Lesson 73 — Quartiles, percentiles, and boxplots

Data: 1, 3, 5, 7, 9. Calculate median, $Q_1$, and $Q_3$.

Data: 2, 4, 6, 8, 10, 12. Calculate the 5-number summary.

Grades: 4, 5, 6, 6, 7, 7, 8, 8, 9, 10. Calculate $Q_1$, $Q_2$, $Q_3$.

Calculate the $IQR$ of the data: 12, 14, 18, 22, 25, 28, 32.

Ages: 18, 20, 21, 22, 23, 24, 25, 27, 30, 35, 60. Apply the 1.5 IQR rule. Is there an outlier?

Salaries ($k): 2, 3, 3, 4, 4, 5, 5, 6, 8, 50. Calculate median and$IQR$.

For $n = 100$ sorted data points, what is the position of $Q_3$ by the linear interpolation method?

Times (s): 10, 11, 11, 12, 13, 13, 14, 14, 15, 100. Calculate Tukey limits and identify the outlier(s).

Weights (kg): 60, 62, 64, 65, 65, 67, 70, 72, 75, 80. Describe all elements of the boxplot.

For $Z \sim \mathcal N(0,1)$, $Q_3 - Q_1 = ?$

Data with $IQR = 6{,}7$. Using the robust estimator $\hat\sigma = IQR/1{,}349$, calculate $\hat\sigma$.

How many points above $Q_3 + 3 \cdot IQR$ would we expect in a sample of 1000 normal observations?

Boxplot A: narrow box, centered median. Boxplot B: wide box, median close to $Q_1$. Compare dispersion and symmetry of the two sets.

Distribution with a long right tail. Where is the mean in relation to the median?

Set A has $IQR = 5$, set B has $IQR = 20$. Which has more dispersion in the central data?

Median of $A = B = 50$. $Q_3$ of $A = 55$, of $B = 80$. Which has a more right-skewed distribution?

$P_{90}$ of company salaries = $30k. Interpret this information.

A student is at the $P_{85}$ of the exam. What does this mean?

If $Q_1 = Q_2 = Q_3$, what can be concluded about the data?

Is the statement "the 1.5 IQR rule flags 5% of data as outliers" correct for normal data?

Ages (years): 40, 52, 55, 58, 62, 66, 72. Calculate the 5-number summary and check for outliers.

Grades of 10 students: 3, 5, 6, 7, 7, 8, 8, 9, 10, 10. Complete boxplot (with outlier check).

Class of 100 students: $Q_1 = 5$, $Q_3 = 8$. A student scored 9.5—are they in the top 25%?

Why do statistical agencies report median income, rather than just the mean, in inequality reports?

Parts produced with diameter: $Q_1 = 9{,}98$ mm, $Q_3 = 10{,}02$ mm. Specification: $10{,}00 \pm 0{,}05$ mm. Is the process centered? Is there significant risk of rejection?

A/B test of a site: variant A has median 1.2 s and $IQR = 0{,}3$; variant B has median 1.1 s and $IQR = 1{,}5$. Which do you prefer for production? Justify using dispersion statistics.

You detect an outlier in financial transactions that appears to be fraud. Should you remove it before analyzing the data? Justify with statistical arguments.

Response times (ms): 120, 130, 135, 140, 142, 145, 148, 150, 155, 380. Calculate the 5-number summary and evaluate if the system meets a 200 ms SLA based on quartiles.

Hospital with 4 wings. Stay times (days): Wing A: 5, 8, 9, 10, 12; Wing B: 3, 4, 4, 5, 20; Wing C: 7, 8, 8, 9, 10; Wing D: 2, 3, 15, 18, 25. Construct 5-number summaries and identify which wing is most predictable for bed management.

Exam scores by school. School A: median 650, $IQR = 80$. School B: median 620, $IQR = 200$. Which school has more uniform performance? What does each pattern suggest for pedagogical policy?

Average monthly precipitation (mm): 234, 181, 130, 83, 68, 52, 44, 47, 82, 122, 145, 201. Calculate the 5-number summary and interpret seasonality.

Real estate prices in a neighborhood ($k): 250, 280, 310, 320, 340, 350, 380, 390, 420, 1800. Calculate median and mean. Why should a buyer use the median as a reference for typical price?

Explain, in your own words, why median and IQR are "robust" while mean and standard deviation are not. Use a concrete example.

Can a boxplot hide a bimodal distribution? Construct a concrete example of a bimodal distribution that has the same boxplot as a unimodal one.

For $X \sim \text{Uniform}(0, 1)$, the $IQR$ is:

Calculate analytically the $IQR$ of $X \sim \text{Exponential}(\lambda)$. Express in terms of $\lambda$.

Argue why the breakdown point of the $IQR$ is 25%, the median is 50%, and the mean is 0%.

Demonstrate: if $X$ is a continuous r.v. with density symmetric around $\mu$, then $\mu$ is the median of $X$.

Show that for $n \to \infty$ and iid samples from Uniform(0,1), the sample estimator of $Q_1$ converges to 0.25. Use properties of order statistics.

Demonstrate that the median minimizes $E[|X - c|]$ over all values $c \in \mathbb{R}$.