Microsoft Word - Exercise_E
or Propagation.docx
1
E
or Propagation: Volumes
Introduction
Physics is an empirical science. Every theoretical concept has to be supported with the
experiment to be fully accepted. Experience has shown that no measurement is exact. All
measurements have some degree of uncertainty due to the limits of instruments and the
people using them. In science it is critically important to specify that uncertainty and
include its reliable estimate in the final statement of the result of the measurement. The
concept of the e
or propagation becomes a vital part of each physics experiment.
There are two concepts associated with measurements: accuracy and precision. The
accuracy of the measurement refers to how close the measured value is to the true or
accepted value. For example, if you used a balance to find the mass of a known standard
100.00 g mass, and you got a reading of 78.55 g, your measurement would not be very
accurate because the percent discrepancy between the measured and “true value” is 21%.
Precision refers to how close together a group of measurements actually are to each other.
In the example above, if the three measurements of mass that obtained are 78.55g,
78.60 g and 78.50 g the measurements can be considered precise. Precision has nothing
to do with the true or accepted value of a measurement, so it is quite possible to be very
precise and totally inaccurate as in the example with an object’s mass.
In many cases, when precision is high and accuracy is low, the fault can lie with the
instrument. If a balance or a thermometer is not working co
ectly, they might
consistently give inaccurate answers, resulting in high precision and low accuracy. One
important distinction between accuracy and precision is that accuracy can be determined
y only one measurement, while precision can only be determined with multiple
measurements. A measurement can be accurate but not precise, precise but not accurate,
neither, or both. For example, if an experiment contains a systematic e
or*, then
increasing the sample size generally increases precision but does not improve accuracy.
Eliminating the systematic e
or improves accuracy but does not change precision.
Commonly the best estimate of the true value of the measured quantity x can be obtained
y calculating the mean (average) value ( x ) of some significant number of measurements
(N) of that quantity repeated using the same equipment and procedures:
N
x
x
N
i
i∑
== XXXXXXXXXX)
* Systematic e
ors: Commonly caused by a flaw in the experimental apparatus. For
example bad cali
ation in the instrumentation will give systematic e
or.
2
In such case the measure of the uncertainty (random e
or*) in the mean value will be the
standard deviation of the mean:
Nx
σ
σ = (2)
where
xσ is the standard deviation of the mean, σ – is the standard deviation and N – is
the number of measurements. The standard form for reporting a measurement of a
physical quantity x is:
xxx σ±= (3)
This statement expresses our confidence that the co
ect value of x probably lies in (or
close to) the range from
xxx σ−= to xxx σ+=
There are two rules that will be followed in this course for stating the final result: a)
uncertainty should be rounded to one sig.fig. unless that sig. fig. equals one; if the very
first significant figure is one than the following digit should be reported; b) The last
sig.fig. in the mean value should be in the same decimal place as the first significant
figure in uncertainty. For example: l = XXXXXXXXXX ± .006) mm
To estimate the accuracy of the final result the percent discrepancy or percent difference
is calculated.
%100*%
exp
theo
ertheo
x
xx
disc
−
= XXXXXXXXXX)
%100*%
2exp1exp
ave
ere
x
xx
diffe
−
= (5)
A measure of the accuracy can only be determined if some prior knowledge of the true
value is available.
The precision of the measurements shows the quality of the measurements and can be
expressed as the relative uncertainty (the standard deviation of the mean divided by the
mean value):
elative uncertainty %100*
x
xσ= (6)
* Random or statistical e
or: Class of e
ors produced by unpredictable or unknown
variations in the measuring process. The effect of the random e
or may be reduced by
epetition of the experiment.
3
The mean value of the physical quantity, as well as the standard deviation of the mean,
can be evaluated after a relatively large number of independent similar measurements
have been ca
ied out. These measured quantities in many cases serve as a basis for the
calculation of other physical quantities of interest. In such situations the uncertainties
associated with the directly measured quantities affect the overall uncertainty in the final
quantity of interest. To estimate the end uncertainty we use the rules of e
or propagation.
