Question

the bird watcher counted the number of birds in each flock that passed overhead.
number of birds number of flocks
14 1
27 1
35 2
58 3
65 1
66 2
x is the number of birds that a randomly chosen flock had. what is the standard deviation of x?
round your answer to the nearest hundredth.

Explanation:

Step1: Calculate the total number of flocks

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

Step2: Calculate the mean $\mu$

The sum of the products of the number of birds and the number of flocks is $14\times1+27\times1 + 35\times2+58\times3+65\times1+66\times2=14 + 27+70+174+65+132 = 482$.
$\mu=\frac{482}{10}=48.2$

Step3: Calculate the squared - differences from the mean for each data - point, weighted by the number of flocks

For $x = 14$: $(14 - 48.2)^2\times1=( - 34.2)^2\times1 = 1169.64$
For $x = 27$: $(27 - 48.2)^2\times1=( - 21.2)^2\times1 = 449.44$
For $x = 35$: $(35 - 48.2)^2\times2=( - 13.2)^2\times2=174.24\times2 = 348.48$
For $x = 58$: $(58 - 48.2)^2\times3=(9.8)^2\times3 = 96.04\times3=288.12$
For $x = 65$: $(65 - 48.2)^2\times1=(16.8)^2\times1 = 282.24$
For $x = 66$: $(66 - 48.2)^2\times2=(17.8)^2\times2 = 316.84\times2 = 633.68$
The sum of these weighted squared - differences is $1169.64+449.44 + 348.48+288.12+282.24+633.68=3171.6$

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

$\sigma^{2}=\frac{3171.6}{10}=317.16$

Step5: Calculate the standard deviation $\sigma$

$\sigma=\sqrt{317.16}\approx17.81$

Answer:

$17.81$