Mean, median and mode

Four ways to describe a list of numbers

How to solve them

1. 1

   The mean: add everything, then divide by how many

   mean = sum ÷ count

   The sum is what you get when you add all the numbers. The count is how many numbers there are. When a question says “average”, it means the mean.

   A quick check: the mean is always between the smallest and the largest number. If your answer is outside them, something went wrong.

   1. (12 + 9 + 15 + 8) ÷ 4
   2. 44 ÷ 4
   3. 11

   Shortcut

   Check: the ups and downs cancel

   From 11: 12 is 1 above, 9 is 2 below, 15 is 4 above and 8 is 3 below. Above: 1 + 4 = 5. Below: 2 + 3 = 5. They balance, so 11 is right.

   Next step: Missing value ↓
2. 2

   Reuse the mean: find a missing value

   total = mean × count

   You already know this move from the Averages lesson: total = mean × count. If you know the mean and the count, you know the total. Find the total the mean needs, then take away the numbers you already have. What is left is the missing number.

   Example

   A driver wants to average 150 miles a day over 4 days. The first three days: 140, 165 and 130 miles. How far must she drive on day 4?

   1. The total the average needs: 150 × 4 = 600 miles.
   2. Already driven: 140 + 165 + 130 = 435 miles.
   3. Day 4: 600 − 435 = 165 miles.

   Answer 165 miles

   1. 150 × 4 − (140 + 165 + 130)
   2. 600 − (140 + 165 + 130)
   3. 600 − 435
   4. 165

   Shortcut

   Check by averaging again

   Put the 165 back in the list: 140 + 165 + 130 + 165 is 600, and 600 ÷ 4 = 150. That is the target.

   Next step: The median ↓
3. 3

   The median: sort first, then take the middle

   The median is the middle number of the list after you put it in order. Half of the numbers are below it and half are above it.

   With an odd count (5, 7, 9 numbers) there is one middle number. With an even count (4, 6, 8 numbers) there are two. Then the median is halfway between them: add the two middle numbers and divide by 2.

   Shortcut

   Where is the middle?

   For a sorted list of n numbers, the middle is at position (n + 1) ÷ 2. For 7 numbers: position 4. For 6 numbers: position 3.5, which means halfway between the 3rd and the 4th.

   1. 31, 18, 25, 40, 22, 18, 27
   2. 18, 18, 22, 25, 27, 31, 40
   3. 18, 18, 22, 25, 27, 31, 40

   1. 14, 9, 20, 11, 16, 12
   2. 9, 11, 12, 14, 16, 20
   3. 9, 11, 12, 14, 16, 20
   4. (12 + 14) ÷ 2 = 13

   Next step: Mode & range ↓
4. 4

   Mode and range: count, then subtract

   The mode is the number that appears most often. A list can have one mode, two modes, or no mode at all if nothing repeats. The mode is the number itself, not how many times it appears.

   One mode

   3, 5, 5, 7

   5

   Two modes

   2, 2, 6, 6, 9

   2 and 6

   No mode

   1, 4, 7, 8

   Nothing repeats: no mode

   The range is the largest number minus the smallest. Find them in the sorted list, not at the two ends of the list as it was written.

   Example

   Hours of sleep logged by 8 recruits: 6, 7, 5, 7, 8, 6, 7, 9. Find the mode and the range.

   1. Sort them: 5, 6, 6, 7, 7, 7, 8, 9. Equal numbers now sit together, so they are easy to count.
   2. 7 appears three times, more than any other number: the mode is 7 hours.
   3. Range: largest minus smallest, 9 − 5 = 4 hours.

   Answer Mode 7 hours, range 4 hours

   Next step: Which one? ↓
5. 5

   Read which one the question asks for

   The four answers are different numbers, and a test question often puts the other three in the choices. So find the key word before you calculate.

   “average”, “mean”Meanadd, then divide

   “middle”, “median”Mediansort, take the middle

   “most often”, “most common”Modecount the repeats

   “spread”, “highest minus lowest”Rangelargest − smallest
   Mean

   Five mechanics worked 12, 5, 9, 12 and 2 hours of overtime. What is the average?

   40 ÷ 5 = 8 hours

   Median

   What is the middle number of hours?

   2, 5, 9, 12, 12 → 9 hours

   Mode

   Which number of hours is most common?

   12 hours, twice

   For the same overtime hours, the range is 12 − 2 = 10: a fourth number, different from all three answers.

   Next step: One big number ↓
6. 6

   One extreme value moves the mean, not the median

   An outlier is a number far away from the rest of the list. The mean uses every number, so one outlier pulls it toward itself. The median only looks at the middle spot, so it hardly moves.

   Five people at a small garage earn $16, $17, $18, $19 and $20 an hour. The mean and the median are both $18. Now the $20 worker is replaced by the owner, who takes $70 an hour.

   1. (16 + 17 + 18 + 19 + 70) ÷ 5
   2. 140 ÷ 5
   3. 28

   One change, two results

   Hourly pay	Mean	Median
   16, 17, 18, 19, 20	$18	$18
   16, 17, 18, 19, 70	$28	$18

   Next step: Three examples ↓
7. 7

   Three worked examples, from easy to test level

   Each one starts with the first try many people make, crossed out, and then the way that works. Read the wrong try too: it is the kind of answer the test puts in the choices.

   The test-level one needs two moves in a row: first find a missing number from the mean, then find the median of the full list. Write the missing number down before you sort.

   Example · Easy

   What is the median of 14, 6, 11, 3, 9?

   First tryThe middle of the list as written: 11

   1. The median needs the list in order. As written, 11 only happens to be in the middle spot.
   2. Sort from smallest to largest: 3, 6, 9, 11, 14.
   3. Five numbers, so the middle is the 3rd: (5 + 1) ÷ 2 = 3.
   4. The 3rd number is 9.

   Answer 9

   Example · Medium

   The ages of six volunteers at a food bank are 34, 21, 28, 45, 19 and 30. What is the median age?

   First try(28 + 45) ÷ 2 = 36.5

   1. The first try took the two middle numbers before sorting. Sort first: 19, 21, 28, 30, 34, 45.
   2. Six numbers: no single middle. The middle pair is the 3rd and 4th, 28 and 30.
   3. Halfway between them: (28 + 30) ÷ 2, and 58 ÷ 2 = 29.

   Answer 29 years

   Example · Test level

   Six soldiers ran a 2-mile course. Five of the times were 15, 17, 14, 16 and 18 minutes. The mean of all six times was 17 minutes. What is the median of all six times?

   First tryMedian of the five known times: 14, 15, 16, 17, 18 → 16

   1. The median must include the sixth time, so find it first. Total of all six: 17 × 6 = 102 minutes.
   2. The five known times add to 15 + 17 + 14 + 16 + 18 = 80.
   3. The sixth time: 102 − 80 = 22 minutes.
   4. Sort all six: 14, 15, 16, 17, 18, 22. The middle pair is 16 and 17.
   5. Median: (16 + 17) ÷ 2 = 16.5 minutes. It is below the mean of 17, because the slow 22 pulls the mean up.

   Answer 16.5 minutes

4 traps in the answer choices

The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.

15 questions in three levels

Warm-up first, then core, then test level. Pick an answer, then read why.

12 flashcards

Tap a card to flip it. Keep the ones you miss until they stick.