Question

ruben went on 5 hikes. the hikes were: 1 mile 2 miles 8 miles 4 miles 1 mile what was the standard deviation of the lengths of rubens hikes? if the answer is a decimal, round it to the nearest tenth. standard deviation (σ): miles

Explanation:

Step1: Calculate the mean

The mean formula is $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$. Here, $n = 5$, $x_{1}=1$, $x_{2}=2$, $x_{3}=8$, $x_{4}=4$, $x_{5}=1$.
$$\bar{x}=\frac{1 + 2+8 + 4+1}{5}=\frac{16}{5}=3.2$$

Step2: Calculate the squared - differences from the mean

For $x_{1}=1$: $(1 - 3.2)^{2}=(-2.2)^{2}=4.84$
For $x_{2}=2$: $(2 - 3.2)^{2}=(-1.2)^{2}=1.44$
For $x_{3}=8$: $(8 - 3.2)^{2}=(4.8)^{2}=23.04$
For $x_{4}=4$: $(4 - 3.2)^{2}=(0.8)^{2}=0.64$
For $x_{5}=1$: $(1 - 3.2)^{2}=(-2.2)^{2}=4.84$

Step3: Calculate the variance

The variance formula for a population is $\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}$.
$$\sigma^{2}=\frac{4.84+1.44 + 23.04+0.64+4.84}{5}=\frac{34.8}{5}=6.96$$

Step4: Calculate the standard deviation

The standard deviation formula is $\sigma=\sqrt{\sigma^{2}}$.
$$\sigma=\sqrt{6.96}\approx2.6$$

Answer:

$2.6$