Fallacy of the two standard deviation formulas

Vanice

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Apr 15, 2016
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14
There are two formulas in determining standard devation

One goes like this

σ=(ΣfXm^2)/n - X^2


And the other like this

S.D = NΣfXm^2 - (ΣfXm)^2 / N(N-1)

There is a table we have to solve for standard deviation

IntervalfXmfXmXm^2fXm^2
41-501445.5773.32070.2528,983.5
31-402535.5887.51260.2531,506. 25
21-302425.5612650.2515,606
11-201615.5248240.253,844
1-1065.53330.25181.5

N=88 ΣfXm= 2553.8 ΣfXm^2= 80,121.25

First formula solution

80,121.25 / 88 = 910.46875

910.46875 - 842.3186983471074 = 68.15

Second formula solution

80,121.5 * 88 = 7,050,670

2553.8 ^2= 6,521,894.44

7,050,670 - 6521,894.44 = 528,775.56

528,775.56 / 7656 = 69.06

Why are the two solutions different? Is there something wrong?
 
There are two formulas in determining standard devation

One goes like this

σ=(ΣfXm^2)/n - X^2


And the other like this

S.D = NΣfXm^2 - (ΣfXm)^2 / N(N-1)

There is a table we have to solve for standard deviation

IntervalfXmfXmXm^2fXm^2
41-501445.5773.32070.2528,983.5
31-402535.5887.51260.2531,506. 25
21-302425.5612650.2515,606
11-201615.5248240.253,844
1-1065.53330.25181.5

N=88 ΣfXm= 2553.8 ΣfXm^2= 80,121.25

First formula solution

80,121.25 / 88 = 910.46875

910.46875 - 842.3186983471074 = 68.15

Second formula solution

80,121.5 * 88 = 7,050,670

2553.8 ^2= 6,521,894.44

7,050,670 - 6521,894.44 = 528,775.56

528,775.56 / 7656 = 69.06

Why are the two solutions different? Is there something wrong?

It took me a long time to check your work by putting it into a spreadsheet; apparently for the first frequency you used 17 in some places instead of 14. And your formulas aren't quite written correctly, and you didn't take the square roots. But once I got all that out of the way ...

The basic issue here is that the first formula is for a population, and the second is for a sample. Do you know the difference? Please check your sources and see how the formulas were stated.
 
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