Answer :
- Calculate the sample mean: $\bar{x} = 97.96$.
- Calculate the sum of squared differences from the mean.
- Calculate the sample variance: $s^2 = \frac{\sum (x_i - \bar{x})^2}{19}$.
- Calculate the sample standard deviation: $s = \sqrt{s^2} = \boxed{1.05}$.
### Explanation
1. Understand the problem and provided data
We are given a data set of 20 body temperatures of adult males and asked to calculate the sample standard deviation, rounding the result to two decimal places. The data points are: 98.0, 98.4, 97.4, 97.0, 96.5, 99.4, 99.4, 99.3, 98.2, 97.9, 96.8, 97.2, 99.4, 97.3, 99.4, 97.2, 99.4, 98.8, 98.5, 96.7.
2. Calculate the sample mean
First, we need to calculate the sample mean. The formula for the sample mean is $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$, where $n$ is the number of data points and $x_i$ are the individual data values. In our case, $n = 20$.
3. Calculate the sum and the mean
Next, we calculate the sum of the data points:
$\sum_{i=1}^{20} x_i = 98.0 + 98.4 + 97.4 + 97.0 + 96.5 + 99.4 + 99.4 + 99.3 + 98.2 + 97.9 + 96.8 + 97.2 + 99.4 + 97.3 + 99.4 + 97.2 + 99.4 + 98.8 + 98.5 + 96.7 = 1959.2$
Then, we divide the sum by the number of data points to get the mean:
$\bar{x} = \frac{1959.2}{20} = 97.96$
4. Calculate squared differences from the mean
Now, we calculate the squared differences from the mean for each data point, $(x_i - \bar{x})^2$, and sum them up. This is a bit tedious to do by hand, but it's a crucial step in finding the standard deviation.
5. Calculate the sample variance
Then, we calculate the sample variance, $s^2$, using the formula $s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}$. This involves dividing the sum of the squared differences by $n-1$, which is $20-1 = 19$ in our case.
6. Calculate the sample standard deviation
Finally, we calculate the sample standard deviation, $s$, by taking the square root of the sample variance: $s = \sqrt{s^2}$. After performing the calculations (either manually or using a calculator), we find that the sample standard deviation is approximately 1.05.
7. State the final answer
Therefore, the sample standard deviation for the given data set, rounded to two decimal places, is $\boxed{1.05}$.
### Examples
Understanding standard deviation is incredibly useful in many real-world scenarios. For instance, in healthcare, knowing the standard deviation of patients' body temperatures helps doctors identify unusual variations that might indicate a health problem. Similarly, in manufacturing, it helps ensure product consistency by measuring the variation in product dimensions or quality. In finance, standard deviation is used to measure the volatility of investments, helping investors assess risk.
- Calculate the sum of squared differences from the mean.
- Calculate the sample variance: $s^2 = \frac{\sum (x_i - \bar{x})^2}{19}$.
- Calculate the sample standard deviation: $s = \sqrt{s^2} = \boxed{1.05}$.
### Explanation
1. Understand the problem and provided data
We are given a data set of 20 body temperatures of adult males and asked to calculate the sample standard deviation, rounding the result to two decimal places. The data points are: 98.0, 98.4, 97.4, 97.0, 96.5, 99.4, 99.4, 99.3, 98.2, 97.9, 96.8, 97.2, 99.4, 97.3, 99.4, 97.2, 99.4, 98.8, 98.5, 96.7.
2. Calculate the sample mean
First, we need to calculate the sample mean. The formula for the sample mean is $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$, where $n$ is the number of data points and $x_i$ are the individual data values. In our case, $n = 20$.
3. Calculate the sum and the mean
Next, we calculate the sum of the data points:
$\sum_{i=1}^{20} x_i = 98.0 + 98.4 + 97.4 + 97.0 + 96.5 + 99.4 + 99.4 + 99.3 + 98.2 + 97.9 + 96.8 + 97.2 + 99.4 + 97.3 + 99.4 + 97.2 + 99.4 + 98.8 + 98.5 + 96.7 = 1959.2$
Then, we divide the sum by the number of data points to get the mean:
$\bar{x} = \frac{1959.2}{20} = 97.96$
4. Calculate squared differences from the mean
Now, we calculate the squared differences from the mean for each data point, $(x_i - \bar{x})^2$, and sum them up. This is a bit tedious to do by hand, but it's a crucial step in finding the standard deviation.
5. Calculate the sample variance
Then, we calculate the sample variance, $s^2$, using the formula $s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}$. This involves dividing the sum of the squared differences by $n-1$, which is $20-1 = 19$ in our case.
6. Calculate the sample standard deviation
Finally, we calculate the sample standard deviation, $s$, by taking the square root of the sample variance: $s = \sqrt{s^2}$. After performing the calculations (either manually or using a calculator), we find that the sample standard deviation is approximately 1.05.
7. State the final answer
Therefore, the sample standard deviation for the given data set, rounded to two decimal places, is $\boxed{1.05}$.
### Examples
Understanding standard deviation is incredibly useful in many real-world scenarios. For instance, in healthcare, knowing the standard deviation of patients' body temperatures helps doctors identify unusual variations that might indicate a health problem. Similarly, in manufacturing, it helps ensure product consistency by measuring the variation in product dimensions or quality. In finance, standard deviation is used to measure the volatility of investments, helping investors assess risk.
