Box Plot

Study guide: Exam MAS-I

A Box Plot (box-and-whisker plot) is an Exploratory Data Analysis display that summarizes a distribution with five numbers: the minimum, first quartile $Q_1$ , median $Q_2$ , third quartile $Q_3$ , and maximum. The box spans the middle half of the data and the whiskers reach the most extreme observations that are not flagged as outliers.

$\text{IQR} = Q_3 - Q_1$

$\text{potential outlier if } x < Q_1 - 1.5\,\text{IQR} \ \text{ or } \ x > Q_3 + 1.5\,\text{IQR}$

The box runs from $Q_1$ to $Q_3$ and holds 50% of the observations; the line inside it is the median, not the mean
The whiskers extend to the furthest observation within $1.5 \times \text{IQR}$ of the box; points beyond are drawn individually as outliers
The convention varies. The $1.5\,\text{IQR}$ limits are the inner fences; NIST calls a point beyond one a mild outlier and a point beyond an outer fence, $Q_1 - 3\,\text{IQR}$ or $Q_3 + 3\,\text{IQR}$ , an extreme one, and its basic box plot has no fences at all — the whiskers run to the minimum and maximum. Quartiles have several definitions too: this page takes the medians of the lower and upper halves (the median itself left out when $n$ is odd), which in the example below matches NIST's $(n+1)p$ -th ordered value, while R's boxplot uses Tukey's hinges — so software can report different quartiles and fences for the same data
A median sitting left of centre with a long right whisker signals right skew — the usual shape for claim severity
Box plots are read side by side: one box per level of a Categorical Predictor shows whether a rating variable separates the response, which is exactly the evidence needed before adding that variable to a Generalized Linear Model
A box plot hides multimodality — two clusters and one broad spread can give the same five numbers, so pair it with a Histogram before concluding anything about shape
Because it is built from Order Statistics, it is resistant to extreme values: one enormous claim moves the maximum but not the box

Building a Box Plot from a Sample Eleven claim amounts (in thousands) are: $2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 40$ . Find the five-number summary and identify any outliers.

Answer With $n = 11$ , the median is the 6th value, $Q_2 = 7$ . The lower half $\{2,3,4,5,6\}$ has median $Q_1 = 4$ ; the upper half $\{8,9,11,14,40\}$ has median $Q_3 = 11$ . $\text{IQR} = 11 - 4 = 7$ $\text{upper fence} = Q_3 + 1.5(7) = 11 + 10.5 = 21.5$ $\text{lower fence} = Q_1 - 1.5(7) = 4 - 10.5 = -6.5$ The value $40 > 21.5$ is an outlier; the upper whisker therefore stops at $14$ . The five-number summary is $(2, 4, 7, 11, 40)$ with $40$ plotted separately. It also lies beyond the outer fence $Q_3 + 3(7) = 32$ , so NIST would call it an extreme outlier. R's boxplot would draw $Q_1 = 4.5$ and $Q_3 = 10$ (fences $-3.75$ and $18.25$ ) — different numbers, the same outlier and whisker.

Comparing Vehicle Classes Box plots of severity for three vehicle classes show medians of $3.1$ , $3.3$ , and $6.8$ (in thousands), with the first two boxes overlapping heavily and the third sitting well above them with several high outliers. What does this suggest for the model?

Answer The first two classes are not distinguished by the data and could be grouped into one level, saving a parameter. The third class needs its own coefficient. The high outliers and the widening spread as the median rises indicate variance growing with the mean — a Gamma Generalized Linear Model with a log Link Function fits that pattern better than ordinary Linear Regression.