Let's now spend some time clarifying the distinction between a population mean and a sample mean, and between a population variance and a sample variance.
Situation
Suppose we are interested in determining \(\mu\), the mean number of hours slept nightly by American college students. Because the population of American college students is so large, we can't possibly record the number of hours slept by each American college student.
Let's take a look!
Now, all we need to do is define the sample mean and sample variance!
 Sample Mean

The sample mean, denoted \(\bar{x}\) and read “xbar,” is simply the average of the \(n\) data points \(x_1, x_2, \ldots, x_n\):
\(\bar{x}=\dfrac{x_1+x_2+\cdots+x_n}{n}=\dfrac{1}{n} \sum\limits_{i=1}^n x_i\)
The sample mean summarizes the "location" or "center" of the data.
Example 818 Section
A random sample of 10 American college students reported sleeping 7, 6, 8, 4, 2, 7, 6, 7, 6, 5 hours, respectively. What is the sample mean?
Solution
The sample mean is:
\(\bar{x}=\dfrac{7+6+8+4+2+7+6+7+6+5}{10}=5.8\)
 Sample Variance

The sample variance, denoted \(s^2\) and read "ssquared," summarizes the "spread" or "variation" of the data:
\(s^2=\dfrac{(x_1\bar{x})^2+(x_2\bar{x})^2+\cdots+(x_n\bar{x})^2}{n1}=\dfrac{1}{n1}\sum\limits_{i=1}^n (x_i\bar{x})^2\)
 Sample Standard Deviation

The sample standard deviation, denoted \(s\) is simply the positive square root of the sample variance. That is:
\(s=\sqrt{s^2}\)
Example 819 Section
A random sample of 10 American college students reported sleeping 7, 6, 8, 4, 2, 7, 6, 7, 6, 5 hours, respectively. What is the sample standard deviation?
Solution
The sample variance is:
\(s^2=\dfrac{1}{9}\left[(75.8)^2+(65.8)^2+\cdots+(55.8)^2\right]=\dfrac{1}{9}(27.6)=3.067\)
Therefore, the sample standard deviation is:
\(s=\sqrt{3.067}=1.75\)
An easier way to calculate the sample variance is:
\(s^2=\dfrac{1}{n1}\left[\sum\limits_{i=1}^n x^2_in{\bar{x}}^2\right]\)
Proof
Example 820 Section
A random sample of 10 American college students reported sleeping 7, 6, 8, 4, 2, 7, 6, 7, 6, 5 hours, respectively. What is the sample standard deviation?
Solution
The sample variance is:
\(s^2=\dfrac{1}{9}\left[(7^2+6^2+\cdots+6^2+5^2)10(5.8)^2\right]=3.067\)
Therefore, the sample standard deviation is:
\(s=\sqrt{3.067}=1.75\)
We will get a better feel for what the sample standard deviation tells us later on in our studies. For now, you can roughly think of it as the average distance of the data values \(x_1, x_2, \ldots, x_n\) from their sample mean.