The link between the arithmetic mean in mathematics and the use of the word "average" in day to day life

Question

The arithmetic mean of a data set $\{ x_i \}$ with $N$ elements is given by $\frac{1}{N} \sum_{i = 1}^{N} x_i$ in mathematics. What does it mean then, or how is this directly related, to notions of "averages" over time used in day to day life? For example, suppose someone spends a total of 120 dollars over a period of 7 days, most people would say that that person spent an average of $\frac{120}{7}$ dollars per day. But how is this linked to the definition of arithmetic mean above? I can't see a direct correlation.

Answer 1

Suppose the person spent $20$ dollars the first day, none the second or third days, $60$ dollars the fourth day, $10$ the fifth, and $15$ on each of the sixth and seventh. That's a total of $120$.

Now the associated data set is $\{20,0,0,60,10,15,15\}$, and it has $7$ entries (one datum for each day). The arithmetic mean is $\frac17(20+0+0+60+10+15+15) = \frac{120}7$, as expected from the colloquial usage of "average." You can quickly note that the arithmetic mean depends only on the sum of the data, not the individual values, so any way we break down the $120$ dollars will give an arithmetic mean of $\frac{120}7$.

Not every colloquial usage of "average" matches arithmetic mean -- it depends a bit on context. But often it will.