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 find the number(s) that would result in the smallest possible standard deviation after rolling the die five times, we need to ensure that the numbers are as close to each other as possible. This means that we should aim for a set of consecutive numbers.
To achieve this, we can roll the die and record the numbers 1, 2, 3, 4, 5, and 6 in any order. For example, if the results of the five rolls are 2, 4, 1, 3, and 6, then we have a set of consecutive numbers.
To find the number(s) that would result in the largest possible standard deviation after rolling the die five times, we need to ensure that the numbers are as far apart from each other as possible. This means that we should aim for a set of numbers that are spread out across the entire range of possibilities.
To achieve this, we can roll the die and record any combination of numbers that covers the entire range from 1 to 6. For example, if the results of the five rolls are 1, 2, 3, 4, and 6, then we have a set of numbers that covers the entire range, resulting in the largest possible standard deviation.
If no repeats are allowed in the first question, meaning we ignore a repeat number and roll again, then we need to adjust our approach.
To find the number(s) that would result in the smallest possible standard deviation with no repeats, we need to ensure that we have a set of distinct numbers that are as close to each other as possible.
One possible approach is to roll the die until we obtain a set of five distinct consecutive numbers. For example, we could roll the die and record the following sequence: 1, 2, 4, 3, 5. In this case, we ignore the repeat number (4) and roll again until we achieve five different numbers. This would result in the smallest possible standard deviation with no repeats.
Similarly, if we want to find the number(s) that would result in the largest possible standard deviation with no repeats, we need to ensure that we have a set of distinct numbers that cover the entire range. Again, we can roll the die until we obtain a set of five distinct numbers that cover the range from 1 to 6. For example, we could roll the die and record the following sequence: 1, 2, 3, 5, 6. In this case, we ignore the repeat number (4) and roll again until we achieve five different numbers. This would result in the largest possible standard deviation with no repeats.