Read five-number summaries informally

A five-number summary represents the minimum, lower quartile, median, upper quartile, and maximum of an ordered data set; the quartiles and median divide the data into four groups with approximately equal numbers of values, while the numerical distances between them describe spread. Reading a summary or box plot includes identifying the center, range, and interquartile range, comparing distributions, and recognizing that unequal segment lengths reflect differing variability rather than unequal numbers of data values.

Detailed Explanation: Read five-number summaries informally

A five-number summary describes a data set using five values in order:

\text{minimum},\quad Q_1,\quad \text{median},\quad Q_3,\quad \text{maximum}

Minimum: smallest value
Lower quartile, $(Q_1)$ : about one-fourth of the data are below this value
Median: middle of the data
Upper quartile, $(Q_3)$ : about three-fourths of the data are below this value
Maximum: largest value

The quartiles and median divide the data into about four equal groups. The distances between the values show how spread out the data are.

Worked example

The numbers of pages read by students in a week have this five-number summary:

6,\quad 10,\quad 14,\quad 18,\quad 30

Step 1: Identify the center

The median is the middle number:

\text{Median}=14

So, the center of the data is about $(14)$ pages.

Step 2: Find the range

The range tells how far the data go from the minimum to the maximum:

\text{Range}=30-6=24

The data values spread across $(24)$ pages.

Step 3: Find the interquartile range

The interquartile range, or IQR, describes the spread of the middle half of the data:

\text{IQR}=Q_3-Q_1=18-10=8

The middle half of the students read between $(10)$ and $(18)$ pages, a spread of $8$ pages.

Step 4: Compare the four sections

Look at the distances between consecutive summary values:

10-6=4,\qquad 14-10=4,\qquad 18-14=4,\qquad 30-18=12

The last section is much longer. This means the highest quarter of the data is more spread out. It does not mean that this section contains more students. Each section represents about one-fourth of the data, even when the section lengths are different.

So, this data set has a median of $(14)$ , a range of $(24)$ , and an IQR of $8$ .

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Statistics - Quartiles - Data Set (With Outliers) to Box Plot

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