I have a 6 sided dice with numbers 1 to 6 written on faces.
What number(s) would have to show up if you rolled the die five times in order to have the smallest possible standard deviation?
What number(s) would have to show up if you rolled the die 5 times in order to have the largest possible standard deviation?
If no repeats were allowed in first question what would answer be? Meaning you ignore a repeat number and roll again.
To determine the numbers that would yield the smallest possible standard deviation, we need to choose the most consistent set of numbers. This means selecting numbers that are closest to each other.
1. Smallest possible standard deviation: In order to minimize the standard deviation, we want to choose consecutive numbers. Rolling the die five times and obtaining the numbers 1, 2, 3, 4, and 5 (in any order) would result in the smallest possible standard deviation. This set of numbers has the least variability since they are the closest together in a numerical sense.
2. Largest possible standard deviation: To maximize the standard deviation, we should choose numbers that are far apart from each other. Rolling the die five times and obtaining the numbers 1, 1, 6, 6, and 6 (in any order) would result in the largest possible standard deviation. This set of numbers has the highest variability since it includes the maximum possible spread between numbers.
3. No repeats allowed: In this case, if we disregard repeating numbers and roll again, it is not possible to determine the exact numbers that would result in the smallest possible standard deviation. However, we can still state that the outcome would be a set of consecutive numbers, such as 1, 2, 3, 4, and 5 (with no repeats). This sequential arrangement would minimize the standard deviation, while adhering to the constraint of no repeated numbers.