For example, the volume of a box is V = LWH. If we measure the width W, height H, and
length L many times to establish e
ors for each quantity: ∆ W,∆ H,∆ L, what will the
e
or in the volume ∆ V be when we multiply the average values of the three quantities
together? Or, in the other words, how the e
or of volume propagates.
In this lab e
or analysis will be practiced by measuring volumes of a few regular shapes
objects. The partial derivatives approach will be used through the course to propagate the
e
or:
XXXXXXXXXX)
where x and y are the measured quantities with the standard deviation of the mean (also
called standard e
ors)Δx and Δy.
Objectives:
• to practice e
or propagation concept and reporting experimental result in the
co
ect format.
Equipment: hollow cylinder, vernier caliper.
Procedure:
A student was asked to use Vernier caliper to measure the necessary dimensions of
hollow cylinder, bullet-shapes object, 3D rectangular wood object to calculate the volume
of each object and its e
or. The Vernier caliper is precise to 0.05 mm, so many of the
measurements gave the same results. Each dimension was measured 6 times. The
collected data is provided in table 1, table 2 and table 3.
Table Hollow cylinder
Height, mm Thickness, mm Inner diameter, mm
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
Based on the acquired data, use Logger Pro to find the mean value of each dimension and
the standard deviation. Using the above measurements to calculate the standard e
or
(standard deviation of the mean) for each dimension1; the mean value of volume for
22
2
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
Δ
∂
∂
+⎟
⎠
⎞
⎜
⎝
⎛ Δ
∂
∂
=Δ y
y
fx
x
ff
4
hollow cylinder and its standard e
or. The final results should be reported in
centimeters, using the co
ect format.
1How to find the mean value of each dimension and the standard deviation of the
mean, using the Logger Pro:
Create columns in Logger Pro and label them appropriately. Copy the acquired data from
the table and paste it to Logger Pro.
Hint: The default file of Logger Pro consists of a two-column table and a graph area.
Double click on the column name to change its name. Enter the co
esponding name and
units for the measured quantity in the appropriate box. Create in this way columns for
each dimension of the object. Name data set by the name of the object.
Use the statistics feature of Logger Pro to find the mean values for each dimension as
well as the standard deviation.
Hint: To get the statistical values (mean, standard deviation etc.) for a set of data you
need to put the same data on both axes of the graph and next press STAT button on
the tool bar. Unless noted otherwise when you present any data in a graph form avoid
lines connecting the data points → in Logger Pro you change graph features by double-
click on the graph window. To add a new graph: INSERT→GRAPH.
To paste the data for the other object add manual columns; paste data and show its
statistics.
Calculate standard deviation of the mean for each of the measured dimensions using
equation (2). Use those values to propagate the e
or in volume with the partial derivative
approach.
Complete the provided Template “E
or Propagation exercise” to submit your work for
grading.
Template_E
or Propagation Exercise
(9 extra points toward final grade)
Template_E
or Propagation Exercise
(9 extra points toward final grade)
At the top of the first page (1 point):
Title of the exercise; Your name (prominent); Date of the Experiment; TA’s name; Lab Section number.
Beginning of the new page:
Experimental Data section (1points)
The raw data that will be used for further calculations in Data Analysis should be presented in the table below.
Hollow cylinde
Dimension
Mean value, unit
Stand. Deviation, unit
E
or (stan.dev.of the mean Logger Pro), unit
(Add/delete rows as needed)
Important: All logger pro graphs need to be attached at the end of the lab report. The graphs can also be inserted as a picture under the tables.
Data analysis section (4 points)
1. Volume calculations and its E
or Propagation
1.1 The equation that was used to calculate volume of the Hollow cylinder is:
Equation:
Calculations:
1.2 The standard deviations