Question

the train conductor made sure to count the number of passengers on each train.
number of passengers number of trains
124 3
166 1
245 3
262 3
x is the number of passengers that a randomly - chosen train had. what is the standard deviation of x?
round your answer to the nearest hundredth.

Explanation:

Step1: Calculate the total number of trains

$n=3 + 1+3 + 3=10$

Step2: Calculate the mean $\mu$

$\mu=\frac{(124\times3)+(166\times1)+(245\times3)+(262\times3)}{10}=\frac{372 + 166+735+786}{10}=\frac{2059}{10}=205.9$

Step3: Calculate the squared - differences and their weighted sum

$(124 - 205.9)^2\times3+(166 - 205.9)^2\times1+(245 - 205.9)^2\times3+(262 - 205.9)^2\times3$
$=(-81.9)^2\times3+(-39.9)^2\times1+(39.1)^2\times3+(56.1)^2\times3$
$=(6707.61\times3)+(1592.01\times1)+(1528.81\times3)+(3147.21\times3)$
$=20122.83+1592.01 + 4586.43+9441.63$
$=35742.9$

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

$\sigma^{2}=\frac{35742.9}{10}=3574.29$

Step5: Calculate the standard deviation $\sigma$

$\sigma=\sqrt{3574.29}\approx59.79$

Answer:

$59.79$