What is Mean?

Example 8.4

For the quiz of Example 8.2:

Example 8.6

Sorted series of $7$ house prices (in thousands): $120$, $150$, $160$, $180$, $210$, $240$, $900$. The median is the $4$th value, $180$. The mean is $\frac{1960}{7} = 280$ — larger than $6$ of the $7$ prices! One extreme value ($900$) pulls the mean far more than the median: the median is robust, the mean is not.

Example 17.3

For the running dataset, the sum is $144$, so $\bar x = \frac{144}{12} = 12$. The two middle values ($6$th and $7$th of the sorted list) are both $12$, so the median is $12$ too. Mean and median need not coincide: the median ignores how far the extreme values are, the mean does not. Replacing the top mark $17$ by $20$ moves the mean to $12.25$ but leaves the median at $12$.

Example 17.8

For the running dataset, the deviations from $\bar x = 12$ are $-4$, $-3$, $-2$, $-2$, $-1$, $0$, $0$, $0$, $1$, $2$, $4$, $5$; their squares sum to

Hence

Roughly speaking, a typical mark sits about $2.6$ marks away from the mean.