Question

suppose you choose 4 numbers 0 - 10, with repeats allowed. which of the following choices will have the largest standard deviation?
a) 1,1,9,9
b) 1,1,1,1
c) 1,4,7,10
d) 0,0,10,10

Explanation:

Step1: Recall standard - deviation formula

The formula for the standard deviation of a sample $x_1,x_2,\cdots,x_n$ is $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}$, where $\bar{x}=\frac{\sum_{i=1}^{n}x_i}{n}$. Here $n = 4$. A larger spread of data points from the mean will result in a larger standard - deviation.

Step2: Analyze option A

For the data set $1,1,9,9$:
The mean $\bar{x}_A=\frac{1 + 1+9+9}{4}=\frac{20}{4}=5$.
The squared differences from the mean are $(1 - 5)^2=16$, $(1 - 5)^2=16$, $(9 - 5)^2=16$, $(9 - 5)^2=16$. The sum of squared differences $\sum_{i = 1}^{4}(x_{iA}-\bar{x}_A)^2=16+16+16+16 = 64$.

Step3: Analyze option B

For the data set $1,1,1,1$:
The mean $\bar{x}_B=\frac{1+1+1+1}{4}=1$.
The squared differences from the mean are $(1 - 1)^2 = 0$, $(1 - 1)^2 = 0$, $(1 - 1)^2 = 0$, $(1 - 1)^2 = 0$. The sum of squared differences $\sum_{i = 1}^{4}(x_{iB}-\bar{x}_B)^2=0$.

Step4: Analyze option C

For the data set $1,4,7,10$:
The mean $\bar{x}_C=\frac{1 + 4+7+10}{4}=\frac{22}{4}=5.5$.
The squared differences are $(1 - 5.5)^2=20.25$, $(4 - 5.5)^2 = 2.25$, $(7 - 5.5)^2=2.25$, $(10 - 5.5)^2=20.25$. The sum of squared differences $\sum_{i = 1}^{4}(x_{iC}-\bar{x}_C)^2=20.25+2.25+2.25+20.25 = 45$.

Step5: Analyze option D

For the data set $0,0,10,10$:
The mean $\bar{x}_D=\frac{0 + 0+10+10}{4}=5$.
The squared differences from the mean are $(0 - 5)^2=25$, $(0 - 5)^2=25$, $(10 - 5)^2=25$, $(10 - 5)^2=25$. The sum of squared differences $\sum_{i = 1}^{4}(x_{iD}-\bar{x}_D)^2=25+25+25+25 = 100$.

Since the sum of squared differences from the mean is largest for option D, option D will have the largest standard - deviation.

Answer:

D. 0, 0, 10, 10