TACHS: Median: sort, count, then find the middle
Find the median of an odd or even set, see how adding or replacing a value changes it, and find a missing value from a known median.
How do you find a median, especially after a new number is added to the data?
Short answer
Put every value in order, including any new one, and count. With an odd count the median is the middle value; with an even count it is the average of the two middle values.
Know first: Ordering whole numbers, Averaging two numbers
Step 1 of 6
The median is the middle of a sorted list
To find a median, list the values from least to greatest first. The middle of an unsorted list means nothing.
Then count. Adding one value changes the count, so an odd set becomes even, and the median can land halfway between two numbers.
| Count | Median | Example |
|---|---|---|
| Odd | The single middle value | 2, 5, 8, 11, 20 → 8 |
| Even | Average of the two middle values | 2, 5, 8, 11 → (5 + 8) ÷ 2 = 6.5 |
To find the middle, cross off one value from each end until one or two remain.
Step 2 of 6
Sort first, every time
Worked example: Adding a value to five numbers
A data set is 3, 9, 6, 14, 11. The number 5 is added. What is the median of the new set?
- A6
- B7.5
- C10
- D15
- 1
Sort all six values
3, 5, 6, 9, 11, 14.
- 2
Count
Six values, an even count, so the median is halfway between the 3rd and 4th values: 6 and 9.
- 3
Average the middle pair
(6 + 9) ÷ 2 = 15 ÷ 2 = 7.5.
- 4
Check by crossing off
Remove 3 and 14, then 5 and 11. You are left with 6 and 9.
- 5
See why the other choices are there
6 picks one middle value instead of averaging two. 10 skips the sort: in 3, 9, 6, 14, 11, 5 the middle pair is 6 and 14. 15 adds the middle pair without dividing by 2.
Answer
The median is 7.5.
A missing value
Six sorted values are 1, 4, x, 12, 15, 18. Their median is 10. What is x?
- 1
There are 6 values, so the median is the average of the 3rd and 4th values.
- 2
Those are x and 12, so x + 12 = 10.
What went wrong
The median is the average of x and 12, so the equation needs a division by 2: (x + 12) ÷ 2 = 10.
- 3
So x = −2.
The fix
(x + 12) ÷ 2 = 10 means x + 12 = 20, so x = 8. Check the order: 4 ≤ 8 ≤ 12, and (8 + 12) ÷ 2 = 10.
Step 3 of 6
What moves the middle, and what does not
Replacing the largest value with an even larger one does not move the middle. The set 2, 8, 13, 17, 21 has median 13. Change 21 to 40 and the median is still 13. The mean would rise, but the median only cares about position.
To reach a target median, think about where the new value must sit. The set 4, 6, 14, 15 has median 10. To make the median of five values equal 11, the 3rd value must be 11, so add 11.
Step 4 of 6
Sort, count, find the middle
Check yourself · Question 1
A data set is 5, 9, 11, 16, 20. The number 13 is added. What is the median of the new set?
Answer: B
Sorted: 5, 9, 11, 13, 16, 20. Six values, so average the 3rd and 4th: (11 + 13) ÷ 2 = 12.
Trap answer: 11
Forgetting the count changed
- Why it looks right
- 13 is close to the middle already, so it seems like the old median should still work.
- Why it's wrong
- With six values there is no single middle value. The two middle values are now 11 and 13.
- The right answer
- 12, the average of 11 and 13.
Check yourself · Question 2
A data set is 6, 8, 11, 15, 19. The largest value is replaced by 49. What is the new median?
Answer: A
Sorted: 6, 8, 11, 15, 49. Five values, so the median is the 3rd: 11. The middle position did not change.
Check yourself · Question 3
Six sorted values are 3, 6, x, 14, 17, 21. Their median is 12. What is x?
Answer: B
The median is the average of the 3rd and 4th values: (x + 14) ÷ 2 = 12. So x + 14 = 24 and x = 10. Check the order: 6 ≤ 10 ≤ 14.
Common mistake
"I found the middle of the list just as it was written."
- Why it's tempting
- The list looks like it is already in order, and sorting feels like an extra step.
- Do this instead
- Sort every list before you look for the middle, including any new value. Then cross off from both ends.
Step 5 of 6
What to remember
Remember
- Sort all the values, including any new one, before looking for the middle.
- Odd count: the middle value. Even count: the average of the two middle values.
- Changing the largest or smallest value does not move the median unless the middle values change.
Step 6 of 6
Practice on a real question
Use what you just learned on this question, then check the explanation.
Try another one
In your own words
In the practice question, how many values are there after the new one is added, and which two values are in the middle?