Averages from frequency tables

In a nutshell

Mode, median and mean can all be calculated from frequency tables. If the data being considered is discrete then all of these averages can be calculated, but for any situation where the data is categoric, only the mode can be found.

Calculating averages

Mode

Procedure

1.	Find the greatest frequency value in the table.
2.	The mode is the data value corresponding to this greatest frequency.

Median

Procedure

1.	Work out the total number of samples taken by summing the frequencies. This value is $n$ .
2.	Ensure the rows of the table are written in ascending order for the data value they represent.
3.	The median is the data value that corresponds to the $\dfrac{n+1}{2} th$ item.

Mean

Procedure

1.	Calculate $n$ and fill in a third column of the table corresponding to frequency multiplied by the data value.
2.	Calculate the sum of the third column. This value is $\text{sum } x$ .
3.	The mean is calculated as $\dfrac{\text{sum } x}{n}$ .

Example 1

Calculate the mode, median and mean from the following frequency table:

Size	Frequency
$6$	$4$
$6.5$	$7$
$7$	$6$
$8$	$3$

Mode: The most frequent value is $\underline{6.5}$

Median: There are $4+7+6+3=20$ pieces of data. Hence the median occurs at the $\dfrac{20+1}{2}=10.5th$ data item. This is the item between the $10th$ piece of data (size $6.5$ ) and the $11th$ piece of data (size $6.5$ ). Hence the median is $\underline{6.5}$

Mean: Create a third column as follows,

SIZE	FREQUENCY	SIZE x FREQUENCY
$6$	$4$	$24$
$6.5$	$7$	$45.5$
$7$	$6$	$42$
$8$	$3$	$24$

This third column has a total of $24+45.5+42+24=135.5$ , and so the mean is:

$\dfrac{135.5}{20}=\underline{6.775}$