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Mean, variance and standard deviation for discrete random variables

‘Mean’ is what we in daily talk often refer to as ‘average’. The variance and standard deviation express the spread in data. In this chapter we will calculate mean, variance and standard deviation for discrete variables.

 

Mean

The average of 2+3+4 is, obviously, 3. Let’s look at a calculation of mean given a certain probability distribution:

Say that a company, for a longer period, has been monitoring their sales and based hereon they can provide the following probability distribution for number of sales per week per salesperson:

Mean, variance and standard deviation for discrete random variables

 

Multiplying the probability P(X) with # of sales per week (X) we get the estimated number of sales per week for each given X. In this way, we can calculate the estimated sales per week and sum it up to get the mean of number of sales per week, which is 2.09:

Mean, variance and standard deviation for discrete random variables

 

As it shows in the table above a salesperson has a 9% chance of making 0 sales per week, 15% chance of doing 1 sale per week, 42% of doing 2; 26% of doing 3 and 8% of doing 4. The probability for each number of sales can be illustrated like this:

Mean, variance and standard deviation for discrete random variables

 

Variance

The variance is calculated by adding up the sum of the squared difference between the mean and each possible outcome multiplied by the probability for each possible outcome which is expressed by this formula:

Mean, variance and standard deviation for discrete random variables

For our sales example, this can be viewed in the spreadsheet like this:

Mean, variance and standard deviation for discrete random variables

 

Standard deviation

As the variance gives us the variance from the mean in a squared unit, we can be interested in knowing the difference from the mean measured in the actual unit that we wish to measure with:

Mean, variance and standard deviation for discrete random variables

 

Mean, variance and standard deviation for discrete random variables in Excel

Calculating mean, v Mean, variance and standard deviation for discrete random variables in Excel can be done applying the standard multiplication and sum functions that can be deduced from my Excel screenshot above (the spreadsheet).

 

This video can be helpful too: David Hays’ youtube video: Excel 2010: Mean, standard deviation and variance of a discrete random variable.

 

Learning statistics

The example above is inspired by this khanacademy video changing a bit on the story and figures.

Carsten Grube

Carsten Grube

Freelance Data Analyst

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