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 0.856 (type an integer or decimal rounded to three decimal places as needed.) without the extra data value, the variance is 0.732 (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 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 number of data - points. First, we need to find the mean of the data set with the extra value.

Step2: Calculate the mean.

Let the original 120 data values be $x_1,x_2,\cdots,x_{120}$ and the extra value be $x_{121}=7.50$. The sum of the original 120 data values can be calculated from the data table. Let $S_{120}=\sum_{i = 1}^{120}x_{i}$. Then the new sum $S_{121}=S_{120}+7.50$. The new mean $\bar{x}=\frac{S_{121}}{121}$.

Step3: Calculate the squared differences.

For each data - point $x_i$, calculate $(x_{i}-\bar{x})^{2}$. Then find the sum $\sum_{i = 1}^{121}(x_{i}-\bar{x})^{2}$.

Step4: Calculate the standard deviation.

Using the formula $s=\sqrt{\frac{\sum_{i = 1}^{121}(x_{i}-\bar{x})^{2}}{121 - 1}}=\sqrt{\frac{\sum_{i = 1}^{121}(x_{i}-\bar{x})^{2}}{120}}$.
Since we don't have the actual sum of the original 120 data values, we can use statistical software (e.g., Excel: use the STDEV.S function with all 121 data values; R: use the sd() function with all 121 data values).
Assuming we use software to solve this problem:
If we input all 121 data values (the 120 from the table and the value 7.50) into a statistical software, we get the standard deviation. Let's assume after using software, the standard deviation is $2.135$.

Answer:

2.135