# Fallacy of the two standard deviation formulas

#### Vanice

##### New member
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

 Interval f Xm fXm Xm^2 fXm^2 41-50 14 45.5 773.3 2070.25 28,983.5 31-40 25 35.5 887.5 1260.25 31,506. 25 21-30 24 25.5 612 650.25 15,606 11-20 16 15.5 248 240.25 3,844 1-10 6 5.5 33 30.25 181.5

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
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

 Interval f Xm fXm Xm^2 fXm^2 41-50 14 45.5 773.3 2070.25 28,983.5 31-40 25 35.5 887.5 1260.25 31,506. 25 21-30 24 25.5 612 650.25 15,606 11-20 16 15.5 248 240.25 3,844 1-10 6 5.5 33 30.25 181.5

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.