= 
+ 
.Advanced Supplement to Statistical Errors
Adapted from Laboratory Manual of Freshmen Physics, National Tsinghua Univ., TAIWAN
This supplement is designed for students with desire to get full credit. However, I think every student in PHYS2LA, B, or C should practice this skill once in his life. So, once, I will ask you to apply this skill in the quarter.
The purpose to analyze experimental errors is to correct them, not just to find them only.
Assume <A>, <B> and <C> are mean values, s A, s B and s C are the errors respectively. Then, we may write:
A = <A> ± s A, B = <B> ± s B, and B = <C> ± s C.
Therefore, statistically,
Equation 1, If C = A ± B, then <C> = <A> ± <B>, and s C2= s A2 + s B2.
Equation 2, If C = A ´
B, then <C> = <A> ´
<B>, and = + .
Equation 3, If C = A ¸
B, then <C> = <A> ¸
<B>, and
=
+ .
Equation 4, If C = ApBq, then <C> = <A>p<B>q, and
= p2
+ q2
.
Example 1,
We want to calculate the volume of a cube, V, and the error of V, s V. Here we use the same result measuring the length of the cube in "Supplement to Significant figures."
What we care here are Average Length, Lave and SDOM, s SDOM.
Mean Value, Average Length, Lave =(S L)/n
SDOM, Standard Deviation of Mean, s SDOM =s /(n)(1/2)=0.16/(16)(1/2)=0.04(mm)
The notation of Average Length, here, Lave=(21.93± 0.04)(mm)
Or, Lave=(21.93± 0.18%)(mm) where (0.04/21.93)´ 100%=0.18%
|
Times of measurement |
Length, L, (mm) |
Deviation, d, (mm) |
Square of d, d2, (mm2) |
|
1 |
22.1 |
+0.17 |
0.029 |
|
2 |
22.0 |
+0.07 |
0.005 |
|
3 |
21.9 |
-0.03 |
0.001 |
|
4 |
21.8 |
-0.13 |
0.017 |
|
5 |
21.8 |
-0.13 |
0.017 |
|
6 |
21.7 |
-0.23 |
0.053 |
|
7 |
21.9 |
-0.03 |
0.001 |
|
8 |
22.0 |
+0.07 |
0.005 |
|
9 |
21.9 |
-0.03 |
0.001 |
|
10 |
22.3 |
+0.37 |
0.137 |
|
11 |
21.9 |
-0.03 |
0.001 |
|
12 |
22.1 |
+0.17 |
0.029 |
|
13 |
21.9 |
-0.03 |
0.001 |
|
14 |
21.8 |
-0.13 |
0.017 |
|
15 |
22.0 |
+0.07 |
0.005 |
|
16 |
21.8 |
-0.13 |
0.017 |
|
Total times, n=16 |
S L=350.9 |
S |d|=1.82 |
S |d|2=0.336 |
|
Average Length, Lave= 350.9/16= 21.93 |
Average Deviation, D= 1.82/16= 0.11 |
Standard Deviation, s = {0.336/(16-1)}(1/2)= 0.16 |
We have the notation, Lave=(21.93± 0.04)(mm) and know that, for a cube, V= L3. However, (s SDOM of Vave) ¹ (s SDOM of Lave,) 3!
We have to use Equation 4,
= p2
+ q2 if C = ApBq.
Since V= L3 = L3´ 10, then, (s V/<V>)2 = 32(s L/<L>)2 + 02(s 1/<1>)2.
Or, (s V/<V>)2 = 32(s L/<L>)2. We may rewrite it as, (s V/<V>) = 3(s L/<L>).
Please notice that the percentage error is %error=(Vmeasured-Vtrue)/Vtrue ´ 100%. You may treat it like (s V/<V>)´ 100% because here (Vmeasured-Vtrue) is somehow like the concept of Deviation, Standard Deviation, or Standard Deviation of Mean.
So, (s V/<V>)´ 100%=3(s L/<L>)´ 100%=3(0.04/21.93)´ 100%=3´ 0.018%=0.54%
With the average volume Vave= (Lave)3 = (21.93)3 = 10546.68 (mm3), we may calculate s SDOM of Vave = 54(mm).
That is, the notation of Vave= (10550± 0.54%)(mm3) or (10550± 54)(mm3)
In other words, the precision of Vave= (Lave)3 = (21.93)3 = 10546.68 (mm3) is not allowed based on the data table of our measurement
Example 2,
IF m = Mass ¸ Length, then <m > = <Mass> ¸ <Length> and (s m /<m >)2 = (s Mass/<Mass>)2 + (s Length/<Length>)2.
Here, if you do things by the book, you should measure Length several times, establish a data table, and find SDOM of Length, s Length. However, it always takes time to do so. Therefore, you may inspect the minimum scale of your ruler or your vernier caliper, and take a reasonable guess about s Length.
For instance, if the minimum scale of your 2-meter-long ruler is 0.1(cm), then you may guess that s Length is possibly ± 0.05(cm). Also, if the minimum scale of your weight balance is 0.1(gram), then SDOM of Mass, s Mass@ ± 0.05(gram).
Example 3,
IF gravity = (2 ´ distance)/(time)2, then <gravity> = 2 ´ <distance>´ <time>-2, and (s gravity/<gravity>)2 = (s distance/<distance>)2 + (-2)2 ´ (s time/<time>)2.
Example 4,
IF v = (2/n) ´ L ´ f , then <v> = (2/n) ´ <L> ´ <f> and (s v/<v>)2 = (s L/<L>)2 + (s f/<f>)2.
Example 5,
IF v = F1/2 ´ m -1/2, then <v> = <F>1/2 ´ <m >-1/2 and (s v/<v>)2 = (1/2)2(s F/<F>)2 + (-1/2)2(s m /<m >)2.
Conclusion:
There is not a single rule, formula, equation, or example to cover all cases. Sometimes, the source of error may be from your declined table or a bad caliper. You should be able to calibrate it once you detected it. Sometimes, the source of error may be the strong wind or the moisture around you and you may find a formula to correct your measured results.
The worst nightmare is that we have so many things to analyze at the same time in one single experiment. The only way is just do it, one by one, step by step.