As we discussed, we assume that our data is a sample from some random variable or a system of random variables. So it is important to discuss the relation between properties of samples like sample average or sample variance and the corresponding properties of random variables. Let us assume that we have a sample of numeric values like x_1, x_2, and so on, x_n. These values are sampled from some random variable X capital. I will refer to the collection of this x's as x small. We can consider the relation between properties of X capital which is a random variable and properties of the sample. We begin with sample average or sample mean. It is related to the expected value of X capital. As we discussed in the probability, the expected value is in a sense an average value of a random variable. Due to law of large numbers, when we increase the size of sample, whether we increase n, sample average tends to the expected value of the corresponding random variable. A similar relation can be established between a simple variance and variance of the random variable, X capital. Both sample variance and unbiased sample variance tends to the variance of the random variable X. Both sample variance and unbiased sample variance can be used to estimate variance of the corresponding random variable X capital. Because as n tends to infinity, so the size of the sample becomes very large. Both of them tend to the corresponding variance of X capital. But this unbiased variance is a little bit better in some statistical terms than biased variance. Then, we can discuss in the same way the standard deviation of sample. Standard deviation is just a square root of the corresponding variance, usually, unbiased sample variance. So the standard deviation of X tends to the standard deviation of X capital which is in turn just a square root of variance of x capital. This also usually denoted by sigma with index X capital. By the way, this variable, X capital, is also usually called a population. So these properties are called properties of population or population statistics. These properties are called sample statistics. Now, let us discuss median. There is a population median which is related to the sample median. For a random variable x, if this random variable is a continuous random variable, we can define median in the following way. Median X is such number. Probability that X is greater than m are equals to probability that X is less than m, and both are equal to one-half. So if we have some distribution that is given by probability density function, something like this one. Then the median is a value that divides the area under this curve, just my two-halves, this is one-half and this is another half. So this sample median corresponds to the corresponding population median which is defined in this way. By the way, in the same way, we can define not only median but any value that divide this area under the curve in a certain proportion. These values are called quantiles. Previously, we discussed quartiles, which are just an example of quantiles. Let me define what is a quantile of a random variable. P-quantile of X where p is some number from zero to one is such a value, x_p, which satisfies the following relation. Probability that X is less than x_p equals to p. So for example, if I have some distribution, and I'm interested in all 0.1 quantile, I have to put a point somewhere here such that the area under this part of the curve is 0.1, and the area under the rest of the curve is 0.9. We can define quantiles for sample as well. For sample, p-quantile is a number x_p such that the following holds. Let us consider the part of our sample which is less than x_p. So this is x_i, all x_i that are less than x_p. I want that this number, the number of elements in the set be approximately equal to p times n, the size of our sample. So it means that, for example, if I'm interested in 0.1 quantile, I have to just sort my sample, then divide it into 10 equal parts. The border of the first part is just x_0.1. So this is 0.1 quantile. Quantiles sometimes can be useful to describe, for example, some of the largest values that belongs to our sample or some of the lowest values. So we see that we have properties of samples and corresponding properties of population, properties of the random variable that are used to generate this sample. In statistics, we use properties of sample to estimate the corresponding properties of population. This is our main goal. We will use statistical tools not only to say that this can be done but also to quantify possible error of this estimation.