Tattoos are increasingly popular among football players. Imagine you want to know how much of their bodies football players cover with tattoos. The dot plots you see here represent the distributions of tattoo density expressed by the percentage of the body covered with tattoos in two football teams. You can see immediately that in the first team, the tattoo density is much less variable than in the second team. This variability can be measured by, for instance, the range or the interquartile range. It can graphically be represented by a box plot. You can see the relevant box plots here. In this video, we'll discuss two other measures of variability that are used very often in statistical studies, the variance and the standard deviation. The huge advantage of the variance and standard deviation over many other measures of variability is that they take into account ALL the values of the variable. Let's start with the variance. This is the formula of the variance. Let me show you how it works step by step. S squared stands for the variance. This part of the formula means that from every observation, x, you have to subtract the mean value of that variable, x bar. Next, you have to square all these values and add them up. The result is what we call the sum of squares. In the following step you divide the sum of squares by the size of your sample, n minus 1. Let's now apply the formula to our tattoo density example to see how it works in practice. These are the team 2 data and this is the formula. The first step is to compute the mean. I won't do that now because I assume that you already know how to do that. The mean of these values equals 15. The second step is to subtract the mean from every single observation. So let's take the first value, 0. We subtract the mean from this value. That gives 0 minus 15 is minus 15. We do that for all our values in the sample. When are finished doing that, we have completed this part of the formula. Notice that you now have negative and positive numbers. This is not strange, as the mean is the middle or the balance point of these values. In fact the negative deviations from the mean, counter balance the positive deviations from the mean. As a result of which the sum of variations equals 0. In other words, the sum of these values equals 0. For that reason, we don't use the original deviations but the squared deviations. That's the next step. We square all these computed values, so -15 squared is -15 times -15 is 225. We do that for all the other observations as well. According to the formula, we now have to add up all these values. After all, this is the sum up symbol. What we now have, is the sum of the squared deviations. Or, in other words, the sum of squares, which equals 639.74. We have to divide the sum of squares by n-1. The n in our case is 11, so n-1=10. 639.74 divided by 10= 63.97. That's our variance. The larger the variance, the larger the variability. That means the larger the variance, the more the values are spread out around the mean. The first team displayed here has a variance of about 6.33. You can see that the larger variability of tattoo density in Team 2 that was already visible from the dot plots and the box plots, is also represented by the larger variance. An important disadvantage of the variance, is that the metric of the variance is the metric of the variable under analysis, but squared. Afterall, we have squared the positive and negative deviations so that they don't cancel each other out. There's a very simple solution to get rid of this problem. We just take the square root of the variance. We call what we get, the standard deviation. The standard deviation can be seen as the average distance of an observation from the mean. The larger the standard deviation, the larger the variability of the data. Because this is the formula of the variance, this is the formula of the standard deviation. So, in our example the standard deviation of team one is the square root of 6.33. That equals 2.52. The standard deviation of team two is the square root of 63.97. That equals 8.0. The standard deviation is the measure of dispersion that is used most often. However, in many statistical methods, the variance plays an important role as well. In this video, you have learned that they are closely related and that you can easily derive the one from the other.