What is the data if the mean is 25 median is 25 mode of 10 and 30 and a range of 40?

Answer 1

It is impossible to give a definite answer as there are many possible data sets.

eg any "multiples" of the data sets:

• {6, 10, 10, 23, 25, 30, 30, 45, 46}
• {6, 10, 10, 24, 25, 30, 30, 44, 46}
• {7, 10, 10, 24, 25, 30, 30, 42, 47}
• {8, 10, 10, 24, 25, 30, 30, 40, 48}
• {9, 10, 10, 24, 25, 30, 30, 38, 49}
• {10, 10, 12, 24, 25, 30, 30, 34, 50}

A "multiple" of a data set is to duplicate each element the required number of times, eg 3 x the first data set would be:

• {6, 6, 6, 10, 10, 10, 10, 10, 10, 23, 23, 23, 25, 25, 25, 30, 30, 30, 30, 30, 30, 45, 45, 45, 46, 46, 46}

which also has the required properties.

The list given is not an exhaustive list; for example adding two numbers whose sum is 50 with one less than 25 (and greater than the first element) and the other greater than 25 (and less than the last element) so that 10 and 30 remain the mode will not change the required properties.

eg adding the numbers 11 & 39 to the first data set to create:

• {6, 10, 10, 11, 23, 25, 30, 30, 39, 45, 46}

creates another base data set that can be multiplied.

To create a base set:

1. start with the mode = 10 & 30 and median = 25 by having two each of 10 & 30 and one 25: {10, 10, 25, 30, 30}
2. Now add in the range of 40 without changing the current median by choosing two numbers that have a difference of 40 and are either side of 10 & 30, eg 6 and 46: {6, 10, 10, 25, 30, 30, 46}
   1. The sum of the data set so far is 6 + 10 + 10 + 25 + 30 + 30 + 46 = 157
   2. The next multiple of 25 is 175, so with the two numbers added, the data set must total 175 + 2 x 25 = 225
   3. Thus the added numbers must total 225 - 257 = 68.
   4. One must be less than 25 and greater than 6; the other greater than 25 and less than 46
   5. Try one at one less than 25 to see what the other could be: 68 - 24 = 44; ok
   6. Required data set is {6, 10, 10, 24, 25, 30, 30, 44, 46}
   7. An alternative pair could have been 23 and 68 - 23 = 45, giving: {6, 10, 10, 23, 25, 30, 30, 45, 46}
3. Now create the required mean by adding two more numbers:

This is by no means the only way to create a data set. Having an even number of elements is possible, just it makes ensuring the median is 25 is not so easy (as consideration then has to be applied to the mean of the middle two numbers), eg {10, 10, 20, 24, 26, 30, 30, 50} is an even data set that satisfies the conditions.