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 3.583 (type an integer or decimal rounded to three decimal places as needed.) without the extra data value, the standard deviation is 0.654 (type an integer or decimal rounded to three decimal places as needed.) without the extra data value, the variance is 0.428 (type an integer or decimal rounded to three decimal places as needed.) with the extra data value, the range is 5.860 (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. First, we need to find the mean of the data set with the extra value.

Step2: Calculate the mean

We have 120 data values in the original set and then add one more value (7.00). The sum of all 121 data values needs to be calculated. Let the original data values be $x_1,x_2,\cdots,x_{120}$ and the new value $x_{121}=7.00$. The mean $\bar{x}=\frac{\sum_{i = 1}^{120}x_{i}+7.00}{121}$. Using a statistical software or calculator (e.g., in Excel, we can use the AVERAGE function with all 121 data values), we find the mean.

Step3: Calculate the squared differences

For each data value $x_i$, calculate $(x_{i}-\bar{x})^{2}$. Then sum these squared differences: $\sum_{i = 1}^{121}(x_{i}-\bar{x})^{2}$.

Step4: Calculate the standard deviation

Using the formula $s=\sqrt{\frac{\sum_{i = 1}^{121}(x_{i}-\bar{x})^{2}}{121 - 1}}=\sqrt{\frac{\sum_{i = 1}^{121}(x_{i}-\bar{x})^{2}}{120}}$. Using a statistical software or calculator (in Excel, we can use the STDEV.S function with all 121 data values), we get the standard deviation. After performing these operations with the 121 - data set (120 original + 7.00), the standard deviation is 0.873 (rounded to three decimal places).

Answer:

0.873