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Experiment 1
Objective: To explain the importance of reporting both a measurement and its error and to address how to properly treat errors in the lab.
DISCUSSION:
Our understanding of nature is based on experimental observation. In many cases, useful observations consist in finding numerical values of physical quantities, such as temperature, mass, electric current, etc. Accurate numerical values of physical quantities are required to establish the validity of physical laws. To determine these numerical values as accurately as possible, measurements are frequently made in the controlled environment of a laboratory. However, strictly speaking, no measurement is ever "exact", i.e. all measurements have some uncertainty associated with them. Thus all measurements consist of two numbers: the value of the measured quantity and its error .
The error reflects the reliability of the measurement. The range of measurement errors varies widely: some quantities, such as the mass of the electron m e
= (9.1093897 ±
0.0000054) x 10
-31
kg, are known to better than one part per million, while other quantities, such as the age of the Earth, T
E
, are only loosely "bounded" i.e. 4 < T
E
< 6 billion years.
Significant Figures:
The electron mass above has eight significant figures (or digits). However, the
Earth's age has only one significant figure and so it is misleading to write the age of the
Earth with more than one figure of accuracy. The number of significant figures reported should be consistent with the measurement accuracy. In general, errors are usually quoted with no more significant figures than the measured result; and the last significant figure of a result should match that of the uncertainty. For example, a measurement of the acceleration due to gravity on the surface of the Earth might be given as g = 9.7±1.2 m/s
2
or g = 9.9±0.5 m/s
2
but not as g = 9.7±1.25 m/s
2
or g = 9.92±0.5 m/s
2
. In the last two cases, the last significant figure of the result and error do not match.
To minimize inaccuracies in calculations due to roundoff, you should generally keep one additional significant figure than is warranted by the errors in each number.
When multiplying or dividing two numbers that have the same number of significant figures but differ in magnitude (e.g. 125 × 5.25), the final results should be quoted to the same number of significant figures (656). When adding and subtracting a two numbers, the lowest number of decimal places is used to specify the number of significant figures.
(10.3 + 11.256 = 21.6) Work out the following examples in your lab write-up:
1.
The length of the base of a large window is measured in two steps. The first section has a length of 
1
=1.22 m and the length of the second section is 
2
=0.7 m. What is the total length of the base of the window?
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2.
A student going to lunch walks a distance of x = 102 m in t = 88.645 s. What is the student's average speed?
Types of errors
A systematic error occurs when all of the individual measurements of a quantity are biased by the same amount. These errors can arise from an incorrect calibration of instruments or by a change in experimental conditions such as slow reflexes on a stopwatch or always holding one's head too far to the left when reading a meter needle with parallax.
Random errors occur when the result of repeated measurements vary due to truly random processes. For example, random errors occur due to small fluctuations in experimental conditions or due to variations in the stability of measurement equipment.
These errors can be estimated from the distribution of values in repeated measurements.
Mistakes can be made in any experiment, either in making the measurements or in calculating the results. Such "human" or "operator" errors are sometimes discovered by repeated measurements but these blunders and major systematic errors can only be avoided by a thoughtful and careful approach to the experiment.
Error Estimates
By eye or reason : Measurement error can often be reasonably estimated from properties of the measurement equipment. For example, the error in the measured length of a sharp edged object using a meter stick with marks every millimeter is likely to be about one quarter of a division on the meter stick, or ±0.25 mm. For an irregular-edged object, the properties of its edges may limit the determination of its length to ±2 mm.
Such an estimate may be quite adequate, especially if the error in this particular quantity does not dominate the overall error in the experiment.
By repeated observation : If a quantity x is measured repeatedly, then the average or mean value of the set of measurements is generally adopted as the "result". If the errors are random, the error in the mean can be derived from the variation in the set of observations. Shortly we will discuss how this is done. (Oddly enough, truly random errors are the easiest to deal with.)
Useful definitions
Here we define some useful terms (with examples) and discuss how errors are reported in the lab.
Absolute error: This is the magnitude of the error assigned to a measured physical quantity. It has the same units as the measured quantity.
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Example 1. Suppose we need 330 ml of methanol to use as a solvent for a chemical dye in an experiment. We measure the volume using a 500 ml graduated cylinder that has markings every 25 ml. A reasonable estimate for the error in our measurements is ½ of the smallest division. Thus we assign an absolute error to our measurement of

