Physics 1140 Summer 2011
Lecture #5
Justification of the Average/Mean as the best estimate.
Section 5.5 in Taylor Page 137.
The Normal Distribution represents an infinite data set in which the average/mean could be calculated and this
would be the “True Value” of the average/mean. Of course it would never be possible for us to make an infinite
number of measurements to then arrive at the “True Value” of what we were measuring. We can only make a
finite (5, 10, 20, 100…) number of measurements, and the goal is to find the number within this data set that
gives the largest probability of being measured.
Let’s suppose we know the “True Value”, X, and we make a set of measurments (x 1, … ,xN) and want to find the
probability of finding a measurement, x, between the small interval x 1 + dx1. The probability would be as follows:
Pr ob(x) =
2
2
1
e-( x1-X ) /2s dx1 .
s 2p
What we are really concerned with is the probability of making the measurements in our data set (x 1, …, xN) so we
simplify the equation above and consider a very small interval, dx1, and obtain the following:
Pr ob(x1 ) µ
1
s
e-( x1-X ) /2s .
2
2
We can do this same thing with every measurement we make in our data set and get the total probability of
obtaining all of the measurements in our set by:
Pr obX,s (x1,..., x N ) = Pr ob(x1 ) ´ Pr ob(x2 ) ´... ´ Pr ob(x N )
Pr obX,s (x1,..., x N ) µ
1
sN
e
-
å( xi -X )2 /2s 2
This would give us the probability IF we knew the values of X and  (the standard deviation) but in general we do
not know these values. The probability above, however, allows us to find the most likely X and  given our data
set. We can guess the values for X and  and when we get the largest probability we have found the best
estimates for X and . The highest probability will occur when the summation in the exponent is at a minimum.
We can find the minimum of the exponent by taking the derivative and setting it equal to zero:
N
å(x - X) = 0
i
i=1
= (x1 + x2 +... + x N ) - N * X = 0
= (x1 + x2 +... + x N ) = N * X
N
å xi
(x1 + x2 +... + x N ) i=1
=X=
=
N
N
Here we get the best estimate for X, which is as close to the “True Value” as we can get with a finite data set.
Justification of the best estimate for the Standard Deviation.
Section 5.5 in Taylor Page 139.
Following from the justification of the average/mean we begin with the probability from above:
Pr obX,s (x1,..., x N ) = Pr ob(x1 ) ´ Pr ob(x2 ) ´... ´ Pr ob(x N )
Pr obX,s (x1,..., x N ) µ
1
sN
e
-
å( xi -X )2 /2s 2
If we take the derivative with respect to  and set it equal to zero we will find where the probability is at a
maximum with respect to the best estimate of the Standard Deviation.
PX,s (xi ) µ
1
s
e
2
-
å( xi -X )2 /2s 2
2
dP
-N å(xi - X) -å( xi -X )2 /2s 2
=(
+
)e
=0
ds s N+1
s N+3
Here we are interested in when the two terms in the parentheses add together to equal zero:
(
-N
å(x - X)
+
s N+1
å(x - X)
s N+3
2
i
s
=
N+3
å(x - X)
i
s2
s=
2
i
)=0
N
s N+1
2
=N
1
(å (xi - X)2
N
Notice that if we know the “True Value” of the average/mean, X, we obtain the first form of the Standard
Deviation as the best estimate. Since we do not know what the “True Value” of the average/mean, we must
replace X, with x and that changes the definition of the Standard Deviation a little as discussed before. The
Standard Deviation is then defined as:
s=
N
1
(å(xi - x )2 .
N -1 i=1
Justification of Addition in Quadrature:
A different look from Taylor:
Your textbook has a different description that what we are going to use. Like most things, there are multiple
ways to get to the same conclusion and neither is better than the next. I encourage you to read section 5.6 in
your book as Taylor justifies addition in quadrature using the normal distribution of measurements made in an
experiment.
We will look at addition in quadrature from a vector perspective, as you may be familiar with from your Intro
Physics classes. First lets look at vector addition:
The vector diagram to the right shows two
displacement vectors (x and y) and the
resultant displacement vector (c). The
displacements x and y are both independent of
each other. Meaning that moving along x does
not affect y in any way, and vice versa. The
vector addition is thus:
c=x+y
and in order to obtain the magnitude of c, we
get:
c
y
x
mag(c) = c 2 = (x + y)*(x + y)
= x 2 + y 2 + 2x * y
Since x and y are orthogonal (perpendicular) or independent to each other, the dot product (x * y) is equal to
zero and we get the familiar Pythagorean addition that we all know and love:
c = x 2 + y2 .
Again this is possible only because x and y are independent to each other.
A similar effect applies to uncertainties contributing to a sum or product. If they are independent of each other,
their net effect is obtained by addition in quadrature as we see above.
In a Sum:
Suppose a and b are some measured quantities. If many measurements are made then these will have some welldefined averages <a> and <b>. If ai and bi are the ith single samples of a and b, then we can define the fluctuations
ai andbi in these samples by:
ai =< a > +d ai
bi =< b > +d bi
where, by definition, the average of the fluctuations in ai and bi, <ai> and <bi>, are zero. The quantities of
< d ai2 >
and
< d bi2 >
are both always positive and are a measure of how big the fluctuations in a and b are.
Now we calculate the fluctuations in c = a + b. Treating the fluctuations like differentials, we have:
dci = d ai + dbi .
