Definition of Mean in Basic Statistics
Definition 1
$$ \overline{x} := {{ 1 } \over { n }} \sum_{k=1}^{n} x_{k} $$
When $n$ quantitative data are given, the value obtained by adding all those values and dividing by $n$, denoted as $\overline{x}$, is called the sample mean, arithmetic mean, or average.
Description
There’s no need to explicitly explain how averages can efficiently summarize data. Anyone studying statistics beyond the undergraduate level should be able to address the following inquiries and know when to be cautious of averages:
- Is the average always reliable? Obviously not. There’s a famous joke going around the internet that the department with the highest average salary at the University of North Carolina is the Geography Department.2 As that amusing anecdote suggests, averages are vulnerable to outliers and may not always be appropriate to use as a representative value.
- When is it particularly risky? When the sample size is too small, there are many outliers, the distribution is not unimodal, amongst others. Although rare, it’s theoretically possible to envisage situations where the population mean does not exist.
- Why is it still considered so important? Because of the Central Limit Theorem. It’s this powerful theorem that states the probability distribution of the sample mean of any random sample from any distribution approaches a normal distribution as the sample size gets larger. Despite its simplicity, it forms the foundation of statistics and thus holds significant value.
Those studying statistics should be mindful of when averages may lose their meaning, and develop the habit of scrutinizing data closely. In essence, knowing when not to rely on averages is just as crucial as knowing how to apply them correctly. The following tweet, albeit exaggerated, warns about how meaningless averages can become if the data is ignored:
See Also
- Three main statistical measures: Mode, Median, Mean
- Mathematical properties of measures of central tendency: The mean has the property of minimizing variance.
