Question 177907
the mean is the sum of the numbers divided by 15.
the sum of the numbers is 1078
the number of numbers is 15
the mean is 1078/15 = 71.86666667
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the standard deviation is: 9.735616171
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i used the built in functions of microsoft excel (stdevp) to calculate the standard deviation after confirming for myself that the formulas they use are consistent with the formulas that you would use.  those formulas are shown below the answer you are looking for if you have an interest in seeing how the standard deviation is developed using a simple example.
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2 standard deviations above the mean is 71.866666667 + (2 * 9.735616171)
3 standard deviations below the mean is 71.866666667 - (3 * 9.735616171)
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your answers become:
mean is 71.86666667
standard deviation is 9.735616171
2 standard deviations above the mean is 91.33789901
3 standard deviations below the mean is 42.65981815
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if you're happy with the answer you can stop here.
if you wish to learn a little more about standard deviation and how it is calculated you can continue.
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there are a couple of ways of getting the standard deviation.
in order to use the first way, you need to find the mean.
the second way does not require you to find the mean.
they both give you the same answer.
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first way is:
get the sum of the numbers.
divide by the number of numbers.
this gets you the mean.
subtract each number from the mean and square it.
get the sum of all these squared differences and divide by the number of numbers.
take the square root of this.
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the second way is:
square each number.
get the sum of each of these squared numbers.
multiply the sum these squared numbers by the number of numbers.
call this result A.
get the sum of the original numbers.
square it.
call this result B.
take A - B and then divide that by the square of the number of numbers.
take the square root of the result.
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both ways will get you the standard deviation.
to show you how this is done, i'll use a simple example:
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take 4 numbers at random:
2
5
8
9
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we will get the standard deviation the first way:
sum of the numbers = 24
number of numbers = 4
mean is 24/4 = 6
each number minus the mean and then squared is:
2-6 = -4 squared = 16
5 - 6 = -1 squared = 1
8 - 6 = 2 squared = 4
9 - 6 = 3 squared = 9
sum of these squared differences from the mean = 30 
30 / 4 = 7.5
square root of 7.5 = 2.7386...
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we will get the standard deviation the second way:
each number squared is:
2 squared = 4
5 squared = 25
8 squared = 64
9 squared = 81
sum of these squared numbers is:
174
4 * 174 = 696
the sum of the original numbers is:
2 + 5 + 8 + 9 = 24
24 squared = 576
696 - 576 = 120
number of numbers squared is (4*4) = 16
120/16 = 7.5
square root of 7.5 = 2.7386...
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comparing these two numbers shows that they are equal so the two methods provide you with the same answer.
that answer is the standard deviation of the population.
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keep in mind that the standard deviation of the population is not calculated in the same way as the standard deviation of a sample of the population.
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since we are dealing with the population here, and not a sample, i used the standard deviation formula for the population.
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