Box plots
Which phone has the more reliable battery: one that almost always lasts between 18 and 22 hours, or one that sometimes lasts 10 and sometimes 30? The mean could be the same for both. A box plot lets you see at a glance how the data are spread out: not just their centre, but also their dispersion and shape.
The five-number summary
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 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 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.
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.
Drawing the box plot
The diagram shows a box from $Q_1$ to $Q_3$ (with the median marked inside) and two "whiskers" that reach out to the minimum and the maximum.
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 = $35$.…
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).
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…
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."
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…
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.
Comparing distributions
The great advantage of box plots is that you can stack several of them to compare centre, spread and symmetry at a glance.
The following graph shows the boxplots for the battery life (in hours) of two mobile phone brands, Alpha and Beta. a) Which brand has a higher median battery life? b)…
The following graph shows the boxplots for the battery life (in hours) of two mobile phone brands, Alpha and Beta.
- a) Which brand has a higher median battery life?
- b) Which of the two brands shows greater variability in the central $50\%$ of its data (larger interquartile range)?
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…
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 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 $2500$ €.…
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.
Outliers
An extremely long whisker can hide an unusual value (an outlier) that is better marked separately than by stretching the whisker all the way to it.
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…
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.
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…
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 $N=8$).
Back to the start: The phone with a battery life of 18 to 22 hours is the more reliable one: even if both have a similar mean, its box plot will have an IQR of only $4$ hours and very short whiskers, which guarantees a much more predictable battery life with no surprises.