AGH University of Science and
Technology in Cracow
Department of Electronics
Laboratory Manual
Physics_1
Title:
Uncertainties of measurements
2009 r.
Experiment
No.
1
AGH University of Science and
Technology in Cracow
Department of Electronics
Laboratory Manual
Physics_1
Title:
Uncertainties of measurements
2009 r.
Experiment
No.
1
1. Standard deviation
To find the uncertainty in measurements, we often calculate the standard deviation, or
σ , of
the measured value. Standard deviation is a measure of the variation of N data points (x
1
...x
N
)
about an average value,
x , and is typically called the uncertainty of a measured result.
The average or mean value,
x , of a set of N measurements is equal to:
Once the mean value of the measurements is determined, it is helpful to define how much the
individual measurements are scattered around about the mean. The deviation,
d
, of any
measurement,
, from the mean
i
x
x
is given by :
Since the deviation may be either positive or negative, it is often more useful to use the mean
deviation, or
d
, to determine the uncertainty of the measurement. This is found by averaging the
absolute deviations,
; that is,
To avoid the use of absolute values we can use the square of the deviation,
, to more accurately
determine the uncertainty of our measurement. The standard deviation,
2
i
d
σ
, (sometimes called the
root-mean square) is given by
It can be shown that for a small number of measurements, this equation becomes:
where N is replaced by N - 1.
2
The experimental result,
E
x
, can then be written as
where,
σ
gives the measure of the precision of the measurement.
Notice the standard deviation is always positive and has the same units as the mean value. It can be
shown that there is a 68% likelihood that an individual measurement will fall within one standard
deviation (
) of the true value. Furthermore, it can be shown that there also exists a 95% likelihood
that an individual measurement will fall within two standard deviations (
) of the true value, and a
99.7% likelihood that it will fall within (
) of the true value.
Some useful expressions:
σ expresses the standard deviation for the single measurement x
i
, so often the notation
x
σ
is
used.
It can be proved, that standard deviation of the mean
x is equal to::
N
x
x
σ
σ
=
When dealing with uncertainties based on a large collection of numbers the manipulation of
measured quantities and the error associated with each quantity will contribute to the error in
the final answer. The following formulae are useful:
y
x
z
+
=
( )
( )
2
2
y
x
z
σ
σ
σ
+
=
3
y
x
z
−
=
( )
( )
2
2
y
x
z
σ
σ
σ
+
=
2. Linear regression
If the relationship between two sets of data (
x and y) is linear and the data is plotted (y versus
x) the result is a straight line. This relationship is known as having a linear correlation and
follows the equation of a straight line
b
ax
y
+
=
.
We can apply a statistical treatment known as linear regression to the data
in order
to determine the constants a and b which are called regression coefficients: a is the slope and
b is the y-intercept for the line which the best fits the data.
Given a set of data
with
n data points, the regression coefficients can be determined
using the following:
a
n
x y
x
y
n
x
x
i
i
i
n
i
i
n
i
i
n
i
i
n
i
i
n
=
⋅
−
⎛
⎝
⎜
⎞
⎠
⎟ ⋅
⎛
⎝
⎜
⎞
⎠
⎟
⋅
−
⎛
⎝
⎜
⎞
⎠
⎟
=
=
=
=
=
∑
∑
∑
∑
∑
1
1
1
2
1
1
2
b
x
y
x
x y
n
x
x
i
i
n
i
i
n
i
i
n
i
i
i
n
i
i
n
i
i
n
=
⎛
⎝
⎜
⎞
⎠
⎟ ⋅
⎛
⎝
⎜
⎞
⎠
⎟ −
⎛
⎝
⎜
⎞
⎠
⎟ ⋅
⎛
⎝
⎜
⎞
⎠
⎟
⋅
−
⎛
⎝
⎜
⎞
⎠
⎟
=
=
=
=
=
=
∑
∑
∑
∑
∑
∑
2
1
1
1
1
2
1
1
2
It is also possible to determine the correlation coefficient, r, which gives us a measure of the
reliability of the linear relationship between the x and y values. A value of r = 1 indicates an
exact linear relationship between
x and y. Values of r close to 1 indicate excellent linear
reliability. If the correlation coefficient is relatively far away from 1, the predictions based on
the linear relationship,
, will be less reliable. The correlation coefficient,
r, can be
determined by:
b
ax
y
+
=
4
The standard deviations of the linear coefficients
σ
a
and
σ
b
are given by expressions:
(
)
[
]
σ
a
i
i
i
n
i
i
n
i
i
n
n
n
y
ax
b
n
x
x
=
−
⋅
−
+
⋅
−
⎛
⎝
⎜
⎞
⎠
⎟
=
=
=
∑
∑
∑
2
2
1
2
1
1
2
σ
σ
b
a
i
i
n
x
n
=
⋅
=
∑
2
1
It may appear that the above equations are quite complicated , however upon inspection, we
can see that their components are nothing more than simple algebraic manipulations of the
columns of data. The table below is useful for these calculations.
x
i
y
i
2
i
x
2
i
y
i
i
y
x
a b
(
)
b
ax
y
i
i
+
−
(
)
(
)
2
b
ax
y
i
i
+
−
x
1
y
1
. . .
