Question

1. Four people each think of an integer 1 to 9. What could have been their four numbers if the SD was zero?
SD = Standard Deviation

2. Four people think of an integer 1 to 9. What would have been their four numbers to have the greatest standard deviation possible?
SD = Standard Deviation

Answer 1

Answer: See below :)

Step-by-step explanation:

1. If the standard deviation is zero, this means that all the values are the same, as there is no variability in the data set. If four integers are chosen from 1 to 9 all four people must have chosen the same number (for the standard deviation to be 0). Therefore, their numbers could be any integer from 1 to 9 as long as they are identical.

For example, their numbers could have been 3, 3, 3, 3 or 6, 6, 6, 6, etc.

2. To find the greatest standard deviation, the values must be as far away from the mean (average) as possible. We can take the greatest (9) and least (1) possible values from the data set to choose the 4 numbers. We can try the set 1, 1, 9, 9.

To find the standard deviation, we first have to find the mean. (1 + 1 + 9 + 9) / 4 gives us our mean of 5.

Next, we have to find the squared difference between each number and the mean:

(1 - 5)^2 = 16

(1 - 5)^2 = 16

(9 - 5)^2 = 16

(9 - 5)^2 = 16

Then, we have to find the mean of these squared differences: (16 + 16 + 16 + 16) / 4 gives us 16.

Finally, we take the square root of our mean (16), giving us 4. Therefore, the numbers 1, 1, 9, 9 give us the greatest standard deviation possible of 4.

Answer 2

Answer:

1. All four numbers must be the same. E.g. 7, 7, 7, and 7.

2. 1, 1, 9, 9

Step-by-step explanation:

Question 1

Standard deviation measures the amount of variation or dispersion in a set of values.

If the standard deviation is zero, then there is no variation and all the values in the data set are the same.

So, if four people think of an integer from 1 to 9, and the standard deviation is zero, all four people must have thought of the same number, for example 7, 7, 7, and 7.

Question 2

To maximize the standard deviation, the numbers chosen by the four people should be as far apart as possible.

In this case, the four numbers should be the extremes of the given range, meaning that two people would both have to choose 1 and the other two people would both have to choose 9. This would result in the greatest standard deviation of 4.