Now if we find the average in the fluctuations of c, we obtain the following:
< dci >=< d ai > + < dbi >= 0
because the average of the fluctuations in a and b are equal to zero. This shows us that the average of the
fluctuations in c really does not tell us much about the magnitude of the ci’s. So instead we calculate the
magnitude in the following way:
< dci2 >= (< d ai > + < d bi >)*(< d ai > + < d bi >)
=< d ai2 > +2 < d ai >< d bi > + < d bi2 >
Since <ai><bi> = 0, we obtain the following:
< dci2 >=< d ai2 > + < dbi2 >
and by taking the square root of both sides, we find that independent uncertainties in a sum add together in
quadrature:
< dci2 = < d ai2 > + < dbi2 > .
This is also holds true for the case of subtraction.
In a Product:
Suppose this time c = ab. Then using the chain rule for differentials:
dc = adb + bda
< d ci >=< a >< d bi > + < b >< d ai >
.
< d ci2 >=< a 2 >< d bi2 > +2 < a >< b >< d ai >< d bi > + < b 2 >< d ai2 >
We know that <ai><bi> is equal to zero so the middle term above disappears and we are left with:
< dci2 >=< a2 >< dbi2 > + < b2 >< d ai2 > .
By substituting c = ab and dividing:
<(
dci
<c>
)2 >=< (
d ai
<a>
)2 > + < (
d bi
<b>
)2 >
And we find that Fractional uncertainties in a product add in quadrature:
<(
d ci
<c>
)2 > = < (
d ai
<a>
)2 > + < (
d bi
<b>
)2 > .
This also holds true for quotients. Think of quotients as just multiplying by a fraction.
The General Case:
Now for the general case, suppose c = f(a, b) where f(a, b) is some general function with variables a, and b. Then:
é ¶f ù
é ¶f ù
d ai + ê úd bi
ú
ë ¶a û
ë ¶b û
dci = ê
And following the same logic as before we find the magnitude of <ci2> and we find the following:
é ¶f ù
é ¶f ù
< d c >= ê ú < d ai2 > +ê ú < d bi2 >
ë ¶a û
ë ¶b û
2
2
2
i
é ¶f ù
é ¶f ù
< d c > = ê ú < d ai2 > +ê ú < d bi2 >
ë ¶a û
ë ¶b û
2
2
2
i
This all depends on the fact that the uncertainties measured in an experiment are independent of each other.
Justification of the Standard Deviation of the Mean.
Taylor section 5.7 page 147.
Suppose we make many measurements of x, (x1, … ,xN) and determine the average/mean for these measurements.
Now suppose we do this again many more times using the function for the average/mean that we are used to:
x=
x1 +... + x N
.
N
The standard deviation for this data set is found in the normal way also:
1 N
sx =
(xi - x )2
å
N i=1
which is the uncertainty on any individual measurement made of the same quantity.
Since all of the our measurements that are made represent the “True Value” X then our averages/means are all
centered around the “True Value” X. Thus using the formula for the average/mean we find that:
X +... + X
=X
N
The average/mean of the true value is the true value.
So our many measurements of the average/mean will be centered on the “True Value” X, and the only thing left is
to find the standard deviation of all of these average/mean measurements. We use the general formula for
adding uncertainties in quadrature as follows:
In this case our function is simply
x=
x1 +... + x N
N
So the Error on the Mean (Standard Deviation of the Mean, etc…) is given by:
2
æ ¶x
ö
æ ¶x ö
s x = ç s x ÷ +... + ç
sx÷ .
è ¶x1 ø
è ¶x N ø
2
Since the function is just the definition of the average/mean, and all of the standard deviations are the same:
¶x
¶x 1
= ... =
=
¶x1
¶x1 N
æ 1 ö
Ns x2
2÷
èN ø
sx = ç
sx =
sx
N
Acceptability of a Measured Answer.
Taylor section 5.8 page 149.
It is often said that the Theory of Error Analysis is really the Theology of Error Analysis. Meaning that what
ever someone comes up with that seems reasonable for the error on a measurement is ok. No where is this more
evident than when we talk about the acceptability of a measurement.
For example: a scientist working at the National Institute of Standards and Technology (NIST) makes a
measurement of the acceleration of gravity here in Boulder. The expected value of gravity here in Boulder is
9.79 m/sec2 and the scientist makes a measurement of 9.89 m/sec 2 with a Standard Deviation of the Mean of
+ 0.05 m/sec2. Comparing this measurement to the expected:
t=
gmeasured - gexpected
s
=2
This means that our scientist has made a measurement that is two times the error on her measurement from the
expected value. The question is… Is this a reasonable measurement? Well the probability of making a
measurement outside of 2 is as follows:
From Appendix A (using the value of t above)
Pr ob(outsid2s ) =100 - Pr ob(inside2s )
=100 - 95.45 = 4.55%
This means that the scientist has a 4.55% chance of making this measurement, not very good.
For a scientist at the National Institute of Standards and Technology we would expect a much better agreement
with the expected value.
On the other hand, an Astrophysicist that is measuring the gravity on a far away planet where the measurements
have very large uncertainties, an agreement of 2 may be extremely good.
So the question still remains… What is considered a “Good” measurement? We will follow the rules that your
textbook lays out. The 5% threshold is where we will draw the line. So if a number you measure is more than 1.9
from the expected value, then a mistake has occurred in the lab and you should go back and see what the problem
is. Generally anything less than 90% agreement with the expected value should be suspect in our lab.