How many Jelly Beans fill a 0'5Liter Bottle

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How many Jelly Beans fill a 0.5-Liter Bottle?

• Take 60 seconds to calculate on your own.

2
Fermi Problems

• Fermi Problems challenge us to ask more
  questions, not just provide an answer.
• Enrico Fermi (1901-1954) Italian physicist best
  known for contributions to nuclear physics and
  the development of quantum theory.
• Fermi used a process of zeroing in on problems
  by saying that the value in question was
  certainly larger than one number and less than
  some other amount yields a quantified answer
  within identified limits.
• The goal is to get an answer to an order of
  magnitude by making reasonable assumption about
  the situation, not necessarily relying upon
  definite knowledge for an exact answer.

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Fermi Problems
Solutions have algorithmic approaches as well as
intuitive approaches. A very rough answer is
better than no answer What was your model for a
jelly bean? For a 0.5-liter bottle? Model is a
partial rather than complete representation The
design of a model depends as much on
circumstances and constraints (, time,
materials, data, personnel, etc.) as it does on
the problem being solved A symbolic
representation is clean and powerful. It
communicates, simply and clearly, what the
modeler thinks is important, what information is
needed, and how that information is used.
(model jelly bean as cylinder or box? Rounded
ends?) What other simplifications or assumptions
did you make?
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Fermi Problems
Get with a partner and come up with a final
estimate.
5
Describe 3 entirely different (but practical)
ways for determining the area (in cm2) of the
darkened region below (design is on a piece of
paper) to within 0.1.

• Superimpose a finely-spaced grid over the figure
  and count squares.
• Cut out figure and weigh it. Compare that weight
  to that of piece of paper. If too light,
  transfer image to another uniformly-dense
  material.
• Divide figure into local regions that can be
  integrated numerically.
• Computer scan image and count pixels.
• Build a container whose cross-section is that of
  the darkened figure. Fill with 1000cc water and
  measure level.

6
Describe 3 entirely different (but practical)
ways for determining the area (in cm2) of the
darkened region below (design is on a piece of
paper) to within 0.1.

• Use a polar planimeter gadget that
  mechanically integrates the area defined by a
  close curve.
• Throw darts. Draw rectangle (of calculable
  area) that encloses image. Pick random points
  within the rectangle and count which ones fall
  within the darkened figure. The ratio can be
  used to estimate area.

7
Why Study Statistics?
Statistics A mathematical science concerned with
data collection, presentation, analysis, and
interpretation.
Statistics can tell us about
Sports
Population
Economy
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Why Study Statistics?
Statistical analysis is also an integral part of
scientific research!
Are your experimental results believable?
Example Tensile Strength of Spaghetti
Data suggests a relationship between Type (size)
and breaking strength
Not perfect have random error.
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Why Study Statistics?
Responses and measurements are variable!
Due to
Systematic Error same error value by using an
instrument the same way
Random Error may vary from observation to
observation
Perhaps due to inability to perform measurements
in exactly the same way every time.
Goal of statistics is to find the model that best
describes a target population by taking sample
data.
Represent randomness using probability.
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Probability
Experiment of chance a phenomena whose outcome
is uncertain.
Probabilities
Chances
Sample Space
Events
Probability Model
Probability of Events
Sample Space Set of all possible outcomes
Event A set of outcomes (a subset of the sample
space). An event E occurs if any of its outcomes
occurs. Rolling dice, measuring, performing an
experiment, etc.
Probability The likelihood that an event will
produce a certain outcome.
Independence Events are independent if the
occurrence of one does not affect the probability
of the occurrence of another. Why important?
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Probability
Consider a deck of playing cards
Sample Space?
Set of 52 cards
Event?
R The card is red. F The card is a face
card. H The card is a heart. 3 The card is a
3.
P(R) 26/52 P(F) 12/52 P(H) 13/52 P(3)
4/52
Probability?
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Events and variables
Can be described as random or deterministic
The outcome of a random event cannot be predicted
The sum of two numbers on two rolled dice. The
time of emission of the ith particle from
radioactive material.
The outcome of a deterministic event can be
predicted
The measured length of a table to the nearest cm.
Motion of macroscopic objects (projectiles,
planets, space craft) as predicted by classical
mechanics.
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Extent of randomness
A variable can be more random or more
deterministic depending on the degree to which
you account for relevant parameters
Mostly deterministic
Only a small fraction of the outcome cannot be
accounted for.
Length of a table

• Temperature/humidity variation
• Measurement resolution
• Instrument/observer error
• Quantum-level intrinsic uncertainty

Mostly Random
Most of the outcome cannot be accounted for.

• Trajectory of a given molecule in a solution

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Random variables
Can be described as discrete or continuous

• A discrete variable has a countable number of
  values.

Number of customers who enter a store before one
purchases a product.

