Question

SD contestThis is a standard deviation contest. You must choose four numbers from the whole numbers 0 to 10, with repeats allowed.

(a) Choose four numbers that have the smallest possible standard deviation.

(b)Choose four numbers that have the largest possible standard deviation.

(c) Is more than one choice possible in either part (a) or (b)? Explain.

Short answer

Part (a) 2, 2, 2, 2

Part (b) 0, 0, 10, 10

Part (c) There are multiple options available for (a) and part (b) unique.

Step-by-step solution

Part (a) Step 1. Given information.

The given statement says that we must select four digits from the range of 0 to 10, we can also repeat numbers.

Part (a) Step 2. Select four numbers that have the smallest possible standard deviation.

From the given set of data, the smallest possible standard deviation is 0 if we choose the four numbers like 2, 2, 2, 2.

We all know that standard deviation is nonnegative.

Therefore, the numbers are 2, 2, 2, and 2.

Part (b) Step 1. Select four numbers that have the largest possible standard deviation.

Getting the largest standard deviation is only possible when we have values that are far from the mean value.

The given numbers range from 0 to 10. Then the largest standard deviation will be possible if we choose two values as 0 and two other values as 10 because our mean will lie exactly in between which is 5.

Therefore, the numbers are 0, 0, 10, and 10.

Part (c) Step 1. State if more choices possible in either part (a) or (b).

Part (a) allows for several options because getting a standard deviation equal to 0 is possible if the sample contains one unique value.

Furthermore, in portion (a), we provided two options, implying that more than one option is available.

In Part (b), the answer is unique because changing only one of the variables changes the mean from 5 to a lower value, lowering the total deviation from the mean.

Because the standard deviation is proportional to the total deviation, the standard deviation will fall.