Measurements are made of the attenuation of radiation as a function of the depth of penetration; for example you could measure the decrease of intensity of red light with depth in sea water by recording the number of photons/second ( counts/sec). Attenuation is well described by the expression
where L is a measure of the attenuation and Io is the initial intensity impinging on the surface. We will assume that x can be measured with high accuracy and that the error in I(x) is given by
The intensity attenuation equation can be linearized by taking the logarithm of each side to give:
and the error in y is found from
In the table below are listed values of I(xi), the calculated y(xi) and the error in y(xi) which is the reciprocal of the square root of I(xi).
i |
x |
I |
y |
|
x2 |
xy |
|
|
|
|
|
cm |
counts/sec |
||||||||||
1 |
1 |
729 |
6.59 | 0.037 |
1 |
6.59 |
730 |
730 |
4813 |
4813 |
730 |
2 |
2 |
500 |
6.21 | 0.045 |
4 |
12.42 |
987 |
1975 |
3066 |
6133 |
493 |
3 |
3 |
360 |
5.89 | 0.053 |
9 |
17.67 |
1067 |
3203 |
2096 |
6290 |
355 |
4 |
4 |
273 |
5.61 | 0.061 |
16 |
22.44 |
1074 |
4299 |
1507 |
6030 |
268 |
5 |
5 |
188 |
5.24 | 0.073 |
25 |
26.20 |
938 |
4691 |
9831 |
4916 |
187 |
6 |
6 |
146 |
4.98 | 0.083 |
36 |
29.88 |
870 |
5225 |
722 |
4337 |
145 |
7 |
7 |
102 |
4.62 | 0.099 |
49 |
32.34 |
714 |
4999 |
471 |
3299 |
102 |
8 |
8 |
61 |
4.11 | 0.128 |
64 |
32.88 |
488 |
3906 |
250 |
2006 |
61 |
9 |
9 |
40 |
3.69 | 0.158 |
81 |
33.21 |
360 |
3244 |
147 |
1330 |
40 |
| SUMS | 7233 |
32277 |
14061 |
39158 |
2384 |
So the relevant sums are:
|
 |
|
|
 |
and as in the previous case, we can solve for b and m, finding b =
6.918 and m = -.337. The errors are then 
and
. We
can summarize the final results for m and b: m = -.34 ±.01 and
b = 6.92 ±.04, from the technique of Least Squares Fitting. Noting that L=1/m
and Io = exp(b); using dL/L = dm/m and dIo = Io
db, we finally find:
Io = 1012 ±40 ( counts/second) and L = 2.97 ±.09 (1/cm)
The plot of y versus x and the fitted line are:
