Question

a volunteer for the local running club counted the number of participants at each race.

number of participants	number of races
47	1
79	3
175	3
179	1

x is the number of participants that a randomly chosen race had. what is the standard deviation of x?
round your answer to the nearest hundredth.

Explanation:

Step1: Calculate the total number of races

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

Step2: Calculate the mean $\mu$

$\mu=\frac{(31\times2 + 47\times1+79\times3 + 175\times3+179\times1)}{10}=\frac{(62 + 47+237+525+179)}{10}=\frac{1050}{10}=105$

Step3: Calculate the squared - differences and multiply by frequencies

$(31 - 105)^2\times2=( - 74)^2\times2 = 5476\times2=10952$
$(47 - 105)^2\times1=( - 58)^2\times1 = 3364\times1=3364$
$(79 - 105)^2\times3=( - 26)^2\times3 = 676\times3=2028$
$(175 - 105)^2\times3=(70)^2\times3 = 4900\times3=14700$
$(179 - 105)^2\times1=(74)^2\times1 = 5476\times1=5476$

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

$\sigma^{2}=\frac{10952 + 3364+2028+14700+5476}{10}=\frac{36520}{10}=3652$

Step5: Calculate the standard deviation $\sigma$

$\sigma=\sqrt{3652}\approx60.43$

Answer:

$60.43$