Difference between Simple Standard Deviation, Population Standard
Deviation, Short- term & Long- Term Standard Deviation
Sample Standard Deviation: A sample standard deviation is an estimate, based on a
sample, of a population standard deviation. Example:
Let's say we wanted to calculate the standard deviation for the amounts of gold coins pirates
on a pirate ship have.
There are 100 pirates on the ship. In statistical terms this means we have a population of
100. If we know the amount of gold coins each of the 100 pirates have, we use the standard
deviation equation for an entire population:
What if we don't know the amount of gold coins each of the 100 pirates have? For example,
we only had enough time to ask 5 pirates how many gold coins they have. In statistical
terms this means we have a sample size of 5 and in this case we use the standard deviation
equation for a sample of a population:
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The rest of this example will be done in the case where we have a sample size of 5 pirates,
therefore we will be using the standard deviation equation for a sample of a population.
Here are the amounts of gold coins the 5 pirates have:
4, 2, 5, 8, 6.
Now, let's calculate the standard deviation:
1. Calculate the mean:
2. Calculate
3. Calculate
for each value in the sample:
:
4. Calculate the standard deviation:
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The standard deviation for the amounts of gold coins the pirates have is 2.24 gold
coins.
Population standard deviation: Example:
You grow 20 crystals from a solution and measure the length of each crystal in millimeters. Here
is your data:
9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4
Calculate the population standard deviation of the length of the crystals.
1. Calculate the mean of the data. Add up all the numbers and divide by the total number of
data points.
(9+2+5+4+12+7+8+11+9+3+7+4+12+5+4+10+9+6+9+4) / 20 = 140/20 = 7
2. Subtract the mean from each data point (or the other way around, if you prefer... you will be
squaring this number, so it does not matter if it is positive or negative).
(9 - 7)2 = (2)2 = 4
(2 - 7)2 = (-5)2 = 25
(5 - 7)2 = (-2)2 = 4
(4 - 7)2 = (-3)2 = 9
(12 - 7)2 = (5)2 = 25
(7 - 7)2 = (0)2 = 0
(8 - 7)2 = (1)2 = 1
(11 - 7)2 = (4)22 = 16
(9 - 7)2 = (2)2 = 4
(3 - 7)2 = (-4)22 = 16
(7 - 7)2 = (0)2 = 0
(4 - 7)2 = (-3)2 = 9
(12 - 7)2 = (5)2 = 25
(5 - 7)2 = (-2)2 = 4
(4 - 7)2 = (-3)2 = 9
(10 - 7)2 = (3)2 = 9
(9 - 7)2 = (2)2 = 4
(6 - 7)2 = (-1)2 = 1
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(9 - 7)2 = (2)2 = 4
(4 - 7)2 = (-3)22 = 9
3. Calculate the mean of the squared differences.
(4+25+4+9+25+0+1+16+4+16+0+9+25+4+9+9+4+1+4+9) / 20 = 178/20 = 8.9
This value is the variance. The variance is 8.9
4. The population standard deviation is the square root of the variance. Use a calculator to
obtain this number.
(8.9)1/2 = 2.983
The population standard deviation is 2.983
Short term Standard Deviation:
The Short-term standard deviation estimates standard deviation within, and is a measure of how
widely values are dispersed from the average value (the mean) due only to the effect of white
noise.
It is used to calculate Short-term Process Capability (i.e.: Cp and Cpk).
Long term standard deviation:
The standard deviation long-term estimates total standard deviation of the process, and is a
measure of how widely values are dispersed from the overall average value (the grand mean) due
to white noise and black noise.
Used to calculate process long-term capability (i.e.: Pp and Ppk) and PPM defective.
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