V = 12 ml. Hence we state the volume of the solvent (before mixing) as V = 330±12 ml.
Fractional error: This is the ratio of the absolute error and the value of the measured quantity. It has no units, that is, it is dimensionless. It is also called the relative error or, when appropriate, the percent error.
Example 2. In the example above the fractional error is:

V
V
 12 ml
330 ml

0 .
0303

3 .
03 %
(1)
Treating Random Errors Arising From Repeated Observation
The error assigned to a physical quantity that is the result of a large set of repeated measurements can be obtained from the ideas of statistics. An initial example will help illustrate the important ideas.
Suppose we want to determine the mass of a standard U.S. penny. We measure the mass of a single penny many times using a balance and interpolate between divisions by eye.
The results of 17 measurements on the same penny are summarized in Table 1.
The mean value ¯ ,m of the measurements is defined to be m

1
N i
N


1 m i
 1
17
 m
1
 m
2
 m
3
 m
17


2 .
518 g where the symbol

denotes a sum over all the measurements i = 1 ,. . . , N and the total
(2) number of measurement is N = 17. The deviation d i of the i th measurement m i
from the mean value
¯ ,m is defined to be
(3) d i
 m i
 m
The standard deviation of the measurements
 m is defined to be
 m
 i
N


1
 m i

N
 m

1


2
 i
N


1
 m i
 m

2
16

0 .
061 g
(4)
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Table 1. Data recorded measuring the mass of a US penny.
11
12
13
14
7
8
9
10
Number
(i)
1
2
3
4
5
6
15
16
17
Mass of a US Penny
Mass (g) m i
2.43
Deviation (g) d i
=m i
-¯ ,m
-0.09
2.46
2.52
2.40
2.58
2.61
2.49
2.52
2.46
2.49
2.49
2.58
2.52
2.55
2.52
2.64
2.55
-0.06
0.00
-0.12
0.06
0.09
-0.03
0.00
-0.06
-0.03
-0.03
0.06
0.00
0.03
0.00
0.12
0.03 i
17 

1 m i m


42 .
81 g
2 .
52 g
Squared Deviation (g 2 ) d i
2
0.0081
0.0009
0.0009
0.0036
0.0000
0.0009
0.0000
0.0144
0.0009
0.0036
0.0000
0.0144
0.0036
0.0081
0.0009
0.0000
0.0036
17
 i

1 d i
2 
0 .
0594 g
2
Fig. 1 shows a histogram plot of the data on the mass of a U. S. penny. Also on the graph is a plot of a smooth, bell-shaped curve that represents what the distribution of measured values would look like if we took many, many measurements. The result of a large set of repeated measurements subject only to small random errors will always approached a limiting distribution called the normal or Gaussian distribution. The larger the number of measurements, the closer the data will approach the normal distribution.
This ideal curve has the mathematical form:
Number(m)

Δm
N
2
π e

 m
 m
 2
2
Δm
2
(5) where N is the total number of measurements.
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Figure 1. The Gaussian or normal distribution for the mass of a penny N=17, ¯ ,m =2.518 g,
 m=0.061 g.
The normal distribution is symmetrical about
¯ ,m
. It can be shown that 68% of the measurements fall within one standard deviation (±  m ) of the mean value
¯ ,m
.
This implies that 68% of the time the result of an individual measurement of m would be within ±  m of the mean value
¯ ,m
. For random errors, the standard deviation
 m is the experimental error for an individual measurement of m . In other words, the standard deviation given in Eq. (4) is the uncertainty associated with a single measurement of m .
On a practical basis, this means that if we were to make a single measurement of the mass of a penny there is a 68% chance that it will be in the range of
¯ ,m ±  m .
To summarize, if we make N measurements of the mass of a penny, we obtain a distribution that looks like the bell-shaped curve in Fig. 1. The most probable value for the penny's mass is our measured mean value
¯ ,m
. How close is this mean value to the
"true" value of the penny's mass? To answer this we could take another set of N measurements of the penny's mass and obtain a second value, then do this a third time and so on. After ten repetitions of this process, we could then plot our 10 values for the mean ¯ ,m . We could then compute an overall mean and a standard deviation of the means about the mean of the means! This is one experiment we will not carry out since the result has been predicted by statistics and confirmed by others . The standard deviation of the means
 ¯ ,m can be obtained from the standard deviation of a single set of N measurements. The standard deviation in the mean, or error, or uncertainty is defined to be
Δ m

