Physics 1140 Summer 2011
Lecture #3
Uncertainties in Products and Quotients.
Equation 3.18 in Taylor page 61.
Suppose that x, . . . , w are measured with uncertainties x, . . . , w, and the measured values are used to
compute:
If the uncertainties in x, . . . , w are and independent and random, then the fractional uncertainty in q is the sum
in quadrature of the original fractional uncertainties,
In any case, it is never larger than their ordinary sum,
Uncertainty in a Power.
Equation 3.26 in Taylor page 66.
If x is measured with uncertainty x and is used to calculate the power q = xn (where n is a fixed known number),
then the fractional uncertainty in q is:
q
q
n
x
x
.
Uncertainty in Any Function of One Variable.

Equation 3.23 in Taylor Page 65.
If x is measured with uncertainty x and is used to calculate the function q(x), then the uncertainty q is:
Uncertainty in a Function of Several Variables. (THE MASTER RULE!)
Equation 3.47 in Taylor page 75.
Suppose that x, . . . , z are measured with uncertainties x, . . . ,z and the measured values are used to compute
the function q(x, . . . , z). If the uncertainties in x, . . . , z are independent and random, then the uncertainty in q
is
q  (
q 2
q
x)  ... ( z) 2 .
x
z
In any case, it is never larger than the ordinary sum

q 
q
q
x  ... z.
x
z
NOTE: Remember equation 3.47 can always be used to propagate the uncertainty in a function due to
random and independent uncertainties.

Examples:
A = 10.1 + 0.2 m
B = 15.91 + 0.04 m/sec
C = 1.04 + 0.02 m/sec
D. 2.3 + 0.2 sec
Do the following calculations along with the error propagation:
A 10.1m
m

 4.39
D 2.3sec
sec
v
v
1
A
1
10.1m
2
v  ( A) 2  ( D) 2  ( A) 2  ( 2 D) 2  (
* 0.2m) 2  (
2 * 0.2sec)
A
D
D
D
2.3m
(2.3sec)
m
v  0.392
sec
m
Final : v  4.4  0.4
sec
v
1.

15.91
m
m
1.04
sec
sec
2.3sec
B C

D
m
a  6.465 2
sec
a
a
a
1C
B 1
B C
a  ( B) 2  ( C) 2  ( D) 2  (
B) 2  (
C) 2  ( 2 D) 2
B
C
D
D
D
D
a
2.
1 1.04
15.91 1
15.91 1.04
* 0.04) 2  (
* 0.02) 2  (
* 0.2) 2
2.3
2.3
(2.3) 2
a  (
a  0.577
m
sec 2
Final : a  6.5  0.6

m
sec 2
A
10.1m
m
m
 2(B  C) 
 2(15.91
1.04
)
D
2.3sec
sec
sec
m
F  41.7
sec
F
F
F
F
F  ( A) 2  ( D) 2  ( B) 2  ( C) 2
A
D
B
C
F
3.
1
A
 ( A) 2  ( 2 D) 2  (2B) 2  (2CC) 2
D
D
1
10.1
* 0.2) 2  ( 2 * 0.2) 2  (2 * 0.04) 2  (2 * 0.02) 2
2.3
2.3
m
F  0.887
sec
 (
Final : F  41.7  0.9
m
sec
Each of these calculations can be done using the uncertainty equations specifically for multiplication/division, and
addition/subtraction (equations 3.16 and 3.18 in Taylor) and will result in the same uncertainty values as above.
You should do this as an exercise.
Random vs. Systematic Errors
Random Errors: Also called stochastic, or uncorrelated errors. These errors are just as likely to make the
measurement larger (+) than it really is, or smaller (-) than it really is. These can be made while interpolating the
measurement of length between two millimeter marks. You are just as likely to estimate the measurement high,
as you are to estimate it low. Taking many measurements of the same value can reduce these errors.
Systematic Errors: These are errors that do not reduce by taking many measurements and show up by skewing
all of the data either too high or too low from the actual measurements. For example; voltage data could be
skewed if there is an initial offset voltage that is not accounted for when making the measurement. By not
subtracting the offset voltage, the measurements will be skewed too high. Systematic Errors can be very
problematic if the source of the error is unknown.
Random Error: Small
Systematic Error: Small
Random Error: Large
Systematic Error: Small
Random Error: Small
Systematic Error Large
Random Error: Large
Systematic Error: Large
In our lab we will work with Random Errors but not Systematic Errors.
Statistical Uncertainty
In this course, there are three general situations in which you need to determine the uncertainty of
measurements.
1. Uncertainty in a direct measurement. This uncertainty is estimated when making a direct measurement
of a quantity such as length with a meter stick, time with a stopwatch, mass with a scale, etc… All of
these measurements have some uncertainty associated with them that is dependent on the tool used to
make the measurement.
2. Propagation of error. In this case, a calculation f(x, y, …) is made using measured values that have known
uncertainties with them. The propagation of the uncertainty in this calculation is found by using the
“MASTER RULE”.
3. Statistical uncertainty. In this case you make a set of repeated measurements of the same quantity, and
each measurement has some random error. Question: what is the best estimate of that quantity and what
is the uncertainty? Let’s talk about this now:
Make measurements of the same quantity (x) several times (N trials). Get the following results:
xi = x1, x2, x3, … ,xN
The Average or Mean:
The best estimate of the true value of this quantity is found by finding the average (mean) value.
N
x best  x avg  x mean  x 


