Standard Deviation
The Standard Deviation is a measure of how spread out numbers
are.
Its symbol is ฯ (the
greek letter sigma)
The formula is easy: it is the square root of
the Variance. So now you ask, "What is the
Variance?"
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Formulas
Here are the two
formulas, explained at Standard Deviation Formulas if
you want to know more:
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The “Sample Standard Deviation”: |
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![square root of [ (1/(N-1)) times Sigma i=1 to N of (xi - xbar)^2 ]](file:///C:/Users/arabi/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif)
Looks
complicated, but the important change is to
divide by N-1 (instead of N) when calculating a
Sample Variance.
When
you have "N" data values that are:
- The
Population: divide by N when calculating
Variance (like we did)
- A
Sample: divide by N-1 when calculating
Variance
Variance
The Variance is defined as:
The
average of the squared differences from the Mean.
To calculate the variance follow these steps:
- Work
out the Mean (the
simple average of the numbers)
Then for each
number: subtract the Mean and square the result (the squared
difference).
Then work out
the average of those squared differences. (Why Square?)
Example of Standard Deviation vs. Variance
To
demonstrate how both principles work, let's look at an example of standard
deviation and variance.
Suppose
you have a series of numbers and you want to figure out the standard deviation
for the group. The numbers are 4, 34, 11, 12, 2, and 26. We need to determine
the mean or the average of the numbers. In this case, we determine the mean by
adding the numbers up and dividing it by the total count in the group:
(4 + 34 + 18 + 12 + 2 + 26) ÷ 6 = 16
![square root of [ (1/N) times Sigma i=1 to N of (xi - mu)^2 ]](file:///C:/Users/arabi/AppData/Local/Temp/msohtmlclip1/01/clip_image003.gif)
So the mean
is 16. Now subtract the mean from each number then square the result:
- (4 - 16)2 = 144
(34 - 16)2 = 324- (18 - 16)2 = 4
- (12 - 16)2 = 16
- (2 - 16)2 = 196
- (26 - 16)2 = 100
Now we have
to figure out the average or mean of these squared values to get the variance.
This is done by adding up the squared results from above, then dividing it by
the total count in the group:
(144 + 324 + 4 + 16 + 196 + 100) ÷ 6 =
130.67
This
means we end up with a variance of 130.67. To figure out the standard
deviation, we have to take the square root of the
variance, which is 11.43
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