variance


 * __Variance__**

The variance is the sum of the probabilities that various outcomes will occur multiplied by the squared deviations from the average of the random variable. When trying to determine the risk associated with a given set of options, the variance is a very useful tool (Baye, 436).

Calculating the variance begins with finding the mean. Once the mean is known, the variance is calculated by finding the average squared deviation of each number in the sample from the mean. David M. Lane demonstrates a simple example using a few numbers. For the numbers 1, 2, 3, 4, and 5, the mean is 5:

The calculation for finding the mean follows: 1 + 2 + 3 + 4 + 5/5 = 15/5 = 3

Once the mean is known, the variance can be calculated. The variance is for this simple set of numbers is calculated as:

σ2 = __(1-3)² + (2-3)² + (3-3)² + (4-3)² + (5-3)²__ 5

σ2 = __(-2)² + (-1)² + (0)² + (1)² + (2)²__ 5

σ2 = __4 + 1 + 0 + 1 + 4__ = 2 5

Baye, Micheal R. __Managerial Economics and Business Strategy__. 2006, pp. 436-437 Lane, David M. http://davidmlane.com/hyperstat/A16252.html
 * __References__**


 * __Sample Test Questions__**

Steve is an economics student who has the opportunity to buy a single share of one of three different stocks. The stocks are in three different sectors. An energy stock is priced at $5.00, a telecommunications stock is priced at $10.00, and a newspaper publisher's stock is priced at $15.00. Steve tracks the stocks for a month and finds that the price of each has gone up by $2.00 over the course of the month. Before the month long tracking period began, Steve was equally likely to buy each stock, so the likelihood of each individual outcome is 1/3. What is the variance for each of the stock prices over the month? a. 15 b. 23 c. 80.66 d. 50

The answer is c, 80.66. The variance is calculated by finding the sum of probabilities that one of the possible outcomes will occur multiplied by the squared deviations from the mean of the random variable. In this case, the mean is the average price increase for each of the three stocks, which is $2 because each stock's price went up by $2.00. In calculating the variance, we'll determine the weight of the uniform ($2.00) change in value of each of the three stocks:

σ² = 1/3(5-2)² + 1/3(10-2)² + 1/3(15-2)² = 1/3(3)² + 1/3(8)² + 1/3(13)² = 1/3(9) + 1/3(64) + 1/3(169) = 3 + 21.33 + 56.33 = 80.66

We now know the variance. Based on this, which stock should Steve choose to buy?

a. The energy stock for $5 b. The telecommunications stock for $10 c. The publisher's stock for $15 The correct answer is a, the energy stock at an initial price of $5. To see this, we can use the variance that we calculated earlier to try to determine which stock will most likely vary the most. From here, we can calculate the standard deviation simply by finding the square root of the variance. The standard deviation is:

√80.66 = 8.98

Each of the three stock prices will vary from our expected value (mean) by 8.98. We were fortunate in this example to experience a rise no matter which stock we chose, but knowing the standard deviation helps us by telling us that the $2 rise in price for any of the three stocks means that the energy stock experienced the largest gain relative to the initial price. It began at $5 and rose to $7.