Variance and Standard Deviation of a Random Variable
Suppose the variable you are measuring is reported in millions of dollars, so the mean is reported as, say, 120 (meaning $120 million).
What unit would be used to describe the variance?
The standard deviation for this probability distribution, $$\sigma(X)$$, is the square root of the variance:
$$\displaystyle \sigma(X) = \sqrt{\sigma^2(X)}.$$
So, for this version of Mike’s Random Runner Program, the standard deviation is
$$\displaystyle \sigma(X) = \sqrt{\mbox{9.19 miles}^2} = \mbox{3.03 miles}.$$
Notice that the units are now just miles, not miles squared.
Mike has a second Random Runner Program, given by the following probability distribution. The expected value has already been calculated as 7.3 miles.
|Distance, X| P(X)|P(X)X|
|---|---|---|
|0|0.10|0.00|
|2|0.25|0.50|
|8|0.25|2.00|
|12|0.40|4.80|
|Sum|1.0|7.30|
What is the variance for this distribution?
What is the standard deviation?
Incorrect.
You went a step further than required and calculated the standard deviation, which would have the unit of miles, not miles2.
Incorrect.
This adds the sum of squared deviations from the expected values, but neglects to weight those by their corresponding probabilities.
Correct.
Incorrect.
This is the square root of the range, which is 12 miles.
In summary:
[[summary]]
No, consider that as you square a million dollars, the million becomes a trillion as the unit of dollars becomes dollars squared. This isn't the same as millions of dollars squared.
Yes!
An observation of 125, for example, would be a deviation of 5, meaning $5,000,000, and the square of this is 25, meaning 25,000,000,000,000 squared dollars, or actually 25 trillions of 'dollars squared'. This is one reason why the standard deviation is so nice. It has the same units as the random variable.
Incorrect.
The variance will be the units used for the random variable, squared.
Your friend Mike has started selling his patented Random Runner Program. Using this system, a runner's daily distance is randomly drawn from a specific probability distribution. While Mike knows how to convey information about average daily running distances using expected values, he needs help in describing the variation, or dispersion, around the expected value.
The two most commonly used measures of dispersion are __variance__ and __standard deviation__. Variance is calculated first, followed by standard deviation.
The variance of a distribution, denoted by either $$\sigma^2(X)$$ or Var(_X_), is another example of an expected value. More specifically, it is the expected value of the squared deviation from the expected value, which can be written as:
$$\displaystyle \sigma^2(X) = E([X-E(X)]^2)$$
Here is one of the distributions available from Mike’s Random Runner Program:
|Distance, X| P(X)|P(X)X|
|---|---|---|
|0|0.15| 0.00|
|5|0.25|1.25|
|8|0.50|4.00|
|10|0.10|1.00|
|Sum|1.0|6.25|
The expected value is 6.25 miles. The variance can be calculated by adding three more columns, one for the deviation from the expected value, X-E(X), a second that squares those values, and a third that multiplies the squared deviations from the expected values by their respective probabilities:
|Distance, X| P(X)|P(X)X|X-E(X)|[X-E(X)]2|P(X)[X-E(X)]2|
|---|---|---|---|---|---|
|0|0.15| 0.00|-6.25| 39.06| 5.86|
|5|0.25|1.25|-1.25| 1.56 | 0.39 |
|8|0.50|4.00|1.75| 3.06| 1.53|
|10|0.10|1.00|3.75|14.06|1.41|
|Sum|1.0|6.25| $$ \, $$ | $$ \, $$ |9.19|
The variance for this distribution is 9.19 miles2.
Note that the units attached to the variance are the units from the random variable, squared.
Correct.
Here's the completed table, showing the calculations:
|Distance, X| P(X)|P(X)X|X-E(X)|[X-E(X)]2|P(X)[X-E(X)]2|
|---|---|---|---|---|---|
|0|0.10| 0.0|-7.3| 53.29| 5.33|
|2|0.25|0.5|-5.3| 28.09 | 7.02 |
|8|0.25|2.0|0.7| 0.49| 0.12|
|12|0.4|4.8|4.7|22.09|8.84|
|Sum|1.0|7.3| $$ \, $$ | $$ \, $$ |21.31|
The variance is 21.31 miles2.
Once you have the variance, which is 21.31 miles2, take the square root:
$$\displaystyle \sigma(X) =\sqrt{\mbox{21.31 miles}^2}=\mbox{4.62 miles}.$$
Millions of dollars
Millions of 'dollars squared'
'Millions of dollars' squared
4.62 miles2
21.31 miles2
103.96 miles2
3.46 miles
4.62 miles
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