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 5.560 (type an integer or decimal rounded to three decimal places as needed.) without the extra data value, the standard deviation is 0.868 (type an integer or decimal rounded to three decimal places as needed.) without the extra data value, the variance is 0.754 (type an integer or decimal rounded to three decimal places as needed.) with the extra data value, the range is 6.360 (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 formula for standard - deviation

The formula for the sample standard deviation 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 number of data - points. First, we need to find the mean of the data set with the extra value (7.50).

Step2: Calculate the mean

Add all the data values (including 7.50) and divide by the total number of data values. Let the data set be $x_1,x_2,\cdots,x_n$. The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$. After adding 7.50 to the original data set, calculate the sum of all values and divide by the new number of data points.

Step3: Calculate the squared differences

For each data value $x_i$ in the new data set, calculate $(x_{i}-\bar{x})^{2}$.

Step4: Calculate the variance

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

Step5: Calculate the standard deviation

The standard deviation $s=\sqrt{s^{2}}$. Using a statistical software or a calculator with statistical functions (e.g., TI - 84 Plus: enter the data into a list, use the 1 - Var Stats function), we can find the standard deviation of the new data set.

Answer:

(You need to use a statistical software or calculator to input the full data set including 7.50 to get the exact value. Without actually performing the calculation with the full data set, we can't provide a numerical answer. But the steps above show how to calculate it.)