Fallacy of the two standard deviation formulas

Vanice

New member
Joined
Apr 15, 2016
Messages
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?
 

Dr.Peterson

Elite Member
Joined
Nov 12, 2017
Messages
6,663
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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