Exercises: Box plots

The ages of 7 students in a drama workshop have been recorded: $12, 14, 15, 18, 20, 22, 25$.

Calculate the minimum, the first quartile ($Q_1$), the median ($Me$), the third quartile ($Q_3$) and the maximum. Also determine the interquartile range ($IQR$).

Observe the following box plot representing the points scored by a basketball player across several matches.

Identify the values of the five-number summary (minimum, $Q_1$, median, $Q_3$, maximum) and calculate the width of the box (interquartile range).

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

In a study on the weight of school backpacks, it was found that the first quartile is $Q_1 = 4,5$ kg and the third quartile is $Q_3 = 7,2$ kg.

Calculate the interquartile range ($IQR$). If the minimum weight recorded is $2$ kg, what is the length of the left whisker in the box plot?

Given the following list of temperatures in degrees Celsius recorded over 8 days, in no particular order: $15, 12, 19, 14, 18, 10, 21, 16$.

Order the data from smallest to largest and calculate the five-number summary required to draw a box plot.

Answer the following conceptual questions about a box plot:

• What approximate percentage of the data lies inside the box (between $Q_1$ and $Q_3$)?

• What percentage of the data lies in each of the whiskers (assuming there are no outliers)?

• If the box is very narrow, what does this indicate about the spread of the central data?

A group of researchers summarised the height (in cm) of a plant population with the following five numbers: Minimum = $110$, $Q_1 = 125$, $Me = 135$, $Q_3 = 142$, Maximum = $160$.

Draw the corresponding box plot in your notebook, clearly showing the scale used on the horizontal axis.

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

The time (in minutes) taken by 9 people to solve a puzzle has been measured: $10, 12, 14, 15, 18, 20, 21, 25, 30$.

Calculate the median and the quartiles $Q_1$ and $Q_3$. Is the value $30$ considered an outlier according to the $1,5 \cdot IQR$ rule?

State whether the following statements are true or false, justifying your answer in each case:

• The line marking the median is always located at the exact geometric centre of the box.

• The two whiskers of a box plot always have the same length.

• The interquartile range is not affected by extreme values (maximum and minimum).

Marcos analyzes the ages of the members of a youth chess club. After ordering the data, he obtains the following five-number summary: $\text{Mínimo} = 8$ years, $Q_1 = 12$ years, $\text{Mediana} = 15$ years, $Q_3 = 18$ years, and $\text{Máximo} = 30$ years.

To check whether the $30$-year-old person should be considered an outlier, Marcos writes in his notebook:

"$RIC = 18 - 12 = 6$. Since the median is $15$, the boundary beyond which a data point is an outlier is $\text{Mediana} + 1{,}5 \cdot RIC = 15 + 1{,}5 \cdot 6 = 15 + 9 = 24$. Since $30 > 24$, the age of $30$ is an outlier."

1. a) Identify and explain the conceptual error Marcos made in the formula for the upper bound.
2. b) Correct the calculation by applying the proper criterion and determine what the actual upper bound is beyond which an age is considered an outlier in this group. Is $30$ still considered an outlier after the correction?

In a speed trap on an urban road with a $50\text{ km/h}$ speed limit, the Local Police record the speeds (in $\text{km/h}$) of $9$ offending vehicles:

$$52,\; 53,\; 54,\; 55,\; 56,\; 58,\; 60,\; 85,\; 95$$

1. a) Estimate without calculating the mean: Observe the position of the central value and the distance of the extreme values relative to it. Do you expect the box plot to be symmetric or skewed? Based on this shape, explain whether you estimate that the arithmetic mean will be noticeably greater than, less than, or equal to the median.
2. b) Verify with calculations: Find the five-number summary (minimum, $Q_1$, median, $Q_3$, maximum) and the interquartile range ($IQR$). Then, calculate the exact arithmetic mean and check whether it confirms your previous estimate about the skewness.

The following graph shows the box plots of resting heart rates in beats per minute for two groups of athletes, A (distance runners) and B (weightlifters).

Based on observing the graphs, answer the following:

• Which group has a lower median?

• Which group shows greater spread in the central $50\%$ of its data?

• Calculate the interquartile range for both groups and compare them.

A group of families were asked about the number of pets they have, obtaining the following frequency table:

• $0$ pets: $8$ families

• $1$ pet: $12$ families

• $2$ pets: $10$ families

• $3$ pets: $6$ families

• $4$ pets: $4$ families

Calculate the quartiles $Q_1$, $Me$ and $Q_3$ from the cumulative frequencies and describe in detail what the corresponding box plot would look like.

In a statistical distribution, the first quartile is $Q_1 = 40$ and the third quartile is $Q_3 = 64$.

