Graphical Representations of Summary Statistics

A five-number picture

• Some summaries are best shown as a compact graph: the boxplot 箱线图.
• It's built from the five-number summary 五数概括: minimum, $Q_1$, median, $Q_3$, maximum.
• A box spans the middle $50\%$ ($Q_1$ to $Q_3$) with a line at the median; whiskers reach the extremes.
• One glance shows center, spread, and skew.

Reading a boxplot

• The box goes from $Q_1$ to $Q_3$ — its length is the IQR.
• The line inside the box is the median; whiskers extend to the min and max (or to the last non-outlier).
• A longer whisker or a median off-center in the box signals skew toward the longer side.
• The box holds the middle half of the data; the whiskers show the tails.

A box-and-whisker plot shows the quartiles and the range

Modified boxplots and outliers

• A modified boxplot plots outliers as separate dots, using the $1.5\times\text{IQR}$ rule.
• The whiskers then stop at the most extreme non-outlier values.
• This keeps outliers from stretching the whiskers and hiding the real spread.
• Marked dots make unusual values obvious at a glance.

z-scores: locating a value

• A $z$-score 标准分数 measures how many standard deviations a value is from the mean:
• $$z=\frac{x-\bar{x}}{s_x}$$
• $z=+2$ means "$2$ standard deviations above the mean"; $z=-1$ means "$1$ SD below."
• $z$-scores let you compare positions across different distributions (a common exam use).

Write from 31 as an outlier

• Q1 is 10 and Q3 is 18, so the IQR is 8.
• A value of 31 is in the data. The lesson fence uses 1.5 times the IQR.