What makes the standard deviation helpful in the calculation


The Variance is one of the most important Dispersion parameters in the statistics. Find out how the Variance defined is what value it describes and what the Difference from the standard deviation is.

With our Video do you understand the topic without any problems - lean back and let ‘explain it to you! What are you waiting for?

You want to understand how exactly the Calculate variance leaves or what the Standard deviation is? Then take a look at our separate article! Also on the topic Empirical variance we have a contribution.

Variance explained simply

Finding the variance for the distribution of a random variable (population variance) is easier when you understand what it means. Let's first look at how it is defined.

The variance is the mean squared deviation of all values ​​in a random experiment from their expected value. The formula for the variance is:

You practically estimate how far the individual values ​​of the random experiment are from the expected value. Then you square the deviation. The whole thing is best illustrated graphically. In this context is also the Standard deviation important. It is the root of the variance. Another important distinction is whether it is the variance of a random variable (i.e. the corresponding probability distribution is known) or whether it is a sample (the probability distribution is unknown). In the latter case you have to use the Sample variance to calculate.

Variance example

Is that too abstract for you? Then imagine the following example:
You have two different games of chance: With the first you can either win or lose € 100 with the same probability, with the second you win exactly one euro or lose one euro. Although both games of chance have exactly the same expected value, namely 0, their variance is very different. This is because the possible results are differently far from the expected value.

Calculate variance

There is a simple procedure to calculate the variance: First you have to determine the expected value, then insert the individual values ​​in the formula and then calculate the variance. In our article Calculate variance let's go into more detail about the procedure and the formula of the variance.

In this example we can easily determine the variance: First we need the expected value. In both cases it is 0. You calculate this by calculating the individual values ​​times their probability of occurrence and adding them together. If you are unsure how to come up with it, have a look at our Video on the expected value at. Then we can insert the values ​​into the formula for the variance and get two different values ​​of the variance for our game of chance:

Bank note:


So you see: Although the expected value is the same, the variance can be very different. This is because the possible events, in the case of the bank note, are further away from the expected value than in the case of the coin.

Variance in statistics

The variance is a measure of statistics and stochastics, which the Spread of the data around the mean indicates. Since there is a difference in the formula, it can only be calculated for cardinally scaled data. You can also tell from the formula that the values ​​are squared, which makes it difficult to interpret. Therefore, the standard deviation is usually used to interpret the spread of the data. If you do not know the probability of occurrence for the events, we will use the sample variance. This weights the individual values ​​equally and forms a distorted or undistorted estimator of the variance. If you want to learn more about it, check out our article Sample variance at!

Difference variance standard deviation

So if we want to interpret the dispersion around the mean, it is not so easy to do with the variance. Instead, we can do the Standard deviation use. But what is the difference between these two values?

The standard deviation is the square root of the variance

You can see from the formula: The standard deviation is nothing other than the root of the variance. Based on our example, we get a standard deviation of € 100 and € 1 - this is how far the values ​​are on average from the mean.

In order to be able to compare individual random experiments with one another and to be able to interpret the values ​​better, it is therefore often helpful to calculate the standard deviation.