Calculate the interquartile range ($IQR$). Next, determine the lower and upper limits to identify outliers (using the $1,5 \cdot IQR$ rule). Would the value $105$ be considered an outlier? What about the value $3$?

In a class of 32 students, the box plot of the final exam marks shows that $Q_1 = 4$, $Me = 6$ and $Q_3 = 7,5$.

Knowing that there are no tied marks matching the quartiles exactly, approximately how many students scored a mark less than or equal to $4$? How many students scored a mark between $4$ and $7,5$?

The following table shows the time (in hours) spent per week playing video games by a group of young people, grouped into intervals:

• $[0, 5)$: $10$ young people

• $[5, 10)$: $25$ young people

• $[10, 15)$: $30$ young people

• $[15, 20)$: $15$ young people

Calculate the position of the quartiles $Q_1$, $Me$ and $Q_3$ by interpolating within the corresponding classes.

The salaries (in euros) of 5 employees in a small company are: $1100, 1200, 1300, 1500, 2500$.

Calculate the five-number summary. If the employee earning $2500$ € receives a pay rise of $500$ € (now earning $3000$ €), how does this affect the median and the interquartile range? What advantage does the box plot have over the arithmetic mean in this case?

A box plot has the following values: Minimum = $10$, $Q_1 = 15$, $Me = 18$, $Q_3 = 35$, Maximum = $40$.

Analyse the skewness of the distribution. Is it skewed to the right or to the left? Justify your answer by comparing the distances between the quartiles and the median.

The marks for an exam out of 10 points have the following summary: Minimum = $3$, $Q_1 = 5$, $Me = 6$, $Q_3 = 8$, Maximum = $10$.

The teacher decides to transform the marks by multiplying them by $0,8$ and adding $2$ points. Calculate the new five-number summary and the new interquartile range.

The customer service department of a telecommunications company records the waiting time (in minutes) of $100$ users across two different channels:

• Phone channel: $\text{Minimum} = 1$, $Q_1 = 3$, $\text{Median} = 4$, $Q_3 = 5$, $\text{Maximum} = 45$. Mean $= 5{,}8\text{ min}$; Range $= 44\text{ min}$; $IQR = 2\text{ min}$. (The maximum of $45$ was due to a one-off IT system outage on a specific day).

• Web chat channel: $\text{Minimum} = 3$, $Q_1 = 6$, $\text{Median} = 8$, $Q_3 = 10$, $\text{Maximum} = 12$. Mean $= 7{,}9\text{ min}$; Range $= 9\text{ min}$; $IQR = 4\text{ min}$.

Management is debating which method to use to compare the consistency and quality of the service:

• Method 1: Compare dispersion using the total range ($\text{Maximum} - \text{Minimum}$).

• Method 2: Compare dispersion using the interquartile range ($IQR$) and the median.

1. a) If the company were to use Method 1, which channel would appear much more dispersed and unpredictable? Does that conclusion reflect the typical experience of the central $50\,\%$ of users?
2. b) Choose with justification which of the two statistical methods is more representative for assessing the everyday quality of the service, and explain why the $IQR$ is more robust against isolated technical incidents.

The State Meteorological Agency (AEMET) records the following minimum temperatures (in degrees Celsius) over $15$ days in January at a high mountain weather station:

$$-12,\; -10,\; -9,\; -8,\; -6,\; -5,\; -4,\; -3,\; -2,\; -1,\; 0,\; 2,\; 3,\; 4,\; 18$$

1. a) Model the distribution of these temperatures by calculating the five-number summary (minimum, $Q_1$, $\text{Median}$, $Q_3$, maximum) and the interquartile range ($IQR$).
2. b) Determine using Tukey's mathematical rule whether the record of $18\;^\circ\text{C}$ (caused by an exceptional warm wind event) should be classified as an outlier. In light of the result, up to what numerical value would the right whisker of the box plot be drawn?
3. c) If all temperatures are transformed to the Fahrenheit scale using the linear function $F = 1{,}8 \cdot C + 32$, calculate the new median and the new interquartile range ($IQR_F$) by applying the properties of linear transformations, without converting the $15$ data points one by one.

Reverse engineering. Create a data set consisting of 7 integers that simultaneously satisfies the following conditions for its box plot:

• Minimum = $2$

• $Q_1 = 4$

• Median = $7$

• $Q_3 = 10$

• Maximum = $15$

Show step by step that your data set meets the required values.

The following histogram shows the age distribution of visitors to a museum during an afternoon.

From the frequencies shown in the histogram, determine the intervals containing the first quartile ($Q_1$), the median ($Me$), and the third quartile ($Q_3$). Justify your calculations using cumulative frequencies.

