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 when rolling the die five times, we need to consider the concept of "range" in statistics. The range is the difference between the largest and smallest values in a dataset. In this case, we want to minimize the range to reduce the standard deviation.
1. Smallest possible standard deviation:
To achieve the smallest possible standard deviation, we need the range to be as small as possible. This means we should aim for all the numbers on the die to be as close together as possible. To achieve this, we want all the numbers to appear with an equal frequency. Therefore, we would roll the die in such a way that each number shows up exactly once. So, for the smallest possible standard deviation, the numbers 1, 2, 3, 4, 5, 6 should show up after rolling the die five times.
2. Largest possible standard deviation:
To achieve the largest possible standard deviation, we need to maximize the range. This means we want the numbers to be as spread out as possible. To achieve this, we want one or more numbers to show up more frequently than others. For example, by rolling a large number of times, we can increase the chances of getting one or more numbers more than others. So, the numbers that would have to show up if we want to have the largest possible standard deviation could be any combination of multiple occurrences of one or more numbers.
3. Smallest possible standard deviation without repeats:
If we have to roll the die five times without allowing any repeats, the numbers that would result in the smallest possible standard deviation would be a combination of five different numbers from 1 to 6. This is because we are excluding the possibility of any numbers showing up more than once. So, any combination of five different numbers from 1 to 6 would give us the smallest possible standard deviation when rolling the die five times without repeats.