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How to Identify the Notation for the Mean and Variance of a Discrete Random Variable

Two of the most important terms in statistics are mean and variance, and so you need to be able to identify their notations when working with discrete random variables.

The mean of a random variable is the average of all the outcomes you would expect in the long term (over all possible samples). For example, if you roll a die a billion times and record the outcomes, the average of those outcomes is expected to be 3.5. (Each outcome happens with equal chance, so you average the numbers 1 through 6 to get 3.5.) However, if the die is loaded and you roll a 1 more often than anything else, the average outcome from a billion rolls is expected to be less than 3.5, as it will be pulled closer to 1.

The notation for the mean of a random variable X is

image0.png

(pronounced "mu sub x"; or just "mu x"). Because you are looking at all the outcomes in the long term, it's the same as looking at the mean of an entire population of values, which is why you denote it

image1.png

(The latter represents the mean of a sample of values.) You put the X in the subscript to remind you that the variable this mean belongs to is the X variable (as opposed to a Y variable or some other letter).

The variance of a random variable is roughly interpreted as the average squared distance from the mean for all the outcomes you would get in the long term, over all possible samples. This is the same as the variance of the population of all possible values. The notation for variance of a random variable X is

image2.png

You say "sigma sub x, squared" or just "sigma squared."

The standard deviation of a random variable X is the square root of the variance, denoted by

image3.png

(say "sigma x" or just "sigma"). It roughly represents the average distance the set of outcomes is from the mean.

Just like for the mean, you use the Greek notation to denote the variance and standard deviation of a random variable. The English notation s2 and s represent the variance and standard deviation of a sample of individuals, not the entire population.

The variance is in square units, so it can't be easily interpreted. You use standard deviation for interpretation because it is in the original units of X. The standard deviation can be roughly interpreted as the average distance the set of outcomes is from the mean.

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