Question

an education researcher recorded the number of students at each of the schools in his county.

number of students	number of schools
164	1
384	2
877	4
920	1

x is the number of students that a randomly chosen school has. what is the standard deviation of x?

round your answer to the nearest hundredth.

Explanation:

Step1: Calculate the total number of schools

$n=2 + 1+2 + 4+1=10$

Step2: Calculate the mean $\mu$

$\mu=\frac{(113\times2)+(164\times1)+(384\times2)+(877\times4)+(920\times1)}{10}=\frac{226 + 164+768+3508+920}{10}=\frac{5586}{10}=558.6$

Step3: Calculate the squared - differences and their weighted sum

$(113 - 558.6)^2\times2+(164 - 558.6)^2\times1+(384 - 558.6)^2\times2+(877 - 558.6)^2\times4+(920 - 558.6)^2\times1$
$=( - 445.6)^2\times2+( - 394.6)^2\times1+( - 174.6)^2\times2+(318.4)^2\times4+(361.4)^2\times1$
$=(198569.36\times2)+(155609.16\times1)+(30485.16\times2)+(101378.56\times4)+(130609.96\times1)$
$=397138.72+155609.16 + 60970.32+405514.24+130609.96$
$=1150842.4$

Step4: Calculate the variance $\sigma^{2}$

$\sigma^{2}=\frac{1150842.4}{10}=115084.24$

Step5: Calculate the standard deviation $\sigma$

$\sigma=\sqrt{115084.24}\approx339.24$

Answer:

$339.24$