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Lecture 4, PHYS1140
Statistical uncertainty.
Given a set of repeated measurements of the same quantity, with each measurement having some
random error, what is the best estimate of that quantity and what is the uncertainty?
Measure same quantity (x) several times (N trials). Get results: xi = x1, x2, x3 … xN
What is the best estimate of the true value of the quantity?
x best
Answer:


x avg
What is the uncertainty in xbest?
x

x
i
N
i
(IF all measurements equally good.)
x best  ??
Answer (all to be justified later!):
x best



x avg

mean

N

 x
standard deviation
=

i
 x
standard deviation of the mean = SDOM
2
i
N 1
If you make a histogram of the results of the repeated measurements, you almost always find that the
distribution of results has a special shape, called the “gaussian” or “normal” distribution.
frequency

x
2
The standard deviation  is a measure of the width of this distribution. You can think of  as the
uncertainty of a single measurement of x. If you make one measurement of x (call it x1) then the
uncertainty in x1 is  That is, after 1 measurement, your best estimate of the true value of x is
x = x1 ± s . Generally, the standard deviation  is unknown unless you make lots of measurements
and then compute it with the formula above.
frequency
frequency
less precise measurement:
larger 
x
more precise measurement:
smaller 
x
For a given experiment, the standard deviation  does not change as you take more and more
measurements.  is a measure of the uncertainty of the measurement apparatus, and that only depends
on the apparatus, not on how many time you use the apparatus.

is the uncertainty in the average (or mean)
N
value of all N trials. After N trials, the best estimate of the true value of x is x = x ± s mean Notice that
mean gets smaller and smaller as the number of trials increases. You get a more precise average when
you average lots of trials.
The standard deviation of the mean mean

The exact shape of the gaussian or normal distribution is given by
f x,s (x) =
2
1
- (x- x ) /(2s 2 )
e
s 2p
The values of the mean, x , and the standard deviation, , depend on the details of the experiment: x
depends on what’s being measured and  depends on the measurement apparatus. But the shape of the
curve is universal. It can be shown that this shape will always be found if the error in the measurements
is due to the net effect of a large number of smaller, uncorrelated errors. (This is called the “Central
Limit Theorem” in statistics – to be discussed later.)
3
A subtle point about the formula for the standard deviation: The underlying distribution function f(x)
has some “true” mean and “true” standard deviation. The formula 
estimate of the “true” value of . Just as x

x
 x

i
 x
i
N 1
2
is our best
i
i
N
is our best estimate of the “true” value of x.
+¥
This function f(x) is “normalized” to 1, meaning
ò f (x)dx
= 1
- ¥
f(x)
area under part between x +  = 0.68
total area under curve = 1
x- x
x+
x
The normal function f gives the probability distribution of a measurement of x:
f(x)
x1
x2
x
x2
Probability(single measurement gives value in the range x1 < x < x2)
=
ò f (x) dx
x1
Note that mean (standard deviation of the mean is always) less than standard deviation . Uncertainty in
average of several measurements is always less than the uncertainty of a single measurement.
4
Proof that uncertainty of average =
s
N
:
Regard xavg as a function of (x1 , x2, x2, ... xN ): x

x
i
i
N

1
 x1  x 2  x 3  ...  x N 
N
Apply master formula for error propagation:
2
2
2
æ¶ x
ö
æ¶ x
ö
æ
ö
¶
x
÷
÷
çç
çç
÷
÷
÷
dx = ççç
dx1 ÷
+
d
x
+
K
+
d
x
N÷
÷
÷
çè¶ x
÷ èç¶ x 2 2 ø
÷
÷
è¶ x 1
ø
ø
N
¶x
1
=
for any i.
¶ xi N
So
dx =
1
dx12 + dx 22 + K + dx2N
N
=
1
s2 + s2 + K + s2
N
Remember:  is the uncertainty in one measurement, and all measurements are assumed equally good,
so dx1 = dx2 = ... = dx N = s . Notice there are N terms are under the square-root sign, so
dx =
1
N s2
N
=
s
N
5
PHYS1140 Males, N = 588 trials
avg = 1.060, stdev=0.085, SDOM=0.003
Frequency
120
100
80
60
40
20
1.27
1.21
1.15
1.09
1.03
0.97
0.91
0.85
0.79
0.73
0
normalized time (1.00 = perfect)
PHYS1140 Females, N = 177 trials
avg = 1.058, stdev=0.092, SDOM=0.007
Frequency
35
30
25
20
15
10
5
1.27
1.21
1.15
1.09
1.03
0.97
0.91
0.85
0.79
0.73
0
normalized time(1.00 = perfect)
Results of time trials:
All time measurements were normalized to 1. If student estimated 12.6 s, and correct time was 11.7 s,
then student normalized estimate is 12.6 s/11.7 s = 1.08
Men: average normalized time = tavg = 1.060 + 0.003
Women: tavg = 1.058 + 0.007 (larger uncertainty due only to smaller N)
Men and women have same tavg within uncertainties.
6
Mathcad plot of (normalized) frequency of male times, and gaussian function (solid curve) with same
mean and same standard deviation.
Notice!

Same systematic error in both men and women results: everyone estimated about 6% too
high (Do you know why?)

Both men and women have the same mean and same standard deviation. (No gender
difference in estimating times)

Smaller number (N) of trials for women, so larger mean (SDOM) for the women.

Bigger number (N) of trials for men, so have better statistics, better fit to normal distribution.
Bigger number of trials gives smaller random fluctuations away from normal distribution.