Question 51·Hard·One-Variable Data Distributions; Measures of Center and Spread
A data set consists of ten distinct positive integers. The mean of the ten integers is 15 and the median is 12. Nine of the integers are
2, 5, 7, 9, 11, 13, 14, 18, and 20
What is the remaining integer?
For data-set problems with a given mean and one missing value, first convert the mean into a total sum by multiplying the mean by the number of data points. Then add the known values and subtract this sum from the total to get the missing value. Finally, if additional conditions like the median or distinctness are given, quickly check them by ordering the numbers and verifying that the specified median or property holds. This approach minimizes algebra and keeps the work organized and fast.
Hints
Connect the mean to the total
You are told the mean of 10 integers is 15. How can you use this to find the sum of all 10 integers?
Use the 9 known integers
Add the nine given integers together. How does their sum relate to the total sum of all 10 integers?
Isolate the missing number
Let the unknown integer be and write an equation: (sum of 9 known integers) (total sum from the mean). Solve for .
Check the median
Once you find a candidate for the missing integer, insert it into the sorted list and identify the 5th and 6th numbers. Does their average equal 12?
Desmos Guide
Compute totals needed for the difference
In Desmos, enter the two expressions S_total = 10*15 and S_known = 2+5+7+9+11+13+14+18+20 to compute the overall total and the sum of the nine known integers.
Find the missing value (and optionally verify the median)
Now compute x = S_total - S_known (or directly 10*15 - (2+5+7+9+11+13+14+18+20)). The output is the missing integer, which equals 51. Optionally, verify the median by listing the ten numbers in order and computing (11+13)/2 to confirm it equals 12.
Step-by-step Explanation
Use the mean to find the total sum
The mean of the ten integers is , so the total sum of all ten integers is
So the ten integers together must add up to .
Find the sum of the 9 known integers
Add the nine given integers:
So the nine known integers have a total sum of .
Set up and solve symbolically for the missing integer
Let the remaining integer be . Then
Subtract from both sides:
We will verify the median and finalize the value in the next step.
Compute the value and check the median condition and distinctness
Now compute : , and check that this value keeps the median at and that all integers are distinct.
List all ten numbers in order:
For numbers, the median is the average of the th and th numbers. Here the th is and the th is , so the median is
All numbers are distinct, and both the mean and median match the problem, so the remaining integer is 51.