Math Core

Averages are a favorite on MATHCOUNTS and the AMC 8: test scores, heights, a number that was removed from a list. Almost every one of them turns on a single move. Don't work with the average; work with the sum.

Sum = average × count

The mean (average) of a list is its sum divided by how many numbers it has. Turn that around:

Think in sums

$\text{sum} = \text{mean} \times \text{count}$

When numbers are added to or removed from a list, the sums change in simple ways, even though the averages don't. Convert every average to a sum, do the arithmetic with sums, and convert back at the end.

The balance point

The mean is the balance point of the numbers on a number line: the distances of the numbers above the mean exactly cancel the distances of the numbers below it. For $2, 5, 11$ , the mean is $6$ . The numbers sit $4$ and $1$ below it and $5$ above it, and $4 + 1 = 5$ .

This gives a quick way to average numbers near some round value: pick a guess, average the differences from it, and adjust. For $87, 92, 78, 91$ , compare with $85$ : the differences are $+2, +7, -7, +6$ , which average $\dfrac{8}{4} = 2$ . The mean is $85 + 2 = 87$ .

Weighted averages

When two groups are combined, the overall average is not the average of the two averages (unless the groups are the same size). Add up the total of each group instead. The combined average always lands between the two group averages, closer to the bigger group's average.

Common mistake

If $20$ students average $80$ and $30$ students average $90$ , the combined average is not $85$ . The larger group pulls it toward $90$ : it's $\dfrac{1600 + 2700}{50} = 86$ .

Evenly spaced numbers

The average of an arithmetic sequence is the average of its first and last terms (which is also the middle term). The average of $13, 15, 17, \dots, 51$ is $\dfrac{13 + 51}{2} = 32$ .

Median and mode

The median is the middle number when the list is in order (or the average of the two middle numbers when the count is even). The mode is the value that appears most often. Problems mixing mean, median and mode are usually solved by writing the list in order with unknowns and using each condition in turn.

Worked example: What do I need on the next test?

Your average on four tests is $85$ . What score on the fifth test would bring your average up to $87$ ?

You need a total of $5 \cdot 87 = 435$ points and have $4 \cdot 85 = 340$ . So you need $435 - 340 = 95$ .

Worked example: Combining groups

In a gym class, the $12$ boys average $60$ inches tall and the $8$ girls average $55$ inches. What is the average height of the whole class?

The total height is $12 \cdot 60 + 8 \cdot 55 = 720 + 440 = 1160$ inches for $20$ students, so the average is $58$ inches.

Worked example: A removed number

The average of $10$ numbers is $50$ . When one number is removed, the average of the rest is $48$ . What number was removed?

The sum was $500$ and is now $9 \cdot 48 = 432$ . The removed number is $500 - 432 = 68$ .

Worked example: Mean, median and mode together

A list of five positive integers has mean $10$ , median $9$ and a unique mode of $8$ . What is the largest possible number in the list?

In order, the list is $a, b, 9, d, e$ with sum $50$ . For $8$ to be the unique mode, it must appear at least twice, and it can only sit below the median, so $a = b = 8$ . Then $d + e = 50 - 25 = 25$ . Also $d$ can't be $9$ (then $9$ would tie $8$ as a mode), so $d \ge 10$ and $e \le 15$ . The list $8, 8, 9, 10, 15$ works, so the answer is $15$ .

Tip

After a mean-median-mode problem, write out the final list and check every condition. It takes ten seconds and catches most mistakes.