Grouped frequency tables
In a nutshell
Sometimes data may be split into different classes to make it more manageable. This means that exact values are not given in the frequency table, instead only a range of values that each one could take.
Different kinds of grouped frequency table
Discrete or continuous grouped frequency tables
For discrete grouped frequency tables there are 'gaps' between each of the classes  for example ages might be split into the classes 1315 and 1618.
For continuous data there should be no 'gap' and inequalities should be used instead.
Example 1
In this example Mike has been asked to record the heights of his classmates.
Height ($cm$)  Frequency 
$140\lt h \le 145$  $2$ 
$145\lt h \le150$  $3$

$150\lt h \le 155$  $13$

$155\lt h \le 160$  $21$ 
$160\lt h \le 165$  $5$ 
Finding averages from a grouped frequency table
Similarly to nongrouped frequency tables, finding averages from a grouped frequency table means finding out which classes contain values. However in this case, you will be asked to find the modal class and class containing the median, instead of an exact value.
Definition
Midinterval value  The value in the middle of a class. 
PROCEDURE
Midinterval value   Take the end values of each interval and add them together.
 Divide by two to get the midinterval value.

Mean   Create a new column: 'midinterval value $\times$ frequency'.
 Divide the column total by total frequency to get the mean

Class containing the median   Work out the position of the median by using the formula: $\frac{(n+1)}{2}$ (where n is the total frequency).
 Find the class containing the value in this position.

Modal class   Find the class with the highest frequency.

Range   Find the smallest value and highest value in the classes.
 Work out the difference between these two values.

Note: The mean and range are only estimates as we do not know the exact values within each class.
Example 2
From the table above, work out estimates for the mean (to one decimal place) and range. Find the modal class and the class containing the median.
Mean:
First create the new columns required.
Midinterval value  Midinterval value $\times$ frequency 
$\frac{140+145}{2}=142.5$
 $142.5\times2=285$

$\frac{145+150}{2}=147.5$
 $147.5\times3=442.5$

$\frac{150+155}{2}=152.5$
 $152.5\times13=1982.5$

$\frac{155+160}{2}=157.5$
 $157.5\times21=3307.5$

$\frac{160+165}{2}=162.5$
 $162.5\times5=812.5$

Use the formula.
$\frac{\text{sum of (midinterval value × frequency)} }{\text{total frequency}}=\frac{285+442.5+1982.5+3307.5+812.5}{44}=\frac{6830}{44} = {155.227...}$
Round to one decimal place.
$\underline{155.2}$
Range:
Take the smallest class value: $140$ away from the highest class value: $165$.
$165140=\underline{25}$
Modal class:
The class with the highest frequency is $\underline{155 \lt h \le 160}$
Class containing the median:
Find the position of the median using the formula.
$\frac{n+1}{2}=\frac{44+1}{2}=22.5$
To find the class containing the $22.5th$ value, add up the numbers in the frequency column until you reach $22.5$.
$2+3=5\\5+13=18\\18+21= 39$
Therefore the class containing the $22.5$th value is: $\underline{155 \lt h \le 160}$.