Standard Deviation

by Richard Reid


The Definition of Standard Deviation:
Variance tells us the dispersion (ie Is it a fat or skinny bell shape?) of our Normal Distribution (bell curve). The "Standard Deviation is referred to as the square root of the Variance. Without any doubt, the "Standard Deviation" is the best measure of dispersion around the arithmetic mean. In a phrase, it is the "root mean squared deviation."


Equation for the Standard Deviation: (for ungrouped data)

SD = sqrt { sum(x - mu)^2 / (n-1) },
where x is the data value,
mu is the arithmetic mean of the data items, and
n is the number of data items.


Calculation of the Standard Deviation: (for ungrouped data)
The "Standard Deviation" for ungrouped data can be calculated in the following steps:
  1. all the deviations (differences) from the arithmetic mean of the set of numbers are squared;
  2. the arithmetic mean of these squares is then calculated;
  3. the square root of the mean is the standard deviation
Non-mathematical people always find this a bit complicated at first, but don't despair, there is an example to follow that should take the sting out of it.

Example:
Given the set of numbers {20, 23, 25, 26}, the "Standard Deviation" can be calculated as follows:

Step 1:
The arithmetic mean of these numbers is found to be equal to 23.5
[eg. arithmetic mean = (20+23+25+26)/4 = 23.5]. The deviations from the mean are respectively:
  1. 23.5 - 20 = 3.5
  2. 23.5 - 23 = 0.5
  3. 25 - 23.5 = 1.5
  4. 26 - 23.5 = 2.5
The squares of these deviations are:
  1. 3.5^2 = 12.25
  2. 0.5^2 = 0.25
  3. 1.5^2 = 2.25
  4. 2.5^2 = 6.25
Step 2: The sum of these squares is 12.25 + 0.25 + 2.25 + 6.25 = 21. This is now divided by (n-1), which is 3, to get 7.

NOTE: Some books show division by "n". However, when calculating the Standard Deviation of small sample, a better estimate of the parent group is obtained by dividing by (n-1) instead of dividing by "n". For large "n", the difference between using "n" or "n-1" is small.

Step 3:
Finally, the square root of 7 is approximately 2.6457513

Answer:
In summary, the Standard Deviation of the set of numbers {20, 23, 25, 26} is 2.6457513



Equation for the Standard Deviation: (for grouped data)

There is an alternative method of determining the Standard Deviation when some of the data values are repeated. A good example of grouped data can be found in simulators that collect data for each True Count.

SD = sqrt { sum f(x - mu)^2 / (n-1) },
where f = the number of data items (or alternatively, weighted value of the data items) in the group,
x is the data value,
mu is the arithmetic mean of the data items, and
n is the number of data items for all groups. In other words, n = sum(f).


Calculation of the Standard Deviation: (for grouped data)
The "Standard Deviation" for grouped data can be calculated as in the previous example, but it can also be calculated via the grouped data method. Let's take a look at how this technique can be used per the following steps:
  1. all the deviations (differences) from the arithmetic mean of the set of numbers are squared;
  2. each result (difference) is multiplied by f
  3. the sum of the resulting products in step 2 is then divided by n-1;
  4. the square root of step 3 is taken to arrive at the standard deviation
Once again there is an example to follow that should help.

Example:
Given the set of numbers {20, 23, 25, 26, 26, 23, 25, 25}, the data can be grouped as follows:

fData
120
223
325
226
The "Standard Deviation" can be calculated using the following steps:

Step 1:
The arithmetic mean of the data is found to be equal to 24.125
[eg. arithmetic mean = (1*20+2*23+3*25+2*26)/8 = 24.125]. The deviations from the mean are respectively:
  1. 24.125 - 20 = 4.125
  2. 24.125 - 23 = 1.125
  3. 25 - 24.125 = 0.875
  4. 26 - 24.125 = 1.875
The squares of these deviations are:
  1. 4.125^2 = 17.015625
  2. 1.125^2 = 1.265625
  3. 0.875^2 = 0.765625
  4. 1.875^2 = 3.515625
Step 2:
The sum of the squared results from step 1 are now multiplied by their associated f. This gives us:
  1. for 20, 1 * 17.015625 = 17.015625
  2. for 23, 2 * 1.265625 = 2.53125
  3. for 25, 3 * 0.765625 = 2.296875
  4. for 26, 2 * 3.515625 = 7.03125
Step 3:
The sum of the resulting products is 17.015625 + 2.53125 + 2.296875 + 7.03125 = 28.875. This is now divided by (n-1), which is 7, to get 4.125


Step 4:
Finally, the square root of 4.125 is approximately 2.031

Answer:
In summary, the Standard Deviation of the set of numbers {20, 23, 25, 26, 26, 23, 25, 25} is 2.031


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