Question

8 16 17 6 14

1. calculate the mean.

μ =

1. fill in the table below: fill in the differences of each data value from the mean, then the squared differences.
2. calculate the population standard deviation (σ).

σ = √(σ(x - μ)² / n) = (please round your answer to two decimal places)

Explanation:

Step1: Calculate the mean

The formula for the mean $\mu$ of a set of data $x_1,x_2,\cdots,x_n$ is $\mu=\frac{\sum_{i = 1}^{n}x_i}{n}$. Here $n = 5$, $x_1=8,x_2 = 16,x_3=17,x_4=6,x_5=14$. So $\mu=\frac{8 + 16+17+6+14}{5}=\frac{61}{5}=12.2$.

Step2: Calculate the differences and squared - differences

For $x = 8$: $x-\mu=8 - 12.2=-4.2$, $(x - \mu)^2=(-4.2)^2 = 17.64$.
For $x = 16$: $x-\mu=16 - 12.2 = 3.8$, $(x - \mu)^2=(3.8)^2=14.44$.
For $x = 17$: $x-\mu=17 - 12.2 = 4.8$, $(x - \mu)^2=(4.8)^2 = 23.04$.
For $x = 6$: $x-\mu=6 - 12.2=-6.2$, $(x - \mu)^2=(-6.2)^2 = 38.44$.
For $x = 14$: $x-\mu=14 - 12.2 = 1.8$, $(x - \mu)^2=(1.8)^2=3.24$.
And $\sum(x - \mu)^2=17.64 + 14.44+23.04+38.44+3.24 = 96.8$.

Step3: Calculate the population standard deviation

The formula for the population standard deviation $\sigma=\sqrt{\frac{\sum(x - \mu)^2}{N}}$, where $N = 5$ and $\sum(x - \mu)^2=96.8$. So $\sigma=\sqrt{\frac{96.8}{5}}=\sqrt{19.36}=4.40$.

Answer:

1. $\mu = 12.2$

2.

$x$	$x-\mu$	$(x - \mu)^2$
16	3.8	14.44
17	4.8	23.04
6	-6.2	38.44
14	1.8	3.24
	$\sum(x - \mu)^2=96.8$

1. $\sigma = 4.40$