Question

at their family reunion, the erickson family recorded everyones age. age number of people 18 2 30 3 36 1 41 1 97 3 x is the age of a randomly chosen person. what is the standard deviation of x? round your answer to the nearest hundredth.

Explanation:

Step1: Calculate the total number of people

$n=2 + 3+1+1+3=10$

Step2: Calculate the mean $\mu$

$\mu=\frac{18\times2 + 30\times3+36\times1+41\times1+97\times3}{10}=\frac{36+90 + 36+41+291}{10}=\frac{494}{10}=49.4$

Step3: Calculate the squared - differences from the mean for each age

For age $18$: $(18 - 49.4)^2\times2=( - 31.4)^2\times2=985.96\times2 = 1971.92$
For age $30$: $(30 - 49.4)^2\times3=( - 19.4)^2\times3=376.36\times3 = 1129.08$
For age $36$: $(36 - 49.4)^2\times1=( - 13.4)^2\times1 = 179.56$
For age $41$: $(41 - 49.4)^2\times1=( - 8.4)^2\times1=70.56$
For age $97$: $(97 - 49.4)^2\times3=(47.6)^2\times3=2265.76\times3 = 6797.28$

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

$\sigma^{2}=\frac{1971.92+1129.08 + 179.56+70.56+6797.28}{10}=\frac{10148.4}{10}=1014.84$

Step5: Calculate the standard deviation $\sigma$

$\sigma=\sqrt{1014.84}\approx31.86$

Answer:

$31.86$