Measures of central tendency: Mean, median, mode

In a nutshell

A value that describes the centre of a data set is a measure of central tendency. The mean, mode and median are measures of central tendency.

Measures of central tendency

Measures of location show a single value that describes a particular position in a data set. Measures of central tendency are a special type of measures of location which describe the centre of the data.

Mean

The mean is given by the division of the sum of all the values in the data set by the number of values, $n$ :

${\overline x=\dfrac{\Sigma x}{n}}$

$\overline x$	Mean
$\Sigma x$	Sum of the values in the data set
$n$	Number of values in the data set

Note: For data in a frequency table, the mean is given by:

$\overline {x}=\dfrac{\Sigma xf}{\Sigma f}$

$\overline {x}$	Mean
$\Sigma xf$	Sum of the product of the data values and their frequencies
$\Sigma f$	Sum of the frequencies

Mode

The mode (or modal class) refers to the value (or class) in the data set that appears the most often.

Median

The median, $\tilde {x}$ , is the middle value value in the ordered data set. If there are two values in the middle, then the median is the mean of those two values.

Finding the best measure

Each particular situation will have a measure that fits best. You can find this information on the table below:

Measure	Best use	Characteristics
Mean	Quantitative data	Uses all pieces of data. Affected by extreme values.
Mode	Qualitative or quantitative data	Only useful in cases of a single mode or two modes. Doesn't inform on the frequency of each data.
Median	Quantitative data	Not affected by extreme values.

Example 1

Sophia registered the number of siblings her colleagues have. She noted her results down in the following table:

Number of siblings	$0$	$1$	$2$	$3$	$4$
Number of colleagues	$3$	$5$	$4$	$2$	$1$

Find the mean, the mode and the median for the data shown. Which measure should Sophia consider when taking conclusions?

The mean will be given by:

 $\overline{x}=\dfrac{\Sigma xf}{\Sigma f}=\dfrac{0\times3+1\times5+2\times4+3\times2+4\times1}{3+5+4+2+1}\approx \underline{1.53 \ (2\space d.p.)}$

The mode corresponds to the value that appears most times: $\underline1$ sibling.

The median is the $\dfrac{15+1}{2}=8^{th}$ value: $\underline 1$ .

When taking conclusions, because this is a quantitative data set and there are no extreme values, Sophia should use the mean.