5.2 Mean or expected value and standard deviation

The expected value is often referred to as the "long-term"average or mean . This means that over the long term of doing an experiment over and over, you would expect this average.

The mean of a random variable $X$ is $\mu$ . If we do an experiment many times (for instance, flip a fair coin, as Karl Pearson did, 24,000 times and let $X$ = the number of heads) and record the value of $X$ each time, the average is likely to get closer and closer to $\mu$ as we keep repeating the experiment. This is known as the Law of Large Numbers .

A step-by-step example

To do the problem, first let the random variable $X$ = the number of days the men's soccer team plays soccer per week. $X$ takes on the values 0, 1, 2. Construct a $\mathrm{PDF}$ table, adding a column $\mathrm{xP(x)}$ . In this column, you will multiply each $x$ value by its probability.

Add the last column to find the long term average or expected value: $\mathrm{(0)(0.2)+(1)(0.5)+(2)(0.3)=\; 0\; +\; 0.5\; +\; 0.6\; =\; 1.1}$ .

The expected value is 1.1. The men's soccer team would, on the average, expect to play soccer 1.1 days per week. The number 1.1 is the long term average or expected value if the men's soccer team plays soccer week after week after week. We say $\mathrm{\mu =1.1}$

Like data, probability distributions have standard deviations. To calculate the standard deviation ( $\sigma$ ) of a probability distribution, find each deviation from its expected value, square it, multiply it by its probability, add the products, and take the square root . To understand how to do the calculation, look at the table for thenumber of days per week a men's soccer team plays soccer. To find the standard deviation, add the entries in the column labeled ${(x-\mu )}^2·P\left(x\right)$ and take the square root.

Add the last column in the table. $0.242+0.005+0.243=0.490$ . The standard deviation is the square root of $0.49$ . $\sigma =\sqrt{0.49}=0.7$

Generally for probability distributions, we use a calculator or a computer to calculate $\mu$ and $\sigma$ to reduce roundoff error. For some probability distributions, there are short-cut formulas that calculate $\mu$ and $\sigma$ .