Question

set a set b 1 22 3 24 5 25 7 26 9 27 data set a has a lower standard deviation because the values in the set are more spread out. data set a has a lower standard deviation because the values in the set are less spread out. data set b has a lower standard deviation because the values in the set are more spread out. data set b has a lower standard deviation because the values in the set are less spread out.

Explanation:

Step1: Calculate the mean of Set A

The formula for the mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$. For Set A: $x_1 = 1,x_2=3,x_3 = 5,x_4=7,x_5 = 9,n = 5$.
$\bar{x}_A=\frac{1 + 3+5+7+9}{5}=\frac{25}{5}=5$

Step2: Calculate the variance of Set A

The formula for variance $s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}$.
$(1 - 5)^{2}=(-4)^{2}=16$, $(3 - 5)^{2}=(-2)^{2}=4$, $(5 - 5)^{2}=0$, $(7 - 5)^{2}=4$, $(9 - 5)^{2}=16$
$s_{A}^{2}=\frac{16+4 + 0+4+16}{5}=\frac{40}{5}=8$

Step3: Calculate the standard deviation of Set A

The formula for standard deviation $s=\sqrt{s^{2}}$. So $s_A=\sqrt{8}\approx2.83$

Step4: Calculate the mean of Set B

For Set B: $x_1 = 22,x_2=24,x_3 = 25,x_4=26,x_5 = 27,n = 5$
$\bar{x}_B=\frac{22+24 + 25+26+27}{5}=\frac{124}{5}=24.8$

Step5: Calculate the variance of Set B

$(22-24.8)^{2}=(-2.8)^{2}=7.84$, $(24 - 24.8)^{2}=(-0.8)^{2}=0.64$, $(25 - 24.8)^{2}=0.04$, $(26 - 24.8)^{2}=1.44$, $(27 - 24.8)^{2}=4.84$
$s_{B}^{2}=\frac{7.84+0.64+0.04+1.44+4.84}{5}=\frac{14.8}{5}=2.96$

Step6: Calculate the standard deviation of Set B

$s_B=\sqrt{2.96}\approx1.72$

Step7: Compare the standard deviations

Since $s_A\approx2.83$ and $s_B\approx1.72$, and standard deviation measures spread. A lower standard deviation means less spread.

Answer:

Data set B has a lower standard deviation because the values in the set are less spread out.