Open Middle®
Directions: Using the digits 1 to 9, find a six number data set that has a Mode of 1, Median of 2 and Mean of 3. Digits can be repeated.
Hint
How can you find the median of 6 numbers? How do you calculate the mean of 6 numbers? What is the definition of mode?
Answer
1, 1, 1, 3, 4, 8
1, 1, 1, 3, 3, 9
1, 1, 1, 3, 5, 7
1, 1, 1, 3, 6, 6
1, 1, 1, 3, 3, 9
1, 1, 1, 3, 5, 7
1, 1, 1, 3, 6, 6
Source: Harold Jacobs
1 1 1 3 6 8
The arithmetic mean of this sample is 20/6, which is not 3.
I can confirm that the solutions given are the only 4 solutions. If you add the digits 0, there are another 3 solutions:
[0, 1, 1, 3, 6, 7]
[0, 1, 1, 3, 5, 8]
[0, 1, 1, 3, 4, 9]
Where I come from, the mode can have several values. In the list [1, 1, 2, 2] for example, the mode would be 1 and 2 (it’s multi-modal). If 1 just has to be one of the modes (instead of the only mode), then we get these additional solutions:
[1, 1, 2, 2, 6, 6]
[1, 1, 2, 2, 5, 7]
[1, 1, 2, 2, 4, 8]
[1, 1, 2, 2, 3, 9]
Solutions with 0 do not work with the directions: digits 1 to 9 (not 0). Possible solutions I believe are:
1,1,1,3,6,6
1,1,1,3,5,7
1,1,1,3,4,8
1,1,1,3,3,9
All have a single mode of 1, median is 2 and in order for the mean to be 3, the two largest numbers need to add to 12.
I would like to elucidate that I have a conundrum of esoterica.
In Rudolf Österreicher‘s answers, the mode is not 1, it is equally 1 and 2, so there is no number that occurs the most, therefore making the answers that are “ the only 4 solutions” wrong.
I would just like to express that do not listen to Luca Sign and keep finding answers!
I would like to make a point that esoterica is too tricky of a word
Good Question, i love being challenged amazing work Luca Sign and Rudolf Österreicher keep going you are amzing mathematicians.
I Think that esoterica is too tricky of a word just keep it simple.