Question

use the following distribution to complete parts (a) through (d) below.
1, 6, 8, 12, 13

a) compute the mean and standard deviation of the distribution.
the mean is 8.
the standard deviation is
(round to the nearest hundredth as needed.)

Explanation:

Step1: Calculate the variance

The formula for variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\), where \(n\) is the number of data points, \(x_{i}\) are the data values, and \(\bar{x}\) is the mean.
We have \(n = 5\), \(\bar{x}=8\), \(x_1 = 1\), \(x_2=6\), \(x_3 = 8\), \(x_4=12\), \(x_5 = 13\).
\((x_1-\bar{x})^{2}=(1 - 8)^{2}=(-7)^{2}=49\)
\((x_2-\bar{x})^{2}=(6 - 8)^{2}=(-2)^{2}=4\)
\((x_3-\bar{x})^{2}=(8 - 8)^{2}=0^{2}=0\)
\((x_4-\bar{x})^{2}=(12 - 8)^{2}=4^{2}=16\)
\((x_5-\bar{x})^{2}=(13 - 8)^{2}=5^{2}=25\)
\(\sum_{i = 1}^{5}(x_{i}-\bar{x})^{2}=49 + 4+0 + 16+25=94\)
\(s^{2}=\frac{94}{5 - 1}=\frac{94}{4}=23.5\)

Step2: Calculate the standard deviation

The standard deviation \(s=\sqrt{s^{2}}\)
Since \(s^{2}=23.5\), then \(s=\sqrt{23.5}\approx4.85\)

Answer:

\(4.85\)