Question

the librarian at the public library counted the number of books on each shelf. number of books number of shelves 16 2 24 3 28 2 56 1 63 1 71 1 x is the number of books that a randomly chosen shelf has. what is the standard deviation of x? round your answer to the nearest hundredth.

Explanation:

Step1: Calculate total number of shelves

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

Step2: Calculate the mean $\mu$

$\mu=\frac{(16\times2 + 24\times3+28\times2 + 56\times1+63\times1+71\times1)}{10}=\frac{(32 + 72+56 + 56+63+71)}{10}=\frac{350}{10}=35$

Step3: Calculate the squared - differences

For $x_1 = 16$: $(16 - 35)^2\times2=(-19)^2\times2 = 361\times2=722$
For $x_2 = 24$: $(24 - 35)^2\times3=(-11)^2\times3 = 121\times3 = 363$
For $x_3 = 28$: $(28 - 35)^2\times2=(-7)^2\times2=49\times2 = 98$
For $x_4 = 56$: $(56 - 35)^2\times1=(21)^2\times1 = 441$
For $x_5 = 63$: $(63 - 35)^2\times1=(28)^2\times1 = 784$
For $x_6 = 71$: $(71 - 35)^2\times1=(36)^2\times1 = 1296$

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

$\sigma^{2}=\frac{722+363 + 98+441+784+1296}{10}=\frac{3604}{10}=360.4$

Step5: Calculate the standard - deviation $\sigma$

$\sigma=\sqrt{360.4}\approx18.98$

Answer:

$18.98$