STATISTICS Higher Concepts of MathematicsThis spread, or distance, of each data point from the mean is called the variance. The varianceof each data point is calculated by:Variancexx_{i}wherex_{i}= each data point= meanxThe variance of each data point does not provide us with any useful information. But if themean of the variances is calculated, a very useful number is determined. The mean variance isthe average value of the variances of a set of data. The mean variance is calculated as follows:Mean Variance1nni 1x_{i}xThe mean variance, or mean deviation, can be calculated and used to make judgments byproviding information on the quality of the data. For example, if you were trying to decidewhether to buy stock, and all you knew was that this month’s average price was , and today’sprice is , you might be tempted to buy some. But, if you also knew that the mean variancein the stock’s price over the month was , you would realize the stock had fluctuated widelyduring the month. Therefore, the stock represented a more risky purchase than just the averageprice indicated.It can be seen that to make sound decisions using statistical data, it is important to analyze thedata thoroughly before making any decisions.Example:Calculate the variance and mean variance of the following set of hourly tank levels.Assume the tank is a 100 gal. tank. Based on the mean and the mean variance, wouldyou expect the tank to be able to accept a 40% (40 gal.) increase in level at any time?1:00 - 40% 6:00 - 38% 11:00- 34%2:00 - 38% 7:00 - 34% 12:00- 30%3:00 - 28% 8:00 - 28% 1:00 - 40%4:00 - 28% 9:00 - 40% 2:00 - 36%5:00 - 40% 10:00- 38%MA-05 Page 6 Rev. 0