The standard deviation is usually represented by the σA (sigma) Greek letter, where A is the thing whose error we are finding. For example, if we calculated some value for gravity (let's call it g) the standard deviation of the calculated gravity would be represented as σg. Pretty straight forward, huh?
Equations
Let's first talk about some general equations. For simplicity, let's call our calculated value Z.If Z is represented as a sum or difference of terms, Z=A1+A2+...+An, then σZ=√σ2A1+σ2A2+...+σ2An.
For example:
Z=A+B−C⇒σZ=√σ2A+σ2B+σ2C
If Z is represented as a product or quotient of terms, Z=A1×A2×...×AnB1×B2×...×Bn, then its relative standard deviation is given by σZZ=√(σA1A1)2+...+(σAnAn)2+(σB1B1)2+...+(σBnBn)2. Relative standard deviation is just the standard deviation over the calculated value itself. Ergo to get just the standard deviation we just multiply the whole thing by our calculated value.
For example:
Z=ABC⇒σZ=Z√(σAA)2+(σBB)2+(σCC)2
If Z is represented as another variable to a power, Z=An, then its relative standard deviation is given by σZZ=nσAA. Like before, to get σZ, we just multiply by our calculated Z.
For example:
Z=A7⇒σZ=7ZσAA
Those are the basic equations that you need to find the standard deviation. For more on techniques, see the following example.
Example
Suppose we want to find the standard deviation of a calculated value, let's call it Z, where:Z=A+B2CD
To make things easier, let U=B2CD and also let V=B2. So,
Z=A+U⇒σZ=√σ2A+σ2U
So, we need to find σU, and since V=B2,
U=VCD⇒σU=U√(σVV)2+(σCC)2+(σDD)2
Lastly, we need to find σVV,
V=B2⇒σVV=2σBB
Subbing 2σBB in for σVV, and B2CD for U we get,
U=VCD⇒σU=B2CD√(2σBB)2+(σCC)2+(σDD)2
Finally, subbing this new σU into the equation for σZ, we get,
σZ=√σ2A+(B2CD√(2σBB)2+(σCC)2+(σDD)2)2
Which can be more simply written as,
σZ=√σ2A+(B2CD)2[(2σBB)2+(σCC)2+(σDD)2]
No comments:
Post a Comment