Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea:Sort the data and the median is the value standing in the exact middle, with half below and half above.
Common stuck point:The procedure for median is the easy part; the trap is finding the middle of the unsorted list. Asking "After sorting the data smallest to largest, what value sits exactly in the middle?" first is what keeps a correct-looking calculation from being attached to the wrong concept.
Sense of Study hint:Ask: After sorting the data smallest to largest, what value sits exactly in the middle?
Worked Examples
Example 1
easy
Find the median of {3,7,1,9,5}.
Answer
Median=5
First step
1
Sort the data in ascending order: {1,3,5,7,9}.
Full solution
2
Since n=5 (odd), the median is the middle value at position 25+1=3.
3
The third value is 5.
For an odd number of data points, the median is the middle value after sorting. The median is robust against outliers, unlike the mean.
Example 2
medium
Find the median of {12,4,7,19,3,15}.
Example 3
easy
In a class of 11 students, what position (rank) does the median occupy in sorted order?
Example 4
medium
Data set {x,4,6,9,12} is already sorted with median 6. What constraint does this place on x?
Example 5
medium
A data set has 9 values with median 50. What is the median of {x/2:x∈data}?
Example 6
medium
What is the median of the first 10 positive integers?
Example 7
hard
Three students have ages 12,14,16 (median 14). A fourth student joins, making the group's median 13.5. Find the fourth student's age.
Example 8
hard
A class has 21 quiz scores sorted as s1≤s2≤⋯≤s21 with median s11=80. The teacher adds 4 perfect scores of 100. Express the new median in terms of the original scores.
Example 9
hard
A data set of 7 distinct positive integers has median 10 and the smallest value is 3. Is the mean bounded above?
Example 10
challenge
Show that among any n distinct real numbers x1,…,xn, the median minimizes the sum of absolute deviations f(c)=∑i∣xi−c∣.
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easy
Find the median of {22,8,15,31,10,18,27}.
Example 2
medium
The ordered data set {2,5,8,x,14,18} has median 10.5. Find x.
Example 3
easy
Find the median of 3,7,5.
Example 4
easy
Find the median of 10,2,8,4,6.
Example 5
easy
Find the median of 4,8.
Example 6
easy
Find the median of 1,2,3,4,5,6.
Example 7
easy
Find the median of 9,9,9.
Example 8
easy
Find the median of 12,15,11,14.
Example 9
easy
Find the median of 20,5,13.
Example 10
easy
Find the median of 7,1,7,3,9.
Example 11
medium
A data set sorted is 4,6,x,10 and its median is 7. Find x.
Example 12
medium
Scores are 82,90,76,88,94,79. Find the median.
Example 13
medium
A list has values 5,8,12,15,20. Add the value 100. How does the median change?
Example 14
medium
The median of 3,7,11,x is 9 (set is already sorted). Find x.
Example 15
medium
Find the median of the first eight positive even numbers.
Example 16
medium
A class of 7 students has ages 11,12,12,13,13,13,40 (a teacher included). Find the median age.
Example 17
medium
Two data sets are combined: {2,4,6} and {8,10}. Find the median of the combined set.
Example 18
medium
In 5,9,9,12,15,18, find both the median and state whether it equals any data value.
Example 19
medium
A data set of 9 numbers has median 20. If every value is increased by 5, what is the new median?
Example 20
challenge
A sorted set of 6 distinct integers has median 10. The two middle values are consecutive integers. What are they?
Example 21
challenge
Five distinct positive integers have median 6 and mean 8. What is the largest possible value in the set?
Example 22
challenge
A set of n numbers (n even) has median m. You append the single value m. Is the new median still m? Explain.
Example 23
easy
Find the median of {4,1,7}.
Example 24
easy
Find the median of {15,11,13,19,17}.
Example 25
easy
Find the median of {100,50,75}.
Example 26
easy
Find the median of {−3,−1,0,2,5}.
Example 27
medium
Find the median of the data set {12,4,18,9,7,14,2}.
Example 28
medium
Eight test scores are 72,85,91,68,77,88,94,80. Find the median.
Example 29
medium
A data set of 10 numbers has median 15. If every value is decreased by 7, find the new median.
Example 30
medium
Six house prices in a neighborhood are (in thousands of dollars): 250,290,310,305,280,1500. Compute median and mean; which better represents the typical price?
Example 31
medium
A set of 12 numbers has median 30. You append the value 30 to the set. What is the new median?
Example 32
hard
A set of 5 positive integers has median 7 and mean 7. Give one example of such a set and explain why the median and mean being equal is consistent here.
Example 33
hard
In the sorted set {3,5,7,x,11,13} with median 9, find x.
Example 34
hard
What is the median of the integers from 1 to 100?
Example 35
challenge
Two data sets each of size 5 have medians 8 and 12. When combined (size 10), what range of medians is possible?