Measures of Central Tendency

The $\text{mean}$ is $\bar{x}=\frac{\sum x}{n}$: the sum of the values divided by how many there are.

The $\text{median}$ is the middle value when the data are ordered. For $n$ values it is the $\frac{n+1}{2}$th value; if $n$ is even, average the two middle values.

The $\text{mode}$ is the most frequent value. Grouped data have a $\text{modal class}$ instead. Data can be bimodal, or have no mode.

When values repeat, a frequency table records each value $x$ with its frequency $f$. The mean is $\bar{x}=\frac{\sum fx}{\sum f}$.

For the median, find its position $\frac{n+1}{2}$ (for discrete data) and use cumulative frequencies to locate the value.

With grouped data the exact values are lost, so averages are $\text{estimates}$. For the mean, use the class midpoint as $x$ in $\frac{\sum fx}{\sum f}$.

For the median of continuous grouped data, find the $\frac{n}{2}$th position and use $\text{linear interpolation}$ within the class that contains it, assuming values are spread evenly across the class.

Pay attention to class boundaries. Ages "20–29" in completed years run from 20 up to (but not including) 30.

If data are coded by $y=\frac{x-a}{b}$, then $\bar{y}=\frac{\bar{x}-a}{b}$. Adding a constant shifts the mean; multiplying scales it. The median and mode transform the same way.

An $\text{outlier}$ pulls the mean towards it but hardly affects the median. So for skewed data or data with outliers, the median is usually more representative.

The mean is preferred for roughly symmetric data because it uses every value. The mode is the only average possible for categorical (non-numerical) data, such as method of travel.