Question

How do you find a set of numbers that will satisfy the following conditions?
The median of a set of $20$ numbers is $24$.
(i) the range is $42$
(ii) to the nearest whole number, the mean is $24$
(iii) no more than three numbers are the same.

Answer 1

According to the question, we will see first what values are given, and then figure out how to find the set of numbers. First, we will try to find out the median. Then we will try to find the mean, and we will try to divide the value left and the total number of items left, and we will get the values of all the items.

Complete step by step solution:
We know that the set has $20$ numbers. Now, we can see that the number of items are even so then, we will find the median by taking the average value of the two values which are at the center of the set. To check that the median is coming out as $24$, we can either choose both the numbers to be $24$, or we can choose one number as $23$and the other number to be $25$, and the average for both the numbers comes out to be $24$.
It is given that the range is $42$. We know that the range is the difference between the highest and the lowest value. So, we can say that:
$Maximum - Minimum = 42$
If the mean of the $20$ numbers is going to be $24$, then this means that the sum of the $20$numbers will be:
$20 \times 24 = 480$
If we take the innermost numbers to be $24$, then we are left with:
$480 - 24 - 24 = 432$
This is for $18$ numbers.
Now, we can figure out the maximum and the minimum number at the center by using two values. The two values are $3$ and $45$. This places half of the range above and below.
Mow, we are left with:
$432 - 3 - 45 = 384$
This is for $16$ numbers.
We know that $384$ is completely divisible by 16 and the answer is $24$. But, it is given that no more than the three numbers are the same. So, we can easily pair out the numbers which are around $24$, and we get:
$23\,and\,25$, $22\,and\,26$, $21\,and \,27$, $20\,and\,28$, $19\,and\,29$, $18\,and\,30$, $17\,and\,31$, $16\,and\,32$.
Therefore, the numbers are:
$3,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32\,and\,45$

Note: The above question comes under the measuring center portion in statistics part of Mathematics. There are three main measuring centers that are the mean, median, and mode. In a normal distribution or perfect distribution, all the mean, median and mode are the same.