1. What are standard deviation, mean, and variance for sample and populations and what Greek letters represent them? Give examples please.
The traditional symbol for the sample standard deviation is S (lowercase or uppercase; there is a slight difference between the two) and the equivalent Greek letter sigma is commonly used to denote the population standard deviation. The greek symbol for the population mean is mu; and the traditional symbol for the sample mean is Xbar (represented by an x with a line over it).
The sample variance is "S" squared and the population variance is sigma squared. These two are probably the ones you will work with the most.
Their mathematical definitions are as follows:
S^2 = Sum(i=1 to n)(X_i - Xbar)^2
n - 1
Where X_i denotes the ith value in our sample, the ith realization of the random variable X, and Xbar, written as a X with a bar above it, denotes the sample mean (again, not to be confused with the population mean).
sigma^2 = E((X - mu)^2)
Where E denotes the expected value function, which you may or may not have encountered already. Roughly, the expected value function tells you the value "on average" that we would expect the expression sent as input to the function to take.
For example, the expected value of the random variable X, E(X), would be the value we expect this variable to take on average, which is nothing more than the population ...
This solution explains standard deviations, means, and variances for samples and populations, and provides the Greek letter symbols and illustrative examples for each.