# Developing a Probability Distribution from Empirical Data

In real-world situations, statisticians obtain data by means of observation and experimental methods. Data obtained in this manner is called empirical data.

A probability distribution obtained by means of observation and experimental methods is referred to as an empirical probability distribution , or a relative frequency distribution based on observation.

Example:

Let $X$ be the number of movies a high school student watches in a given month.

A survey conducted at one particular high school in the month of December gives by the following table:

 Number of Movies Watched $1$ $2$ $3$ $4$ $5$ $%$ of students $17$ $28$ $34$ $15$ $6$

If we assume that the students at this high school are typical, and that December is a typical month, then

The probability that a high school student will watch $1$ movie per month is $17%$ ;

The probability that a high school student will watch $2$ movies per month is $28%$ ;

etc.

Use this empirical probability distribution to find the expected value for the number of movies a high school student will watch in a month.

Convert the percentages to decimals.

$17%=0.17$

Use the weighted average formula.

$E\left(x\right)=\sum {x}_{i}\cdot P\left({x}_{i}\right)$

$\begin{array}{l}=\left(1\right)\cdot \left(0.17\right)+\left(2\right)\cdot \left(0.28\right)+\left(3\right)\cdot \left(0.34\right)+\left(4\right)\cdot \left(0.15\right)+\left(5\right)\cdot \left(0.06\right)\\ =0.17+0.56+1.02+0.90+0.30\\ =2.95\end{array}$

So, we can expect the average high school student to watch $2.95$ movies per month.