Box plots
In a nutshell
Box plots are a diagram used to show the spread of data and are a useful way of summarising.
Features of a box plot
A box plot shows minimums, maximums and quartiles but does not show exact values. See the example below for drawing each of these features on your own box plot.
Definitions
Box plot feature  Definition  Formula (where n is total frequency) 
Lower quartile (Q_{1} or LQ)
 Value $25\%$ of the way through the data
 $\frac{n+1}{4}$

Median (Q_{2})
 Value $50\%$ of the way through the data
 $\frac{n+1}{2}$

Upper quartile (Q_{3 }or UQ)
 Value $75\%$ of the way through the data
 $\frac{3(n+1)}{4}$

Interquartile range (IQR)  Contains middle $50\%$ of the data  Upper quartile minus lower quartile: $Q_3Q_1$

Example 1
Olivia is collecting data about the ages of her friends' siblings. The answers she collects are $1,1,2,2,3,4,5,5,5,5,7,7,7,7,8$. The median of these numbers is $5$.
Work out the upper and lower quartiles and the interquartile range.
Lower quartile
 Work out the position of Q_{1}: $\frac{n+1}{2} = \frac{15+1}{4}=4$
The $4$th value is the lower quartile: $\underline2$

Upper quartile  Work out the position of Q_{3}: $\frac{3(n+1)}{2} = \frac{3(15+1)}{4}=12$
The $12$th value is the upper quartile: $\underline7$ 
Interquartile range  $\text{IQR} = 72=\underline5$

Median  $\underline5$

Drawing a box plot
Drawing a box plot involves using all the quantities worked out in the example above.
procedure
1.  Draw a scale.

2.
 Mark on the LQ, UQ and median and draw a box with a line where the median is.

3.
 Mark on the maximum and minimum and connect to the box with a line.

Example 2
Using the values calculated in example $1$, draw a box plot to represent the data.