In a statistical distribution, the quartiles are given by algebraic expressions in terms of a parameter $x > 0$:

• $Q_1 = x$

• $Me = 2x + 1$

• $Q_3 = x^2 - 2$

Knowing that the interquartile range ($IQR$) is equal to $10$, set up an equation, solve it to find the value of $x$, and calculate the numerical values of $Q_1$, $Me$ and $Q_3$.

In a box plot, the right whisker is extremely long compared to the left whisker and the box. For example: Minimum = $10$, $Q_1 = 20$, $Me = 25$, $Q_3 = 35$, Maximum = $120$.

Reason and mathematically justify whether the arithmetic mean of this distribution will be greater than, less than, or approximately equal to the median. Why does this happen?

Consider a data set consisting of 6 numbers: $4, 8, 10, 15, 20, y$.

Knowing that the median of this data set is $12$, find the unknown value of $y$. Once found, calculate the first and third quartiles, and determine whether there are any outliers.

Prove with a counterexample that a box plot is not sufficient to identify whether a distribution is bimodal (that is, whether it has two modes or frequency peaks).

To do this, create two data sets of 8 elements each: one that is clearly unimodal and another that is clearly bimodal, but such that both have exactly the same five-number summary (Minimum, $Q_1$, $Me$, $Q_3$, Maximum).

A company analyses the lifespan (in hours) of two brands of light bulbs, A and B. Both brands have the same median ($Me = 1000$ h) and the same interquartile range ($IQR = 200$ h). However, the overall range for brand A is $400$ h, whereas the overall range for brand B is $1200$ h.

What information does this difference in overall ranges give us regarding the reliability and the presence of outliers in both brands? Which of the two brands would you consider more predictable and why?

The heart rate (in beats per minute) of 11 athletes after a race has been recorded: $110, 115, 120, 122, 125, 130, 135, 140, 145, 150, 200$.

Calculate the five-number summary and identify whether the value $200$ is an outlier using the $1,5 \cdot IQR$ criterion. If we remove this outlier from the data set, how do the mean and median change? Which measure of central tendency is more robust against this change?

Reverse engineering of a salary distribution. In a sociological study, the monthly net salaries of $N = 10$ employees of a cooperative are collected, expressed in hundreds of euros and ordered from smallest to largest: $x_1 \le x_2 \le \dots \le x_{10}$.

Construct a numerical series of $10$ integer values that simultaneously satisfies all of the following conditions in its box plot:

1. $\text{Minimum} = 10$ ($1000\text{ \euro}$).

2. $Q_1 = 13$ and $Q_3 = 19$. (For a sample of $10$ ordered data points, take as quartiles the median of each half: $Q_1 = x_3$ and $Q_3 = x_8$).

3. The median of the sample is exactly $16$ ($1600\text{ \euro}$), but no employee in the central positions earns exactly that amount ($x_5 \ne x_6$).

4. In the distribution there is exactly one upper outlier and no lower outlier.

Explicitly calculate the fences that define the outliers, propose the $10$ numerical values $\{x_1, x_2, \dots, x_{10}\}$ and justify step by step why your set satisfies each of the requirements.

Two urban bus lines (Line A and Line B) connect the train station with a university campus. Based on the records of travel times (in minutes) throughout the school year, their box plots have been drawn, with the following characteristic data:

• Line A: $\text{Minimum} = 12$, $Q_1 = 18$, $\text{Median} = 25$, $Q_3 = 28$, $\text{Maximum} = 34$ (no outliers).

• Line B: $\text{Minimum} = 15$, $Q_1 = 20$, $\text{Median} = 22$, $Q_3 = 24$, $\text{Maximum} = 42$ (with a single outlier at $42$; the last non-outlier value where the whisker ends is $29$).

1. a) A student arrives at the station at $8:30$ and their exam starts at $9:00$ (they have a $30$-minute margin). Deduce from the quartiles what percentage of trips takes $28$ minutes or less on Line A and what percentage takes $24$ minutes or less on Line B. Which line offers a better guarantee that they will arrive on time at least $75\,\%$ of the time?
2. b) Analyze the skewness of each line by comparing the distances $(Me - Q_1)$ versus $(Q_3 - Me)$ and the length of the whiskers. Justify in which of the two lines the arithmetic mean is expected to be clearly lower than the median.
3. c) To speed up traffic, a reserved lane is introduced that reduces travel time according to the relationship $t' = 0{,}8 \cdot t - 1$. Prove algebraically in a general way that if any data point $x$ was an upper outlier in the original variable $t$, its transformed value $x'$ must necessarily still be an upper outlier in the variable $t'$.