x
i1
N
i
Average/Mean: Equation 4.5 in Taylor page 98.
Standard Deviation:
It is natural to look at the difference or deviation of one measured value x i from the average value xavg. If this
deviation is small then xi is close to the average and is a precise measurement. If the deviation is large, then x i is
far from the average and is less precise.
di  x i  x avg
The deviation of a measurement from the average/mean.
Example: We take five measurements of the length of a piece of rope and find the following:
Trial #
1
2
3
4
5
Average: xavg = 4.8 m
Measurement of length (m)
4
3
7
6
4
Deviation (di)
-0.8
-1.8
2.2
1.2
-0.8
Deviations: di = xi - xavg
By calculating the deviations we can see how precise (or less precise) any individual measurement is. Generally we
would like to have the deviations be as small as possible but this depends on the instrument we use to make the
actual measurement. (Remember every measurement has some uncertainty)
To see how precise our average is, we might calculate the “average deviation” from the numbers above. When we
calculate the “average deviation” we find a problem as seen below:
davg




1 N
(0.8)  (1.8)  2.2 1.2  (0.8)
  di 
0
N i1
5
This does not give us much useful information.
Adding the deviations will give a zero answer so the average deviation becomes zero as well. This happens due to
the symmetry of the measurement taking. It is just as likely that we will make a measurement higher than the
average as it is to make a measurement that is lower than the average. So for any number of measurements we
make the “average deviation” as calculated above will result in zero. (See problem 4.4 in Taylor)
To get around this problem we can square the deviations, which will make them all positive numbers and we can
calculate what is called the “variance”.
1 N
  (di ) 2
N i1
2
The variance defined.
We are interested in what is called the “Standard Deviation” which is similar to the “average deviation” that we
tried to calculate above. To find the “Standard Deviation” we just take the square root of the variance to get:
1 N
1 N
2

(d )  N (x i  xavg )2
N i1 i
i1
The Standard Deviation or Root Mean Square (RMS) Deviation
The Standard Deviation gives us the average uncertainty of a single measurement within the set of
measurements used to calculate the average/mean. The definition above however, is not the definition that we
will use in class for the Standard Deviation. For theoretical reasons that are beyond the scope of this class we
will replace the N in the above equation with N-1. This results in:

1 N
(x i  x avg ) 2

N 1 i1
Standard Deviation: Equation 4.9 in Taylor Page 100.
This definition makes sense if you think of making only a single measurement. With the definition before, if you
make only one measurement the standard deviation would be as follows:
1 N
1 N
1 1
0
2
2

(di ) 
(x i  x avg )  (1 1) 2 


N i1
N i1
1 i1
1
This does not really make sense.
This result tells us that a single measurement has no uncertainty, but we know that is not correct. If we use the
“New and Improved” definition for the Standard Deviation we would get the following:


1 N
1 1
0
2
(x

x
)

(1 1) 2 


i
avg
N 1 i1
1 1 i1
0
This gives an undefined due to dividing by zero.
The “New and Improved” definition for the Standard Deviation gives us a result for a single measurement that
says that the uncertainty is undefined. This is more of what we would expect because the experimenter, limited
to the measurement apparatus, estimates the uncertainty of a single measurement.

Standard Deviation of the Mean: (Standard Error on the Mean)
The Standard Deviation above is the average uncertainty on a single measurement within the set of
measurements used to calculate the average/mean or best estimate of the true value. Since the average is found
through some combination of the many measurements made of the same quantity, it is reasonable to believe that
the average should be more reliable than any single measurement. Thus we define the Standard Deviation of the
Mean as follows:
mean 


Standard deviation of the Mean. Equation 4.14 in Taylor page 102.
N
Clicker Questions:
1.
Using the master rule, propagate the uncertainty for the following function given these measurements:
A = 10.1 + 0.2 m
B = 15.91 + 0.04 m/sec
C = 1.04 + 0.02 m/sec
D. 2.3 + 0.2 sec
F

A.
B.
C.
D.
A
D
B
2.91 + 0.4 sec
3.0 + 0.3 sec
2.9 + 0.4 sec
2.916 + 0.089 sec
2. A student measures g, the acceleration due to gravity, five times, with the following results (all in m/s2)
9.9, 9.6, 9.5, 9.7, 9.8
What is the mean and standard deviation of the data set?
A. xavg = 9.70
B. xavg = 9.7
C. xavg = 10
D. xavg = 9.7




= 0.158
= 0.2
= 0.2
= 0.158
3. A data set of 25 measurements has a Standard Deviation of the Mean (Standard Error on the Mean) of
2.5. How many measurements must be made to reduce the Standard Deviation of the Mean to, at most,
0.2?
A. 4000 measurements
B. 3906 measurements
C. 3910 measurements
D. 20000 measurements
E. All of the above will give the desired Standard Deviation of the Mean