Averages and Missing Values

Average questions come in two directions. Forward, you are given the numbers and asked for the average, which is easy. Backward, you are given the average and asked for a number, which is where the subtest actually lives.

Both directions run through the same quantity: the sum.

Forward, and the one thing to carry

The mean is the total divided by how many values there are.

A trainee scores 78, 85, 91 and 86 on four tests. What is the average?

Sum: 78 plus 85 is 163, plus 91 is 254, plus 86 is 340. Divided by 4 is 85.

The reason to care about the sum specifically, rather than treating the average as an atomic operation, is that average x count = sum is the equation every harder question rearranges.

Working backward to a missing value

A trainee has averaged 82 across five tests. Four of the scores are 75, 88, 79 and 90. What was the fifth?

Total needed: 82 times 5 is 410.

Total of the four known: 75 plus 88 is 163, plus 79 is 242, plus 90 is 332.

Missing: 410 minus 332 is 78.

Check: 78 is below the 82 average, and the four known scores average 83, so the fifth should pull it down. It does.

That check is worth running every time. If the known values average above the target, the missing one must be below it, and vice versa. It catches the subtraction done in the wrong order, which is the standard error here.

What score do I still need

The same machinery, phrased as a goal.

Five test scores average 84. There is one test left, and the goal is an average of 86 across all six. What is needed on the last test?

• Current total: 84 times 5 is 420.
• Required total: 86 times 6 is 516.
• Needed: 516 minus 420 is 96.

The intuition that would give you 88 - "two points above, so add two" - is wrong, and 88 will be a choice. Raising an average of five values by two points requires the new value to cover both its own share and the two-point lift for all five previous values: 86 plus 5 times 2, which is 96. Same answer, and it shows why the gap is so much larger than it feels.

Weighted averages

This is the topic's real test.

One class of 12 averages 88. Another class of 28 averages 78. What is the average across both classes?

The answer is not 83. The classes are different sizes, and the bigger class pulls harder.

• Class one total: 12 times 88 is 1,056.
• Class two total: 28 times 78 is 2,184.
• Combined total: 3,240. Combined count: 40.
• Average: 3,240 divided by 40 is 81.

Eighty-one, not 83, and it sits closer to 78 because most of the people are in that class. The weighted average always lands nearer the average of the larger group, which is the sanity check.

The method is always: totals up, counts up, divide once at the end. Never average the averages unless the groups are the same size, in which case both methods agree and it does not matter.

Adding and removing a value

Six readings average 40. A seventh reading of 61 is added. What is the new average?

• Old total: 240.
• New total: 301.
• New average: 301 divided by 7 is 43.

Removing works the same way: subtract the removed value from the total and reduce the count by one.

Direction check: 61 is above 40, so the average must rise. It did.

The median, when it is asked

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About a dozen questions here ask for the median instead of the mean. Sort the values and take the middle one; with an even count, take the mean of the two middle values. For 4, 4, 9, 15, the middle two are 4 and 9, so the median is 6.5. The median is the better "typical" value when one number is far from the rest: in ages 20, 22, 24, 26 and 38, the median of 24 describes the group better than the mean of 26. The Mathematics Knowledge lesson covers the median, mode and range in full.

What you can skip

Across the 108 questions on this topic:

• Mode and range come up occasionally here (8 and 6 questions) and are one-step answers: the most frequent value, and largest minus smallest.
• Weighted averages by name. The word never appears in this topic, though the mixture-of-prices questions are weighted averages in practice, as above.

Where people lose points

Averaging the averages. The big one. Whenever two groups have different sizes, the plain average of their averages is wrong and is always a choice.

Subtracting in the wrong direction when finding a missing value. The check above catches it.

Using the wrong count on a "what do I still need" question. The required total uses the count including the new test. Using the old count is the most common error on that shape.

Forgetting that the target is an average of everything, not of the remaining tests.

Assuming the answer is possible. Some questions are built so the needed score exceeds the maximum, and the correct answer is that the goal cannot be reached. If the arithmetic gives you 108 out of 100, that is the answer, not a mistake.

Work one in under a minute

Four shipments weigh an average of 1,250 pounds. Three of them weigh 1,100, 1,400 and 1,325 pounds. What does the fourth weigh?

Total: 1,250 times 4 is 5,000.

Known: 1,100 plus 1,400 is 2,500, plus 1,325 is 3,825.

Fourth: 5,000 minus 3,825 is 1,175 pounds.

Check: the three known shipments average 1,275, which is above target, so the fourth must be below 1,250. It is.

Where this leads

The weighted-average idea is the same idea as average speed over unequal legs, and the mean is one of the four statistics Mathematics Knowledge asks about directly.