithm for a critical real-time system where it is essential that the execution time never exceeds $60$ ms, but at the same time you want the overall performance (median) to be as fast as possible, which algorithm would you choose and why? Justify your answer by ruling out the other two.

The salaries (in thousands of euros) of $200$ workers are grouped into the following intervals:

• $[10, 20)$: $40$ workers

• $[20, 30)$: $80$ workers

• $[30, 40)$: $60$ workers

• $[40, 50)$: $20$ workers

Estimate the values of $Q_1$, $Me$ and $Q_3$ using linear interpolation within the intervals to sketch its boxplot.

In a boxplot of income for a population, it is observed that the distance from $Q_1$ to the median is $500$ €, while from the median to $Q_3$ it is $2500$ €. Furthermore, the upper whisker is much longer than the lower one.

What relationship do you expect between the arithmetic mean and the median (which will be larger)? Reason your answer based on the skewness shown by the box.

A data set has the following five-number summary: Minimum = $10$, $Q_1 = 20$, $Me = 25$, $Q_3 = 35$, Maximum = $50$.

The linear transformation $y = -2x + 100$ is applied to all the data.

Calculate the new five-number summary for variable $y$ and determine its new interquartile range. Take special care with the order of the quartiles when multiplying by a negative number.

A boxplot summarises the centre and spread using quartiles, but it can hide the actual shape of the distribution.

Propose two different data sets of $8$ elements each: the first must be a uniform distribution (equally spaced values) and the second a strongly bimodal distribution (values concentrated at the extremes), such that BOTH sets have exactly the same Minimum, $Q_1$, Median, $Q_3$ and Maximum.

From a plausible sample of data from the INE Wage Structure Survey on young people under $25$ years of age in Spain, the following five-number summary of their monthly net salary (in euros) has been obtained:

1. a) Calculate the interquartile range ($RIC$) and determine the exact numerical thresholds (outlier fences) to consider a salary an outlier. Reason mathematically whether the maximum salary of $2100\text{ €}$ is an outlier.
2. b) If the second highest salary in the entire sample is $1520\text{ €}$, describe in detail how the upper end of the box plot should be drawn: up to which numerical point does the right whisker reach, and how is the $2100\text{ €}$ salary represented?
3. c) At a panel discussion, a spokesperson states: "Since the right whisker is much longer than the left one and the box is wider upwards ($Q_3 - Me = 170\text{ €}$) than downwards ($Me - Q_1 = 70\text{ €}$), the distribution is skewed to the right, so we can state with absolute mathematical certainty that the mean salary is higher than the median ($1150\text{ €}$)". Based on what you have studied about box plots, critically evaluate this claim: why is the plot's skewness an indication, but not a sufficient guarantee to ensure the position of the mean?

An educational research team is studying the daily time (in minutes) spent on social media by $8$ students in Year 10 of ESO. You are asked to model a dataset that exactly satisfies these four statistical conditions:

1. The median must be exactly $75$ minutes.

2. The interquartile range must be $RIC = 20$ minutes, with a box perfectly symmetrical with respect to the median ($Me - Q_1 = Q_3 - Me$).

3. There must be no lower outliers, but there must be exactly one upper outlier.

4. The minimum value of the sample must be $40$ minutes.

1. a) Deduce the mandatory numerical values of $Q_1$ and $Q_3$, and calculate the exact numerical threshold beyond which a time is considered an upper outlier.
2. b) Construct an ordered list of $8$ integers that satisfies all the conditions, explaining how you calculate the median and quartiles for $N = 8$ (using the mean of the central positions of each half).
3. c) Prove that with that minimum value of $40$ minutes it is impossible for lower outliers to appear, and justify what bound condition the seventh value ($x_7$) must satisfy to ensure that only the eighth value ($x_8$) is an outlier.

Topic challenge. Do screens influence our lifestyle? As a junior data analyst, you will investigate the relationship between mobile phone use and sports in your environment through a short statistical report.

1. Population and sample: Define the study population and survey a sample of at least 15 peers or family members. Collect two pieces of data per person: daily screen hours and whether they regularly practice sports (Yes/No).

2. Contingency table: Group the hours into two intervals (for example, $\le 3,5$ h and $> 3,5$ h) and create a two-way contingency table relating both variables.

3. Parameters and box plot: Enter the continuous hours data into a spreadsheet or GeoGebra. Calculate the mean, median, mode, variance, and standard deviation. Generate the box plot, identifying the interquartile range and any outliers.

4. Reasoned proposal: Write a short report interpreting the dispersion, the representativeness of the sample, and whether an association between both variables is observed.

Assessment criteria:

• Correct identification of population, sample, and accurate preparation of the contingency table.

• Accuracy in the digital calculation of measures of central tendency (mean, median, mode) and dispersion (variance, standard deviation).

• Proper construction and interpretation of the box plot.

• Thoroughness and critical thinking in the conclusions of the final report.