Let us consider a simple example of a small discrete population consisting of the first ten integers $$\{1,2,3,4,5,6,7,8,9,10\}$$. It is fairly simple to calculate the mean and the standard deviation of the given example. We do that in R to refresh our knowledge.

population <- c(1,2,3,4,5,6,7,8,9,10)
mean(population)
##  5.5
sd(population)
##  3.02765

The population mean, denoted by $$\mu$$ and the population standard deviation, denoted by $$\sigma$$ is 5.5 and approximately 3.028, respectively. It is important to realize, that these parameters, the population parameters will not change! They are fixed.

Let us now take one random sample without replacement of size $$n = 3$$ from this population. Once again we apply R to do all the work, by calling the sample function. Recall its form: sample(x, size, replace = FALSE, prob = NULL).

my.sample <- sample(population, size = 3)
my.sample 
##  8 3 1

Now we calculate the mean and the standard deviation of the given sample. However, this time, as we refer to a particular sample, we call the statistical parameter sample statistic or if we relate to the distribution of values (elements) sample distribution. To make this more explicit, the sample mean is denominated as $$\bar x$$ and the sample standard deviation as $$s$$.

x.bar <- mean(my.sample)
x.bar
##  4
s <- sd(my.sample)
s
##  3.605551

The sample mean, $$\bar x$$, and the sample standard deviation, $$s$$, is approximately 4 and 3.606, respectively. Please note, that depending on the actual elements in the sample, the sample statistics will change from sample to sample.