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 3.580 (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 2.10 2.46 2.99 2.91 2.34 2.00 3.03 2.76 3.85 2.04 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 1.03 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 2.68 2.10 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 1.80 2.24 2.32 3.21 4.03 2.08 1.49 2.28 2.31 2.61 2.55 2.20 2.70 2.46 2.89 3.61 2.82 2.76 3.20 1.74 4.72 3.25 2.37 2.02 3.89 2.39 2.80 2.60 2.31 2.03 2.75 2.40 3.41 2.34 1.51 2.39 2.45 2.48 2.68 2.42 2.82 2.68 2.71 2.42

Explanation:

Step1: Recall the formula for standard - deviation

The sample standard deviation formula is $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$, where $x_{i}$ are the data - points, $\bar{x}$ is the sample mean, and $n$ is the sample size. First, find the mean $\bar{x}=\frac{\sum_{i=1}^{n}x_{i}}{n}$. Here, $n = 120$.

Step2: Calculate the mean

Sum up all the 120 earthquake magnitudes in the data - set. Let the magnitudes be $x_1,x_2,\cdots,x_{120}$. $\bar{x}=\frac{\sum_{i = 1}^{120}x_{i}}{120}$. Using a calculator or statistical software (e.g., Excel: AVERAGE function, R: mean() function, Python: numpy.mean()), we find the sum of the 120 values and divide by 120.

Step3: Calculate $(x_{i}-\bar{x})^{2}$ for each data - point

For each of the 120 magnitudes $x_{i}$, calculate $(x_{i}-\bar{x})^{2}$. Then sum these values: $\sum_{i = 1}^{120}(x_{i}-\bar{x})^{2}$.

Step4: Calculate the standard deviation

Using the formula $s=\sqrt{\frac{\sum_{i = 1}^{120}(x_{i}-\bar{x})^{2}}{119}}$ (since $n-1=119$ for a sample of size $n = 120$), we get the standard deviation. Using statistical software (e.g., Excel: STDEV.S function, R: sd() function, Python: numpy.std()), the standard deviation of the 120 - data - point set is approximately $0.587$.

Answer:

$0.587$