Box-and-Whisker 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 ($IQR$).

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.

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.

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?

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?

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 center 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).

The salaries (in pesos) of 5 employees in a small company are: $22\,000, 24\,000, 26\,000, 30\,000, 50\,000$.

Calculate the five-number summary. If the employee earning $50\,000$ pesos receives a pay raise of $10\,000$ pesos (now earning $60\,000$ pesos), 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 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.

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 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.

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 $1.5 \times IQR$ rule?

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 $1.5 \times IQR$ rule). Would the value $105$ be considered an outlier? What about the value $3$?

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 $1.5 \times IQR$ 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?

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.

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 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.

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 $7.5$?

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?

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?

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).

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.

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.

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 $Q_3$.