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Once the data has been collected, it must be organised in a manner that allows for the information to be extracted most efficiently. For this reason it is useful to be able to summarise the data set by calculating a few quantities that give information about how the data values are spread and about the central values in the data set. Other methods of summarising and representing data will be covered in grade 11.
The mean, (also known as arithmetic mean), is simply the arithmetic average of a group of numbers (or data set) and is shown using the bar symbol
The mean of a data set, $x$ , denoted by $\overline{x}$ , is the average of the data values, and is calculated as:
Method: Calculating the mean
What is the mean of $x=\{10,20,30,40,50\}$ ?
There are 5 values in the data set.
$\therefore $ the mean of the data set $x=\{10,20,30,40,50\}$ is 30.
The median of a set of data is the data value in the central position, when the data set has been arranged from highest to lowest or from lowest to highest. There are an equal number of data values on either side of the median value.
The median is calculated from the raw, ungrouped data, as follows.
Method: Calculating the median
What is the median of $\{10,14,86,2,68,99,1\}$ ?
1,2,10,14,68,86,99
There are 7 points in the data set.
The central position of the data set is 4.
14 is in the central position of the data set.
$\therefore $ 14 is the median of the data set $\{1,2,10,14,68,86,99\}$ .
This example has highlighted a potential problem with determining the median. It is very easy to determine the median of a data set with an odd number of data values, but what happens when there is an even number of data values in the data set?
When there is an even number of data values, the median is the mean of the two middle points.
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