David M Harrison Dept of Physics Univ of Toronto May 2014 1 A Perhaps Apocryphal Story In the early 1800s Gauss graduate students were doing astronomical measurements When they repeated the measurements they didnt give exactly the same values ID: 400172
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Slide1
A Review of Bell-Shaped Curves
David M. Harrison, Dept. of Physics, Univ. of Toronto, May 2014
1Slide2
A Perhaps Apocryphal Story
In the early 1800’s Gauss’ “graduate students” were doing astronomical measurementsWhen they repeated the measurements, they didn’t give exactly the same values
Gauss said they were incompetent, and stormed into the observatory to show them how it should be done
Gauss’ repeated measurements didn’t give exactly the same values either!
2Slide3
Final Exam Marks for
PHY131 – Summer 2012
The red curve
n
max
=
maximum value
= value of
m
for which n(m) = nmax = standard deviation
Fit result
3Slide4
Another Approximately Bell-Shaped
Curve:
a Quincunx
Bell-shaped
curve aka
Gaussian aka
Normal distribution
The Gaussian describes the probability that a particular ball will land at a particular position: it is a probability distribution function.
4Slide5
Another Approximately Bell-Shaped
Curve:
a Quincunx
5
For a finite number
n
of balls, their distribution is only approximately Gaussian
If you use balls their distribution will be:
A perfect Gaussian shape
Still only approximately GaussianSlide6
Repeat of an Earlier Slide:
Another Bell-Shaped
C
urve: a Quincunx
Bell-shaped curve
Gaussian
Normal distribution
The Gaussian describes the probability that a particular ball will land at a particular position: it is a probability distribution function.
6
approximatelySlide7
The Standard Deviation
is a Measure of the Width of the Gaussian
7
All
probability distribution functions must have a total area under them of exactly 1
These two curves are properly normalised:
the area under each is = 1Slide8
The Standard Deviation
is a Measure of the Width of the Gaussian
8
Physical scientists tend to
characterise
the width of a distribution by the standard deviation. Social scientists instead often use the
variance
.Slide9
The Shaded Area Under the
Curve Has an Area = 0.68
9
If you choose one measurement of
d
i
at random, the probability that it is within of the true value is:
A. 0 B. 68% C. 95% D. 99% E. 100%
is the
standard uncertainty u
in each individual measurement
diSlide10
Characterising
Repeated Measurements as a Gaussian is Almost Always Only an Approximation
A true Gaussian only approaches zero at
If the number of measurements
random fluctuations mean that the measured values can be too high, or too low, or too scattered, or not scattered enough
Therefore, we may only estimate the mean and the standard deviation
10