Download Lecture 5

Measurement uncertainties
ASTR320
Monday September 12, 2016
Measurements
• Astronomy is a purely observational science—we can’t
do experiments, all we can do is make measurements
remotely
• Often astronomical research is done by looking for
correlations between measured quantities
• How correct the science is depends on how well the
quantities are measured
• Error analysis is very important…
Measurement errors
• Two types of errors:
– systematic errors that limit the precision and accuracy of results
in more or less well-defined ways that can be estimated by
considering how the measurement is made
– random errors introduced by random fluctuations in
measurements that yield different results each time an
experiment is repeated
• Errors are not the same as mistakes!
Accuracy vs. Precision
• Accuracy: a measure
of how close an
experimental result is
to the true value
• Precision: a measure
of how well the result
has been determined,
without reference to
its agreement with the
true value
Populations and distributions
• Parent distribution
– Hypothetical distribution of an infinitely large set of
measurements of a quantity
– Defines the probability of measuring a particular value in a single
measurement
• Sample population
– Actual set of experimental measurements made
– A subset of the parent distribution
• Consider a given dataset to be a sample taken from a
parent population
• When making measurements it is important to carefully
consider how this sample is constructed
– Error analysis provides information about how well the sample is
constructed and how reliable are the results
Mean, median, mode
• Mean
– Arithmetic average
𝑥≡
1
𝑁
𝑁
𝑥𝑖
𝑖=1
• Median
– Value midway in the
frequency distribution
– Half of the measurements
are greater than this value,
half are below
• Mode
– Value that occurs most
frequently in a distribution
– The most probable value
Deviations
• Deviation: the difference of a given measurement from
the mean of the parent distribution
𝑑𝑖 ≡ 𝑥𝑖 − 𝜇
• Average deviation: the average of the absolute values
of the deviations
1
𝛼 ≡ lim
𝑥𝑖 − 𝜇
𝑁→∞ 𝑁
• This is a measure of the dispersion of the measurements
about the mean
Variance of the parent population
• Standard deviation is a better (i.e. mathematically easier)
way to measure the dispersion of the measurements (i.e., the
uncertainty in the measurements)
• Variance: the limit of the average of the squares of the
deviations from the mean μ:
𝜎2
1
≡ lim
𝑁→∞ 𝑁
𝑥𝑖 − 𝜇
2
= lim
𝑁→∞
1
𝑁
𝑥𝑖 2 − 𝜇2
• Standard deviation, σ, is just the square root of the variance
– The uncertainty in determining the mean of the parent distribution is
proportional to the standard deviation of the distribution
Variance of the sample population
• The variance of the sample population is subtly different:
𝑆2
1
≡
𝑁−1
𝑥𝑖 − 𝑥
2
• This accounts for the fact that the average value has
been determined using the data in the sample population
and not independently
• Note that σ is often used when discussing standard
deviation, even though we really mean S
Mean and standard deviation
• Consider the parent population of measurements with
distribution p(x)
• The probability density p(x) is defined so that, in the limit
of large number of observations, the fraction of
measurements of x that yield values between x and x+dx
is given by dN=Np(x) dx
• Then by definition, the mean is the most probable value,
or expectation value, 𝑥 , of the measurements
• The standard deviation is the expectation value of the
deviations from the mean
𝜎 = (𝑥 − 𝜇)2
• The mean and standard deviation are often used to
characterize a set of measurements
Binomial distribution
• The probability distribution of the number of successes in
a sequence of n independent yes/no experiments, each
of which yields success with probability p
• We’ll use this general case to derive some more useful
distributions
Coin toss experiment
• Take a fair coin that has a 50% chance of landing heads
up and 50% chance of landing tails up
– If the coin is tossed an infinite number of times, the fraction of
times that it lands heads up is ½
– But in any given toss, it is impossible to know which way it lands
• Now take n of these coins and toss them all at once (or
toss one coin n times)
• P(x; n) is the probability of success, i.e., that x coins land
heads up (and (n-x) coins land tails up)
– In this case p=(1/2)n
– More generally, we can assign a probability to each ―success‖
case
Probability
• If p is the probability of success and q is the probability
of failure of n trials
– For the coin toss, 𝑝 𝑥 = (1/2)𝑥 ; 𝑞𝑛−𝑥 = (1 − 1/2)𝑛−𝑥 = (1/2)𝑛−𝑥
– For a die, p=1/6, q=1-1/6; so 𝑝 𝑥 𝑞𝑛−𝑥 = (1/6)𝑥 × (5/6)𝑛
• More generally, the probability of observing x of the n
items to be in the state with probability p is called the
―binomial distribution‖
Binomial distribution
• The mean of the binomial distribution comes from the
definition of μ:
1
𝜇 = lim
𝑁→∞ 𝑁
𝑁
𝑖=1
1
𝑥𝑖 = lim
𝑁→∞ 𝑁
𝑁
= lim
𝑁→∞
𝑁
𝑥𝑗 𝑁𝑃(𝑥𝑗 )
𝑗=1
𝑥𝑗 𝑃(𝑥𝑗 )
𝑗=1
• And the binomial distribution probability
𝑛!
