Question

use software or a calculator to find the range, variance, and standard deviation of the following body temperatures, in degrees fahrenheit, taken at 12:00 a.m
click the icon to view the table of body temperatures
the range of the data set is 2.80 °f
(round to two decimal places as needed.)
the standard deviation, s, of the data set is
(round to two decimal places as needed.)

Explanation:

Step1: Recall the formula for standard - deviation

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, find the mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$.

Step2: Calculate the mean

Sum all the data - points in the table. There are $n = 60$ data - points. Let the data - points be $x_1,x_2,\cdots,x_{60}$. $\sum_{i = 1}^{60}x_{i}=98.2 + 98.2+97.7+\cdots+97.3$. Using a calculator or software, $\sum_{i = 1}^{60}x_{i}=5907.7$. Then $\bar{x}=\frac{5907.7}{60}\approx98.46$.

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

For example, if $x_1 = 98.2$, then $(x_1-\bar{x})^{2}=(98.2 - 98.46)^{2}=(-0.26)^{2}=0.0676$. Do this for all 60 data - points and sum them up: $\sum_{i = 1}^{60}(x_{i}-\bar{x})^{2}$. Using a calculator or software, $\sum_{i = 1}^{60}(x_{i}-\bar{x})^{2}=27.979$.

Step4: Calculate the variance

The sample variance $s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}=\frac{27.979}{60 - 1}\approx0.474$.

Step5: Calculate the standard deviation

The sample standard deviation $s=\sqrt{s^{2}}=\sqrt{0.474}\approx0.69$.

Answer:

$0.69$