Sample moments

Summary

Definitions

• A population is a set of similar items such as all individuals in Sweden or all firms in Europe.
• A sample is set of data selected from a population by some defined procedure.
• A sample is a random sample if each item in the population has the same probability of be selected into the sample.
• Each element of the sample is called a sample point . The symbol $n$ is the common symbol for the size of the sample, that is, the number of sample points.
• $x_i$ will denote a particular measurement of sample point $i$ , for example the income of individual $i$ , where $i$ is any number from 1 to $n$ .
• We often refer to the $n$ numbers $x_1,x_2,…,x_n$ as the sample.

Sample mean, sample variance and sample standard deviation

• Review summation sign
• Review summation rules
• Given data $x_1,x_2,…,x_n$ we define
• The sample mean

$$\bar{x}= \frac{1}{n}\sum_{i=1}^{n}{ x_i }$$

• The sample variance

$$s_x^2= \frac{1}{n-1}\sum_{i=1}^{n}{ {\left( x_i-\bar{x} \right)}^2 }$$

• The sample standard deviation

$$s_x=\sqrt{s_x^2}$$

Sample covariance and sample correlation

• Given a population, we can make two distinct measurements from sample point $i$ and denote them by $x_i$ and $y_i$ . For example, $x_i$ could be the total revenue and $y_i$ the total cost for firm $i$ .
• Given data $x_1,x_2,…,x_n$ and $y_1,y_2,…,y_n$ we define
• The sample covariance

$$s_{x,y}^2= \frac{1}{n-1}\sum_{i=1}^{n}{ \left( x_i-\bar{x} \right)\left( y_i-\bar{y} \right) }$$

• The sample correlation

$$r_{x,y}^2= \frac{s_{x,y}^2}{s_xs_y}$$