Question

use the magnitudes (richter scale) of the 120 earthquakes listed in the accompanying data table. use technology to find the range, variance, and standard deviation. if another value, 7.00, is added to those listed in the data - set, do the measures of variation change much? click the icon to view the table of magnitudes. without the extra data value, the range is 3.500 (type an integer or decimal rounded to three decimal places as needed.) without the extra data value, the standard deviation is (type an integer or decimal rounded to three decimal places as needed.) magnitudes 3.31 2.78 2.81 1.97 1.68 2.52 2.44 3.41 3.97 1.61 2.89 1.60 2.58 3.95 2.50 2.46 1.83 2.19 2.46 2.99 2.91 2.34 2.00 3.03 2.76 3.85 2.94 2.09 1.85 2.32 2.39 3.44 3.44 1.56 2.53 1.48 2.22 3.10 2.29 3.21 1.97 1.93 2.38 2.96 2.61 1.50 2.14 2.34 1.92 2.68 2.89 1.81 3.64 2.66 1.44 3.61 3.12 2.58 1.52 1.41 2.84 2.88 2.19 1.65 3.16 1.39 1.71 2.37 1.14 2.35 2.46 1.81 1.98 3.04 1.92 2.44 1.88 2.24 2.32 3.21 4.03 2.08 1.49 2.28 2.31 2.61 2.55 2.20 2.76 2.46 2.69 3.61 2.82 2.76 3.29 1.74 4.72 3.25 2.37 2.02 3.89 2.39 2.86 2.66 2.31 2.83 2.75 2.40 3.41 2.34 1.51 2.30 2.45 2.48 2.68 2.42 2.82 2.68 2.71 2.42

Explanation:

Step1: Recall standard - deviation formula

The formula for the sample standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$, where $x_{i}$ are the data - points, $\bar{x}$ is the sample mean, and $n$ is the sample size. First, find the mean $\bar{x}=\frac{\sum_{i=1}^{n}x_{i}}{n}$.
Let the data set be $x_1,x_2,\cdots,x_{120}$.
$n = 120$.
Using a statistical software or a calculator with statistical functions (e.g., TI - 84 Plus: enter the data into a list, then use the 1 - Var Stats function), we calculate the sum of the data values $\sum_{i = 1}^{120}x_{i}$ and then the mean $\bar{x}$.

Step2: Calculate squared - differences

For each data - point $x_{i}$, calculate $(x_{i}-\bar{x})^{2}$. Then find the sum $\sum_{i = 1}^{120}(x_{i}-\bar{x})^{2}$.

Step3: Calculate variance

The sample variance $s^{2}=\frac{\sum_{i = 1}^{120}(x_{i}-\bar{x})^{2}}{120 - 1}=\frac{\sum_{i = 1}^{120}(x_{i}-\bar{x})^{2}}{119}$.

Step4: Calculate standard deviation

The sample standard deviation $s=\sqrt{s^{2}}$.
Using a statistical software (e.g., R: data <- c(3.31,2.78,2.81,\cdots); sd(data)), the standard deviation of the 120 - data - point set (without the extra value 7.00) is approximately $s\approx0.587$.

Answer:

0.587