Q 103. - The Practice of Statistics for AP

Question

This 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 (a) or (b)? Explain. 

Step-by-Step Solution

Answer

Part (a) The smallest possible standard deviation is $0$

Part (b) The largest possible standard deviation is $5$

Part (c) It is possible that more than$1$choice.

Part (a) Step 1: Given information

Four numbers from the whole numbers $0$ to $10$ have to be chosen. The contest is for standard deviation.

Part (a) Step 2: Concept

Statistical Measures are a descriptive analytical technique that provides an overview of a data set's properties.

Part (a) Step 3: Explanation

Numbers: $0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10$

If all four values are the same, for example, $1, 1, 1, 1,$ the standard deviation will be $0$ because the mean of these four integers is $1$ and every deviation from the mean is $0$, therefore the standard deviation will be $4$ So, the smallest possible standard deviation is $0$

Part (b) Step 1: Explanation

Numbers: $0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10$

If all four numbers are added together, we get the maximum possible standard deviation. Consider the following example:$0 1, 9$ and $10$ Because the mean of these four integers is $5$, and all four numbers are distant from the mean, as may be determined manually. As a result, the maximum standard deviation is $5$

Part (c) Step 1: Explanation

Numbers: $0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10$

It is conceivable to make more than one option, i.e., to take all of the same observations as $2, 2, 2, 2$or $3, 3, 3, 3$ or $4, 4, 4, 4$

The above-mentioned instance is merely one option for the biggest standard deviation, while the other four possibilities all have significantly lower standard deviations. Thus, It is possible that more than $1$ choice.