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, 7.50, 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 3.590 (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 3.28 2.80 2.83 1.97 1.69 2.51 2.45 3.42 3.97 1.59 2.88 1.01 2.55 4.00 2.53 2.44 1.83 2.17 2.43 2.99 2.89 2.36 2.00 3.03 2.76 3.07 2.94 2.06 1.84 2.32 2.38 3.43 3.42 1.53 2.56 1.49 2.21 3.07 2.28 3.26 1.97 1.90 2.39 2.01 2.59 1.49 2.16 2.34 1.91 2.73 2.09 1.82 3.67 2.67 1.44 3.62 3.12 2.59 1.54 1.41 2.85 2.87 2.18 1.64 3.19 1.30 1.75 2.37 1.14 2.37 2.48 1.78 2.00 3.05 1.93 2.43 1.86 2.25 2.36 3.20 4.01 2.08 1.50 2.30 2.31 2.59 2.57 2.19 2.76 2.47 2.73 3.60 2.84 2.78 3.26 1.75 4.73 3.24 2.39 2.01 3.85 2.41 2.87 2.64 2.33 2.83 2.74 2.43 3.42 2.35 1.51 2.39 2.42 2.47 2.67 2.42 2.78 2.06 2.70 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.

Step2: Calculate the mean

First, find the sum of all 120 data - points in the table. Let the data - points be $x_1,x_2,\cdots,x_{120}$. The sum $S=\sum_{i = 1}^{120}x_{i}$. Then the mean $\bar{x}=\frac{S}{120}$.

Step3: Calculate the squared - differences

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

Step4: Calculate the standard deviation

Using the formula $s=\sqrt{\frac{\sum_{i = 1}^{120}(x_{i}-\bar{x})^{2}}{119}}$. 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 find that the standard deviation of the 120 data - points is approximately $0.457$.

Answer:

$0.457$