[MUSIC] In lecture one, we described graphically, the shape of a distribution as symmetric or skewed by examining a histogram. If the center of the distribution is located at the median of the distribution and then divides the graph of the distribution into two mirror images, then the distribution is said to be symmetric. Apart from a visual point of view that is by using graphs, we can also describe the shape of a distribution numerically by computing a measure of skewness. The skewness is positive if the distribution is skewed to the right. The skewness is negative for distributions skewed to the left. Finally, the skewness is equal to zero for distribution, such as the bell shaped distribution, which are symmetric around the mean. If we look at this skewed left distribution, we can spot that the mean is less than the median. If the distribution is skewed right, the mean is usually greater than the median. In a symmetric distribution, the mean and the median are equal. Notice that this relationship between the mean and the median is usually true for continuous numerical variables. It may not be true for discrete numerical variables or for some continuous numerical variables. An example of a typically skewed distribution refers to the incomes. In fact, the distribution of incomes is often right skewed. This is due to relativity small portion of ith values, which compose the distribution. The median is the preferred measure to describe the distribution of incomes in a city, or in a state, or country. On the one hand, we have that large portion of the population has relatively modest incomes. On the other hand, a small proportion of the population have a very high income. In these cases the value of the mean is inflated by the very wealthy portion of the population. However, this is a too optimistic picture of the economic conditions. In this case, the mean of such distributions is well higher than the median. This is why in this case, the median is the perfect measure. This does not mean that we always have to prefer the median to the mean when a population or a sample is skewed. Our reasoning gives us just the insight as to why the choice of the numerical measure is context specific based. There are examples for which the mean should be the preferred measure even if the distribution is skewed. The distributions of claim sizes in an insurance company, it is likely to be right skewed. Now, if the company is interested in knowing the most typical claim size we should perfect the medium as a measure. But if the company is more interested in knowing how much money is needed to cover the claims, then the mean should be preferred. Notice that in spite of the advantage of the median in discounting extreme observations, still, the mean tends to be the preferred measure. As we will see in lecture three the mean has certain properties which make it more attractive than the median. The main reason is that the theoretical development of differential procedures, based on the mean and the measures related to it is small step forward than the procedures based on the median. This makes the mean more efficient than the median. [MUSIC]