𝑃𝐵 𝑥; 𝑛, 𝑝 =
𝑝𝑥 1 − 𝑝
𝑥! 𝑛 − 𝑥 !
• So the mean is
𝑛
𝑛!
𝜇=
𝑥
𝑝 𝑥 1 − 𝑝 𝑛−𝑥
𝑥! 𝑛 − 𝑥 !
𝑥=0
𝑛−𝑥
= 𝑛𝑝
Binomial distribution
• And the variance of a binomial distribution comes from
the definition of σ 2 :
𝑛
𝑛!
2
2
𝜎 =
(𝑥 − 𝜇)
𝑝 𝑥 (1 − 𝑝)𝑛−𝑥 = 𝑛𝑝(1 − 𝑝)
𝑥! 𝑛 − 𝑥 !
𝑥=0
• If 1-p is the probability of failure, q,
• Then for the coin toss, p=q=1/2, then the distribution is
symmetric about the mean, and the median is the most
probable value (and equal to the mean), and σ2 is equal
to μ/2.
• If p is not equal to q, then the distribution is asymmetric
Binomial distribution
• For the coin toss,
p=q=1/2, then the
distribution is symmetric
about the mean, the
median is the most
probable value (and
equal to the mean), and
σ2 is equal to μ/2.
• If p is not equal to q,
then the distribution is
asymmetric
Poisson distribution
• An approximation to
the binomial
distribution for the
special case when the
average number of
successes is much
smaller than the
possible number
• Not generally used in
astronomy, but useful,
e.g., for radioactive
decay
Gaussian (normal) distribution
• Most useful probability
distribution because it
best describes the
actual observed
statistics of
experimental data
• Bonus: it’s simple and
easy to use
• Derived from binomial
distribution for the case
that the number of
observations n
becomes infinitely large
and the probability p is
finitely large, so np>>1
Gaussian distribution
• Gaussian probability density is defined as
1
1 𝑥−𝜇 2
𝑝𝐺 =
𝑒𝑥𝑝 −
2
𝜎
𝜎 2𝜋
• Where μ and σ are the mean and standard deviation.
• Remember that the probability density function is defined
to be the probability dP that the value of a random
observation will fall within an interval dx around x.
• In a Gaussian distribution, the width of the curve is
determined by the value of the standard deviation, σ
• For x=μ+σ, the height of the curve is reduced to 𝑒 −1/2 of
its peak value
FWHM
• Often refer to the full width at half maximum (FWHM) of
a Gaussian distribution
• Defined to be the range of x between which the
Gaussian probability density function
1
1 𝑥−𝜇 2
𝑝𝐺 =
𝑒𝑥𝑝 −
2
𝜎
𝜎 2𝜋
has half of its maximum value.
• The FWHM, Г, is
Г = 2.354𝜎
Gaussian distribution
The mean, standard deviation, FWHM, and probable error
of a Gaussian probability distribution with unit area
Probable error is the 50% probability limit (𝜎𝑃𝐸 = 0.6745𝜎)
Gaussian distribution
• Note the probabilities that a measurement falls within 1-σ
(68%) and 2-σ (95%) of the mean