Exploring Positive Integers with Specific Statistical Properties
Are you looking for positive integers that meet specific statistical criteria such as a mode, median, mean, and range? In this article, we will walk you through a step-by-step process to find five positive integers that have a mode of 4, a median of 6, a mean of 7, and a range of 10. We will also discuss how to solve this problem using a brute force approach with the J programming language.
Step-by-Step Process
Mean Calculation
The mean of the five positive integers is given as 7. The sum of these integers must be:
Mean frac{Sum}{5} implies Sum 7 times 5 35
Median Requirement
The median is 6, meaning that when the numbers are arranged in ascending order, the third number must be 6. Therefore, the numbers can be denoted as:
a b 6 d e
Where a leq b leq 6 leq d leq e.
Mode Requirement
The mode is 4, meaning that 4 must appear more frequently than any other number. To ensure this, we can include at least two occurrences of 4:
a 4, b 4 thus we have:
4 4 6 d e
Range Requirement
The range is 10, which means:
Range e - a 10 implies e a 10
Substituting e 14 into the sum equation:
4 4 6 d 14 35 implies d 28 35 implies d 7
Thus, the five numbers are:
4 4 6 7 14
Verification
Mode: The mode is 4, which appears most frequently.
Median: The median is 6, which is the middle value.
Mean: The mean is:
frac{4 4 6 7 14}{5} frac{35}{5} 7
Range: The range is:
14 - 4 10
Brute Force Solution Using J Programming Language
Using the J programming language, we can find 108 sets of 5 positive integers that meet all the criteria. Here is the snippet used:
n.odo 517ma.6medianmb.4modemc.7meanmd.10-/m.n~108
The code generates 108 sets of 5 positive integers, and the result confirms the existence of multiple solutions.
Conclusion
To find five positive integers that have a mode of 4, a median of 6, a mean of 7, and a range of 10, we followed a systematic approach. By understanding and applying the mean, median, mode, and range principles, we determined the specific values that meet these criteria. The brute force solution also confirmed that 108 such sets exist.