Δm
N
Thus from a set of N measurements, our result is
Mass of penny
 m
 Δ m
 m

Δm
N

2 .
518 g

0 .
061
17 g

2 .
518

0 .
015 g
(6)
(7)
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Thus, if our experiment is subject to random errors in an individual measurement of
 m , we can improve the precision of that measurement by doing it repeatedly and taking the mean of those results. Note, however, that it eventually becomes difficult to make progress because the precision improves only as
1 N
so that to improve by a factor of say, 10, we have to make 100 times as many measurements. We also have to be careful in trying to get better results by letting
N→∞,
because the overall accuracy of our measurements may be limited by systematic errors, which tend not to cancel out the way random errors do.
Combination and propagation of random errors
To obtain a final result for an experiment, we have to measure a variety of quantities (length, time, temperature, etc.) and combine them according to some mathematical formula to obtain the final result (acceleration of gravity, specific heat of a metal, etc.). Therefore we are interested in how the errors in individual quantities combine to produce the error in the final result. Estimating the error in a final result (e.g. a = acceleration) from the measured quantities used to find the result (e.g. a
 2 s t
2 measurements of s = distance and t = time) is known as the propagation of errors .
from
Independence of errors to be combined
For all these formulae, it is important that the quantities being combined are the results of truly independent measurements and that the error
 x assigned to quantity x not be related to the error
 t assigned to quantity t . For example, we may measure the speed of an object by measuring a distance (using a meter stick) and the time it takes to traverse that distance (using a clock). The measurement of time and distance can be truly independent as they are done with two measurement devices and there is no reason to think that if the time measurement is too large, then the distance measurement is also too large.
Error estimates for independent random errors
Here we summarize a number of common cases. For the most part these should take care of what you need to know about how to combine errors.
Errors in sums and differences : If several quantities x , y , z , are measured with independent, random errors
 ¯,x ,  ¯,y ,  ¯,z then the error in ¯,Q where ¯,Q =
¯,x ± ¯,y
± ¯,z
is
(8) 
Q
  x
2   y
2   z
2
In other words, the random errors add as the square root of the sum of the squares, whether the terms are all added, subtracted, or some combination of the two.
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Errors in products and quotients : If several quantities x , y , z , are measured with independent, random errors
 ¯,x
,
 ¯,y
,
 ¯,z then the fractional error in ¯,Q where x
 y
Q
 z
(or any other combination of multiplication and division) is
Δ
Q
Q



Δ x


2 x

Δ y y
2



Δ z


2 z
(9)
(10)
In other words, the fractional uncertainties combine as the square root of the sum of the squares of the individual fractional errors in the component terms.
EXERCISES:
1.
a. Using the vernier calipers 1 , make a set of measurements to determine the surface area of the sphere. b.
Record the mean value of the set of measurements (do at least 10) and also compute the error in the surface area.
2.
a. In the book of Genesis (Chapter 6) it is recorded that God told Noah to build an ark (a large boat) of the following dimensions "The length of the ark shall be
300 cubits, its breadth 50 cubits, and its height 30 cubits". A cubit is the length of the forearm from the elbow to the tip of the middle finger. First determine the mean length of a cubit by measuring the appropriate length on 10 students in the lab. b.
Compute the mean value of a cubit (and its error) from your distribution. What is your best estimate of the volume of Noah's Ark? What systematic errors might contribute to your estimate of the ark's volume?
1 See Appendix A for instructions on how to read the calipers. Ask your TA for further clarification.
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