Exercises: Box plots

The grades of a group of $9$ students in a math 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 the visitors to a museum during a morning.

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

The five-number summary of the weights (in pounds) of the dogs treated in 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 box plot representing the time (in minutes) that students take to get to school, the lower quartile is $15$ and the upper quartile is $30$.

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

A box plot shows that the distance between the lower quartile and the median is $2$ units, while the distance between the median and the upper quartile is $10$ units.

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

The prices (in dollars) 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 their box plot.

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

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

In a box plot, 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 dispersion of the data at the extremes of the distribution?

The following graph shows the box plots of the battery life (in hours) of two brands of cell phones, Alpha and Beta.

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

A basketball team has scored the following points in its last $15$ games:

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

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

A box plot summarizes the heights of $120$ athletes. If the upper quartile ($Q_3$) is $185$ inches and the maximum value is $205$ inches, approximately how many athletes have a height greater than $185$ inches? Assume there are no repeated values that coincide exactly with the quartiles.

In a study on the number of heartbeats per minute of a group of athletes at rest, the following summary was obtained: $Q_1 = 48$, $Me = 52$, $Q_3 = 58$.

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

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 to be able to construct the corresponding box plot.

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

In an engineering company, the $25\%$ of the highest-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 box plot (Minimum, $Q_1$, $Me$, $Q_3$, Maximum) the values $\$3500$, $\$1200$ and $\$2100$ correspond to.

A box plot has been constructed with 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, the following year all the recorded maximum temperatures increase by exactly $3^\circ \text{C}$, what will be the new values of $Q_1$, $Me$ and $Q_3$? Will the value of the interquartile range change?

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 set of data on 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 to be considered an upper outlier in the box plot. (Remember that for this calculation $N=8$).

Make up a set of data consisting of exactly $6$ positive integers whose box plot has the following parameters:

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

(Note: Use the 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 dollars) 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 be able to sketch their box plot.

In a box plot of the income of a population, it is observed that the distance from $Q_1$ to the median is $\$500$, while from the median to $Q_3$ is $\$2500$. In addition, the upper whisker is much longer than the lower one.

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

A set of data 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 the variable $y$ and determine its new interquartile range. Be especially careful with the order of the quartiles when multiplying by a negative number.

A box plot summarizes the center and dispersion using quartiles, but it can hide the true shape of the distribution.

Propose two different sets of data 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.