Quartiles, IQR and box plots

Range, quartiles and interquartile range, the five-number summary, outliers, drawing and reading box plots, and comparing data sets.

Quartiles

Example 1●●

Find the five-number summary of $2, 4, 5, 7, 9, 11, 14$.

Median $= 7$. Lower half $2, 4, 5$: $Q_1 = 4$. Upper half $9, 11, 14$: $Q_3 = 11$.

Min $2$, $Q_1 = 4$, median $7$, $Q_3 = 11$, max $14$. $\text{IQR} = 7$.

Box plots and outliers

A box plot draws the five-number summary: whiskers to the min and max, and a box from $Q_1$ to $Q_3$ with a line at the median. An outlier is a value below $Q_1 - 1.5 \times \text{IQR}$ or above $Q_3 + 1.5 \times \text{IQR}$.

Example 2●●●

For $3, 5, 7, 8, 10, 12, 15, 18$, find the IQR and the upper outlier boundary.

$Q_1 = 6$ (from $3, 5, 7, 8$), $Q_3 = 13.5$ (from $10, 12, 15, 18$), $\text{IQR} = 7.5$.

Upper boundary $= 13.5 + 1.5 \times 7.5 = 24.75$. No outliers.

Flashcards

IQR

$Q_3 - Q_1$: spread of the middle $50\%$

Five-number summary

Min, $Q_1$, median, $Q_3$, max

Outlier test

Beyond $1.5 \times$ IQR from the box

Practice

IQR of $2, 4, 5, 7, 9, 11, 14$?
IQR of $3, 5, 7, 8, 10, 12, 15, 18$?
Upper outlier boundary for that data?
Median of $3, 5, 7, 8, 10, 12, 15, 18$?
About what percentage of data lies inside the box of a box plot?
$Q_1 = 10$, $Q_3 = 20$. Is $36$ an outlier?
Why is the IQR often better than the range?
Find $Q_3$ for $1, 3, 6, 10, 15, 21$.