STATISTICS
Higher Concepts of Mathematics
This spread, or distance, of each data point from the mean is called the variance. The variance
of each data point is calculated by:
Variance
x
xi
where
xi = each data point
= mean
x
The variance of each data point does not provide us with any useful information. But if the
mean of the variances is calculated, a very useful number is determined. The mean variance is
the average value of the variances of a set of data. The mean variance is calculated as follows:
Mean Variance
1
n
n
i 1
xi
x
The mean variance, or mean deviation, can be calculated and used to make judgments by
providing information on the quality of the data. For example, if you were trying to decide
whether to buy stock, and all you knew was that this month’s average price was , and today’s
price is , you might be tempted to buy some. But, if you also knew that the mean variance
in the stock’s price over the month was , you would realize the stock had fluctuated widely
during the month. Therefore, the stock represented a more risky purchase than just the average
price indicated.
It can be seen that to make sound decisions using statistical data, it is important to analyze the
data 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, would
you 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
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