MEASUREMENT UNCERTAINTY
I. Objectives:
1. Become acquainted with the notion of the average value and the standard deviation of a series of
measurements of a parameter (in this experiment the period T of a simple pendulum)
2. Become familiar with the least squares fit method.
3. Measure the elongation x of a spring as function of applied force F. Use the method of least squares to
determine the spring constant k
4. Become familiar with the use of Microsoft EXCEL
II. Equipment:
Stand, spring, mass holder, masses, meter stick, pendulum bob, string, stopwatch, computer with EXCEL
software.
III. Introduction:
When we measure any physical parameter such as length, time, mass, etc an uncertainty in the parameter
value is involved. We know that this is true because if we repeat the same measurement several times we
get slightly different values. There are several factors that contribute to uncertainty in a measurement. One
source is the measuring instrument itself. Consider a length measurement of an object using a meter stick
whose smallest division is one millimeter (1 mm) as shown in fig.1. Each length measurement will have an
uncertainty between 0.3 mm and 0.7 mm. The second factor is the person who is carrying out the
measurement. A person with sharp eye sight will be able to measure with uncertainty 0.3 mm (the smallest
value our meter stick allows). A person with less sharp eyes will have a larger uncertainty. A third factor
that contributes to uncertainty is the measuring procedure. In the measurement of fig.1 for example the use
of a magnifying glass will help us keep the uncertainty to a low 0.3 mm. Another detail is the position of
the observer’s eye during the measurement (see fig.2). If the observer views the ruler at right angles
(position A in fig.2) this minimizes the uncertainty in the measurement. On the other hand placing the
observer’s eye at an angle (positions B and C in fig.2) results in a larger uncertainty. This phenomenon is
known as parallax.
Average and standard deviation:
One method to improve the accuracy of our measurement and get an quantitative estimate of the
uncertainty is to take several (N in number) measurement of the length ( x1 , x2 , x3 , ... , xN ). We define the
average value x of the measurement as:
x
x1  x2  ...  xN
N
(eqs.1)
1
The standard deviation  x of the measurement is given by the equation:
 x  x1    x  x2 
2
x 
 ...   x  xN 
N 1
2
2
(eqs.2)
Using probability theory one can show that there is a 68% probability that the real value of x lies
somewhere between  x   x  and  x   x  . This probability increases to 95.4% for the interval between
 x  2 x  and  x  2 x  .
Thus the standard deviation  x gives us a reasonable estimate of uncertainty in
the measurement of x. The final result of the measurement is given in the form:
 x  x 
(eqs.3)
Consider the following example: A length measurement gave an average value x = 7.584 mm and a
standard deviation  x  0.342 mm . The final result is reported as: x   7.6  0.3 mm. Please note that
we used only one significant figure for  x . The reason is that if we are uncertain about the first significant
figure in  x , it makes no sense to give the second (  x  0.34 mm ) or the third (  x  0.342 mm ). In
addition, we truncate the value for x to the decimal that corresponds to that of  x . Reporting the result as
x   7.58  0.3 mm , or x   7.584  0.3 mm would be meaningless.
Least square fit. Many relationships between two physical parameters x and y are described by an
equation of the form:
y  mx  b
(eqs.4)
Here m and b are constants known and slope and intercept, respectively. If we plot y versus x, we get a
straight line (see fig.3) that cuts the y-axis at b (thus the name intercept). The slope m of the straight line
can be determine by drawing the orthogonal triangle ABC as shown in fig.3. The slope m is given by the
ratio:
y
(eqs.5)
x
Here y and x are the lengths of the vertical (BC) and horizontal (AB) sides of the triangle.
In the course of experimental work we frequently run into the following situation: We measure parameter y
versus parameter x. For example x can be the voltage V applied across a device and y can be the current I
that flows through the device. The experimental points (y, x) are plotted in fig.3. We would like to draw a
straight line that best fits the experimental data. Such a straight line passes between the points leaving
approximately half above and half below. This straight line is known as a least square fit and is define by
the parameters m and b. The calculation of these parameters is a simple problem in calculus. Below we
give the solution. Please do not be turned off by the complex algebra involved. You will use EXCEL to
determine m and b and the corresponding uncertainties  m and  b .
m
2
m
N  y1 x1  y2 x2  ...  y N xN    x1  x2  ...  xN  y1  y2  ...  y N 
N  x12  x22  ...  xN2    x1  x2  ...  xN 
2
 y1  y2  ...  yN   x12  x22  ...  xN2    x1  x2  ...  xN  y1x1  y2 x2  ...  yN xN 
b
2
N  x12  x22  ...  xN2    x1  x2  ...  xN 
(eqs.6)
(eqs.7)
IV. Experimental Method:
In section V-1 you will measure the duration of five pendulum periods using a stopwatch. A period is the
time it takes for the pendulum bob to travel from point A to point B and back to point A (see fig.4). One lab
partner can give a signal and release the pendulum bob. The other partner can operate the stopwatch.
In section V-2 you will measure the elongation x of a spring as function of the force F applied to one end of
the spring (see fig.5a and fig.5b). The spring force is provided by a weight of mass m placed in a mass
holder hanging at the end of the spring. The applied force F = mg. Here g is the acceleration of gravity
( g  981 cm/s2 ). The spring elongation x = x1 – x2
Here x1 is the height of the empty mass holder from the table top; x2 is the height of the holder when it is
loaded with the mass m. The lengths x1 and x2 are measured using a vertical meter stick.
V. Procedure
V-1: Use the pendulum.
Measure the length ℓ′ of the pendulum string (see fig.6) using the meter stick. Measure the duration of 5
periods for the pendulum. Repeat the measurement for a total of 10 times.
V-2: Use the spring
Measure the distance x1 from the empty holder to the table top
Place a mass m on the holder and measure the new distance x2 from the table top
Determine x2 for m = 40, 60, 80, 100, 120, 140, 160, 200, 220, 240, 260, 280, and 300 g
VI. For the report
VI-1: Tabulate the values of T determined in section V-1. Use EXCEL to calculate the average value T
and its standard deviation σT. Report your result numerically in the form: T  T   T 
3
VI-2: Tabulate the values of m and x from section V(b). Plot m versus x. The points should lie on a straight
line that passes through the origin and has slope
k
S  . Use the method of least squares to determine the slope S and its uncertainty σS.
g
Report your result numerically in the form:  S   S  g/m
VII. Questions
VII-1. The period of a pendulum is given by the equation: T  2
gravity. The distance
g is given by: g 
 
g
. Here g is the acceleration of
d
,where d is the diameter of the pendulum bob (see fig.6). The value of
2
4 2
. Calculate g .
T2
VII-2. Calculate the uncertainty σk in the spring constant using the equation:  k  g S . Report your
final result in the form:  k   k 
Object whose length L
we wish to measure
Ruler
Fig.1
C
A
y
B
C
Δy
Object Ruler
b
A
Δx
B
Fig.2:
O
Fig.3: A least squares fit for (x,y)
points that obey the equation:
y  b  mx
4
x
Fig.5a
A
B
Fig.5b
meter
stick
x1
Fig.4: A simple pendulum
m
x2
ℓ
ℓ′
d
Fig.6
5