Question

jaden went on 6 hikes. the hikes were:
7 miles 1 mile 4 miles 5 miles 8 miles 8 miles
what was the standard deviation of the lengths of jadens hikes?
if the answer is a decimal, round it to the nearest tenth.
standard deviation (σ): miles

Explanation:

Step1: Calculate the mean

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\). Here \(n = 6\), \(x_{1}=7\), \(x_{2}=1\), \(x_{3}=4\), \(x_{4}=5\), \(x_{5}=8\), \(x_{6}=8\).
\(\bar{x}=\frac{7 + 1+4 + 5+8 + 8}{6}=\frac{33}{6}=5.5\)

Step2: Calculate the squared - differences from the mean

\((x_{1}-\bar{x})^{2}=(7 - 5.5)^{2}=2.25\)
\((x_{2}-\bar{x})^{2}=(1 - 5.5)^{2}=20.25\)
\((x_{3}-\bar{x})^{2}=(4 - 5.5)^{2}=2.25\)
\((x_{4}-\bar{x})^{2}=(5 - 5.5)^{2}=0.25\)
\((x_{5}-\bar{x})^{2}=(8 - 5.5)^{2}=6.25\)
\((x_{6}-\bar{x})^{2}=(8 - 5.5)^{2}=6.25\)

Step3: Calculate the variance

The formula for the variance \(\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}\)
\(\sum_{i = 1}^{6}(x_{i}-\bar{x})^{2}=2.25+20.25 + 2.25+0.25+6.25+6.25=37.5\)
\(\sigma^{2}=\frac{37.5}{6}=6.25\)

Step4: Calculate the standard deviation

The formula for the standard deviation \(\sigma=\sqrt{\sigma^{2}}\)
\(\sigma=\sqrt{6.25}=2.5\)

Answer:

\(2.5\)