. . .
. . .
. . .
. . .
. .
x
n
y
n
∑
i
x
∑
i
y
∑
2
i
x
∑
2
i
y
i
i
y
x
∑
(
)
(
)
∑
+
−
2
b
ax
y
i
i
3. Total differential method
Suppose A is a function of several independent variables, i.e. A = f(x
1
, x
2
, x
3,
..., x
k
) and
each of these variables has its own uncertainty
Δx
1,
Δx
2,
...,
Δx
k.
.
Absolute error of such measurements
ΔA can be calculated using the method of total
differential of a function A. Take the differential of A with respect to several variables:
Δ
Δ
Δ
A
f
x
x
f
x
x
f
x
x
k
k
=
⋅
+
⋅
+
+
⋅
∂
∂
∂
∂
∂
∂
1
1
2
2
...
Δ
where
k
x
f
x
f
x
f
∂
∂
∂
∂
∂
∂
,....,
,
2
1
are partial derivatives. A partial derivative means that the derivative
is taken with respect to one variable, while all the other variable are considered constant.
5
Relative error of such measurements can be calculated using the relationship:
(
)
σ
∂
∂
∂
∂
=
=
⋅
⋅
+
+
⋅
⎛
⎝
⎜
⎞
⎠
⎟
Δ
Δ
Δ
A
A
f x
f
x
x
f
x
x
k
k
k
1
1
2
1
1
,
, ...,
...
x
x
Total differential method allows us to find the true value of a measurement in the interval (A
-
ΔA, A + ΔA) with the probability equal to 0,999.
5. Logarithmic derivative method
When the function A is the product of variables x
1
.... x
k,
, the log derivative method is used to
find the uncertainty of A.
k
a
k
a
a
x
x
x
A
⋅
⋅
⋅
=
.....
2
2
1
1
Take the natural log of each side:.
k
k
x
a
x
a
x
a
A
ln
...
ln
ln
ln
2
2
1
1
+
+
+
=
Then differentiate:
k
k
k
x
x
a
x
x
a
x
x
a
A
A
Δ
+
+
Δ
+
Δ
=
Δ
.....
2
2
2
1
1
1
Let's take the absolute values of a
1
, ..., a
k
. because uncertainties always sum up and never
subtract.
Relative uncertainty::
σ =
=
+
+ +
Δ
Δ
Δ
Δ
A
A
a
x
x
a
x
x
a
x
x
k
k
k
1
1
1
2
2
2
...
Absolute uncertainty
Δ
Δ
Δ
Δ
A
A a
x
x
a
x
x
a
x
x
k
k
k
=
+
+ +
⎛
⎝
⎜
⎞
⎠
⎟
1
1
1
2
2
2
...
Example 1
Calculate the uncertainty of mobility of the carriers
μ using the method of logarithmic
derivative:
z
dB
al
=
μ
Calculate logarithm of the both sides of the equation:
d
B
l
a
z
ln
ln
ln
ln
ln
−
−
+
=
μ
Derivate both sides of the equation:
6
b
b
B
B
l
l
a
a
z
z
Δ
+
Δ
+
Δ
+
Δ
=
Δ
μ
μ
where:
a
a
σ
=
Δ
is the uncertainty of the regression coefficient a
m
l
3
10
2
,
0
−
⋅
=
Δ
,
are the geometrical uncertainties of the probe,
m
b
3
10
1
,
0
−
⋅
=
Δ
=
Δ
z
B
is the uncertainty of reading B
z
from the electromagnet characteristic.
Example 2
Calculate the uncertainty of concentration of the carriers n
using the total differential method
e
R
n
H
1
=
Remember: e= 1.6 x 10
-19
C.
H
H
R
R
n
n
Δ
⋅
∂
∂
=
Δ
2
1
H
H
R
e
R
n
⋅
−
=
∂
∂
H
H
R
R
e
n
Δ
⋅
⋅
=
Δ
2
1
5. How to draw plots?
1. All graphs should be done by hand in pencil on graph paper. (As the semester
progresses you may be allowed to use a computer to generate graphs once proficiency
with hand-drawn graphs is shown)
2. . The graph should use as much of the graph paper as possible. Carefully choosing the
best scale is necessary to achieve this. The axes should extend beyond the first and last
data points in both directions.
3. All graphs should have a short, descriptive title at the top of each graph, detailing what
is being measured.
4. Each axis should be clearly labeled with titles and units.
5. Clearly label the scale of each axis.
6. Never connect the dots on a graph, but rather give a best-fit line or curve.
7. The best-fit line should be drawn with a ruler or similar straight edge, and should
closely approximate the trend of all the data, not any single point or group of points.
http://www.rit.edu/cos/uphysics/uncertainties/Uncertainties.html
Barbara Dziurdzia, Faculty of Electronics AGH UST; Updated: 14.02.2009
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