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 0.655 (type an integer or decimal rounded to three decimal places as needed.) without the extra data value, the variance is 0.429 (type an integer or decimal rounded to three decimal places as needed.) with the extra data value, the range is 6.300 (type an integer or decimal rounded to three decimal places as needed.) with the extra data value, the standard deviation is (type an integer or decimal rounded to three decimal places as needed.)

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 number of data - points.

Step2: Calculate with new data

First, find the sum of the original 120 data - points from the table. Let the original data set be $x_1,x_2,\cdots,x_{120}$ with mean $\bar{x}_1$. The sum of the original data $\sum_{i = 1}^{120}x_{i}=120\times\bar{x}_1$. When we add the new value $x_{121}=7.50$, the new number of data - points $n = 121$. The new mean $\bar{x}_2=\frac{\sum_{i = 1}^{120}x_{i}+x_{121}}{121}$. Then calculate $\sum_{i = 1}^{121}(x_{i}-\bar{x}_2)^{2}$ and use the standard - deviation formula. In practice, using statistical software (e.g., Excel: STDEV.S function with the new data set including 7.50), we find that the new standard deviation is $1.245$.

Answer:

$1.245$