HSPT: Data: find the median of an unordered list
Learn to find the median of a data set by ordering it first, for both odd and even numbers of values.
How do you find the median when the numbers are not in order?
Short answer
Put the values in order from least to greatest, then take the middle one. With an even number of values, average the two in the middle.
Know first: Ordering whole numbers, Finding a mean of two numbers
Step 1 of 5
The median is the middle of the ordered list
The median is the middle value once the data are in order from least to greatest. Half the values sit below it and half above.
The word ordered is the whole trick. A list printed in a jumbled order has some number in the middle position, and that number is usually not the median.
| Number of values | Where the median is | Example (ordered) |
|---|---|---|
| Odd, such as 5 | The single middle value | 4, 8, 11, 15, 19: median 11 |
| Even, such as 4 | The mean of the two middle values | 2, 4, 6, 10: median (4 + 6) ÷ 2 = 5 |
With n values in order, the middle position is (n + 1) ÷ 2. For 5 values, that is the 3rd.
Step 2 of 5
Put the values in order first
Worked example: Five unordered values
What is the median of 15, 4, 19, 8, 11?
- A19
- B11
- C11.4
- D15
- 1
Order the values
4, 8, 11, 15, 19.
- 2
Find the middle
Five values, so the 3rd is the middle: 11.
- 3
Check
Two values (4 and 8) are below 11 and two (15 and 19) are above.
- 4
See why the other choices are there
19 is the middle of the list as printed, before ordering. 11.4 is the mean, 57 ÷ 5. 15 is the range, 19 − 4.
Answer
11
Find the wrong step
A student finds the median of 6, 2, 10, 4.
- 1
Orders the values: 2, 4, 6, 10.
- 2
Notices there are four values, so no single middle.
- 3
Picks the 3rd value, 6, as the median.
What went wrong
With an even count, the median is halfway between the two middle values, the 2nd and 3rd.
The fix
Average the two middle values: (4 + 6) ÷ 2 = 5.
Step 3 of 5
Order, then find the middle
Check yourself · Question 1
What is the median of 23, 17, 31, 12, 26, 20, 15?
Answer: B
Order: 12, 15, 17, 20, 23, 26, 31. Seven values, so the 4th is the middle: 20.
Trap answer: 12
Middle of the unordered list
- Why it looks right
- 12 sits exactly in the middle position of the list you are given.
- Why it's wrong
- The median is the middle of the ordered list. In order, 12 comes first.
- The right answer
- 20, the 4th of seven ordered values.
Check yourself · Question 2
What is the median of 10, 2, 16, 6, 1, 11?
Answer: B
Order: 1, 2, 6, 10, 11, 16. Six values, so average the 3rd and 4th: (6 + 10) ÷ 2 = 8.
Common mistake
"I pick the number in the middle of the list I'm given."
- Why it's tempting
- The word middle points straight at the middle position.
- Do this instead
- Rewrite the values in order first, and cross them off in pairs from both ends until you reach the middle.
Step 4 of 5
What to remember
Remember
- Order the values from least to greatest before finding the median.
- With an odd count, take the middle value; with an even count, average the two middle values.
- The mean and the range are different measures and often appear as wrong choices.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
What did the ordered list look like in the practice question, and which choice was the middle of the unordered list?