standard+deviation


 * __Standard Deviation__**

The square root of the variance is called the standard deviation. The standard deviation is a measure of how much each outcome varies from the mean or expected value.

Once the variance of a sample of numbers has been calculated, finding the standard deviation is very simple. In a sample of the five numbers from 1 through 5, the variance is 2. The standard deviation for this sample of numbers (1, 2, 3, 4, and 5) is simply √2, which is equal to 1.4142.


 * __References__**

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


 * __Sample Test Questions__**

John, a basketball player for a team called the Cougars, is averaging 10 points per game. If the Cougars played three games last week and John scored 19, 12, and 7 points in the three games, what was his standard deviation?

a. 5.85 b. 8 c. 7 d. 2

The correct answer is a, 5.85. To solve this problem, we first need to calculate the variance. The variance is the sum of the probabilities that each of John's point totals during the week (19, 12, and 7) will occur times the squared deviations from the mean, which is John's scoring average of 10.

σ2 = 1/3(19 - 10)² + 1/3(12 - 10)² + 1/3(7 -10)² = 1/3(9)² + 1/3(2)² + 1/3(-3)² = 81/3 + 4/3 + 9/3 =34.33 is the variance.

The standard deviation is the square root of the variance, which is √34.33 = 5.85.

A employee that works in an office building is constantly running late to work and would like to calculate the chances of getting a speeding ticket. The employee has the choice between taking a road where the speed limit is 25 mph and driving to work at 35 mph. Or the employee could take a road to work that is somewhat longer but has a higher speed limit at 40 mph. The employee would travel at about 55 mph if he takes the longer road and knows that each possible road leads to a commute of 30 minutes. What is the employee's likelihood of getting a speeding ticket for speeding on one of these two roads?

a. The standard deviation is 9 b. The standard deviation is 12.74 c. The standard deviation is 5 d. The standard deviation is 16

The correct answer is b. Since the standard deviation in this case is 12.74, the employee should take the shorter road and drive only 10 miles per hour over the speed limit because he knows that each route takes about half an hour and would like to minimize the chances of getting a speeding ticket. This is calculated by first finding the variance:

σ² **=** ½(35 - 25)² + ½(55 - 40)² σ² = 0.50(10)² + 0.50(15)² σ² = 0.50(100) + 0.50(225) σ² = 50 + 112.50 σ² = 162.50 Once the variance is calculated, we can take the square root to calculate the standard deviation. The standard deviation in this example is:

√162.50 = 12.74