Three ways to say "typical". They disagree, and the disagreement is the point.
Find the missing score
A student's first four quiz scores are 82, 75, 90 and 68. What must the fifth score be for the mean to reach 80?
Five scores averaging 80 need a total of 5 × 80 = 400. The first four add to 315, so the fifth must be 400 − 315 = 85. A assumes the new score equals the target mean, which would only work if the first four already averaged 80 — they average 78.75. Working through the total is what makes these questions easy.
| Measure | Answers | How |
|---|---|---|
| Mean | if it were shared equally | add up, divide by how many |
| Median | what the middle one got | sort, take the centre |
| Mode | what happened most often | count repeats |
4, 7, 7, 9, 13
already sorted
mean = 40 ÷ 5 = 8
the fair share
median = 7
the 3rd of 5
mode = 7
it appears twice
Even count? Average the two middle values.
12, 5, 9, 20, 3, 14
six values, unsorted
3, 5, 9, 12, 14, 20
sort
middle two: 9 and 12
3rd and 4th
median = 10.5
their average
10.5 is not one of the six numbers, and that is perfectly normal for an even count. Taking the middle of the unsorted list, on the other hand, is simply wrong.
The mode is the only one of the three that works for non-numbers — the most popular colour, the commonest surname. You cannot average those.
| Monthly salaries | ₱18,000 × 7 | ₱200,000 × 1 |
|---|---|---|
| Mean | ₱40,750 | |
| Median | ₱18,000 | |
| Mode | ₱18,000 | |
Seven of the eight people earn ₱18,000. The mean of ₱40,750 describes nobody in the room — one large value dragged it upward.
Incomes, house prices and waiting times almost always have a long tail of large values. That is why official statistics quote the median.
total = mean × how many
82, 75, 90, 68 and x
mean must be 80
needed total = 5 × 80 = 400
the key step
current total = 315
add the four
x = 85
400 − 315
Add the same number to every value and the mean moves by that number. Multiply every value and the mean is multiplied.
The mean of eight numbers is 25. Find the new mean if every number is decreased by 3, and if every number is doubled.
total = 8 × 25 = 200
go through the total
each − 3: total falls by 8 × 3 = 24
176 ÷ 8 = 22
each × 2: total doubles to 400
400 ÷ 8 = 50
So the mean simply becomes 25 − 3 = 22, or 25 × 2 = 50. The median and mode move in exactly the same way.
The mean of five numbers is 18. The number 30 is removed. What is the mean of the rest?
total = 5 × 18 = 90
90 − 30 = 60
the four left
60 ÷ 4 = 15
divide by the NEW count
60 ÷ 5 = 12 divides four numbers' total by five. After a value leaves, the count drops to 4.
The three measures again, in less tidy data.
Morning temperatures: −3, 2, −5, 0, 1 (°C). Find the mean and the median.
total = −3 + 2 − 5 + 0 + 1 = −5
negatives subtract
mean = −5 ÷ 5 = −1 °C
sorted: −5, −3, 0, 1, 2 → median 0 °C
−5 is the smallest
A mean can be negative, and 0 is a real value that counts in both the total and the count.
Ten pupils have 1, 2 or 3 siblings: 1 sibling (3 pupils), 2 siblings (5 pupils), 3 siblings (2 pupils). Find the mean, median and mode.
total = 1(3) + 2(5) + 3(2) = 19
value × how many
mean = 19 ÷ 10 = 1.9
divide by 10 pupils, not 3 rows
median: 5th and 6th values are 2 and 2
sorted 1, 1, 1, 2, 2, 2, 2, 2, 3, 3
median = 2, mode = 2
Divide by the number of pupils, never the number of rows in the table.
The ordered data 3, 5, x, 9 has median 6. Find x.
median = 5 + x2
middle two of four
5 + x2 = 6, 5 + x = 12
x = 7
and 5 ≤ 7 ≤ 9 keeps the order ✓
Items ask for one measure of a short list, which measure best describes skewed data such as salaries, or a missing value for a target mean. The fastest route is to sort before taking a median and to go through the total for anything backwards. For a 10-second check, the mean must lie between the smallest and largest values, and an extreme value pulls the mean towards it.
Find the median
Find the median of 14, 8, 22, 5, 17, 11.
Sorted: 5, 8, 11, 14, 17, 22. With six values, the median is the average of the 3rd and 4th: 11 + 142 = 12.5. A takes the 3rd value only, forgetting that an even count has no single middle. C averages the 4th and 5th instead, and D takes the middle of the unsorted list — sorting is the step that has to come first.
Hint: six values means two in the middle. Sort them, find the third and fourth, and average those.
