# Mean, variance and standard deviation of discrete random variable

This online calculator calculates the mean, variance, and standard deviation of random variables entered in the form of a value-probability table.

This calculator can help you to calculate basic discrete random variable metrics: mean or expected value, variance, and standard deviation.

Mean or expected value of discrete random variable is defined as
$E(X)=\mu_X=\sum_{x \in R_x} x f(x)$

Variance of random variable is defined as
$Var(X)=\sigma_{X}^2=E([X-\mu_X]^2)$

An alternative way to compute the variance is
$Var(X)=\sigma_{X}^2=E(X^2)-\mu_{X}^2$

The positive square root of the variance is called the standard deviation $\sigma_{X}$.

As you can see, these metrics have quite simple formulas. Sometimes you need to use them to solve probability theory problems. For discrete random variables, the trick is to find correct value-probability pairs; then, it is just simple math of additions and multiplications. So, this calculator can take care of simple math for you once you enter value-probability pairs into the table. You can find an example of usage below the calculator.

#### Probability table

ValueProbability
Items per page:

Digits after the decimal point: 4
Mean

Variance

Standard deviation

### Example

Problem: A set of 10 microwave ovens includes 3 that are defective. If 5 of the ovens are chosen randomly for shipment to a hotel, how many defective ovens can they expect?

How to use the calculator:

1. Select the current data in the table (if any) by clicking on the top checkbox and delete it by clicking on the "bin" icon on the table header.
2. Add value-probability pairs (you need to determine them, but it is the essence of the problem). Note that the quickest way to do it is to "import" data. Click on the "import" icon on the table header and enter the following values
0;0.0833.
1;0.4167.
2;0.4167.
3;0.0833.

After that, you will get a mean of 1.5. Of course, 1.5 defective ovens do not make any physical sense. Instead, it should be interpreted as an average value if repeated shipments will be made under these conditions.

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PLANETCALC, Mean, variance and standard deviation of discrete random variable