• The values of a continuous variable can not be
  listed

Distance between two oxygen molecules in a room.
Consider data collected for undergraduate
students
Is the height a discrete or continuous variable?
How could you measure height and shoe size to
make them continuous variables?
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Probability Distributions
If a random event is repeated many times, it will
produce a distribution of outcomes (statistical
regularity).
(Think about scores on an exam)
The distribution can be represented in two ways

• Frequency distribution function represents the
  distribution as the number of occurrences of each
  outcome
• Probability distribution function represents
  the distribution as the percentage of occurrences
  of each outcome

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Discrete Probability Distributions
Consider a discrete random variable, X
f(xi) is the probability distribution function
What is the range of values of f(xi)?
Therefore, Pr(Xxi) f(xi)
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Discrete Probability Distributions
Properties of discrete probabilities
for all i
for k possible discrete outcomes
Where
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Discrete Probability Distributions
Example Waiting for a success
Consider an experiment in which we toss a coin
until heads turns up.
Outcomes, w H, TH, TTH, TTTH, TTTTH Let X(w)
be the number of tails before a heads turns up.
For x 0, 1, 2.
Board Example
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Cumulative Discrete Probability Distributions
Where xj is the largest discrete value of X less
than or equal to x
?
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Discrete Probability Distributions
Example Distribution Function for Die/Dice
Distribution function for throwing a die
Outcomes, w 1, 2, 3, 4, 5, 6 ? f(xi)
1/6 for I 1,6
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Discrete Probability Distributions
Example Distribution Function for Die/Dice
Distribution function for the sum of two thrown
dice
f(xi) 1/36 for x1 2 2/36 for
x2 3
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Continuous Probability Density Function
Cumulative Distribution Function (cdf) Gives
the fraction of the total probability that lies
at or to the left of each x
Probability Density (Distribution) Function
(pdf) Gives the density of concentration of
probability at each point x
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Continuous Probability Distributions
Properties of the cumulative distribution
function
Properties of the probability density function
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Continuous Probability Distributions
For continuous variables, the events of interest
are intervals rather than isolated values.
Consider waiting time for a bus which is equally
likely to be anywhere in the next ten minutes
Not interested in probability that the bus will
arrive in 3.451233 minutes, but rather the
probability that the bus will arrive in the
subinterval (a,b) minutes
F(t)
1
t
10
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Continuous Probability Distributions
Example Gaussian (normal) distribution
Each member of the normal distribution family is
described by the mean (µ) and variance (s2).
Standard normal curve µ 0, s 1.
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Normal / Gaussian Distribution
Normal (Gaussian) Distribution
Can be used to approximately describe any
variable that tends to cluster around the mean.
Central Limit Theorem
The sum of a (sufficiently) large number of
independent random variables will be
approximately normally distributed.
Importance
Used as a simple model for complex phenomena
statistics, natural science, social science e.g.,
Observational error assumed to follow normal
distribution
Examples of experiments/measurements that will
produce Gaussian distribution?
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Standard Error
Deviation
Standard Deviation
Variance
Variance is the average squared distance of the
data from the mean. Therefore, the standard
deviation measures the spread of data about the
mean.
Standard Error
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Standard Error

• How do we reduce the size of our standard error?
• Repeated Measurements
• Different Measurement Strategy

Jacob Bernoulli (1731) For even the most
stupid of men, by some instinct of nature, by
himself and without any instruction (which is a
remarkable thing), is convinced the more
observations have been made, the less danger
there is of wandering from ones goal" (Stigler,
1986).
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Moments
Other values in terms of the moments
Skewness

• lopsidedness of the distribution
• a symmetric distribution will have a skewness
  0
• negative skewness, distribution shifted to the
  left
• positive skewness, distribution shifted to the
  right

Kurtosis
? Describes the shape of the distribution with
respect to the height and width of the curve
(peakedness)
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Central Limit Theorem
As the sample size goes to infinity, the
distribution function of the standardized
variable leads to the normal distribution
function!
http//www.jhu.edu/virtlab/prob-distributions/
31
Moments
In physics, the moment refers to the force
applied to a system at a distance from the axis
of rotation (as in a lever).
In mathematics, the moment is a measure of how
far a function is from the origin.
The 1st moment about the origin
(mean)
? Average value of x
The 2nd moment about the mean
(variance)
? A measure of the spread of the data
32
Two teams measure the height of a pole.
Height in cm

• Which team did the better job?
• Why do you think so?

33
If you measure something many times, you get
random error.

• The positive and negative errors should balance
  out.
• The average should be closer to the true value
  than any one measurement might be.
• The deviations from the average for individual
  measurements give an indication of the
  reliability of that average value.
• Standard deviation measures the reliability of
  the average.

Height in cm
34
How do we make lines?
e6
e5
e4
e3
e2
e1
35
Plot ei vs xi
e6
e5
e4
e3
e2
e1
Good lines have random, uncorrelated errors
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Error, Precision, Accuracy

• Error difference between an observed/measured
  value and a true value.
• We usually dont know the true value
• We usually do have an estimate
• Systematic Errors
• Faulty calibration
• User bias
• Change in conditions e.g., temperature rise
• Random Errors
• Statistical variation
• Precision of measurement

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Error, Precision, Accuracy

• Accuracy measure of how close result is to the
  true value
• Measure of correctness
• Precision measure of how well the result is
  determined
• Measure of variation in the data, within itself,
  not relative to true value

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Error, Precision, Accuracy
High Precision Low Accuracy
Low Precision High Accuracy
Low Precision Low Accuracy
39
Calculators and significant digits Let the
uncertain digit determine the precision to which
you quote a result
Calculator 12.6892 Estimated Error /-
0.07 Quote 12.69 /- 0.07
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What is an error?

• In data analysis, engineers use
• error uncertainty
• error ? mistake.
• Mistakes in calculation and measurements should
  always be corrected before calculating
  experimental error.
• Measured value of x xbest ? ?x
• xbest best estimate or measurement of x
• ?x uncertainty or error in the measurements

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All measurements have errors.

• What are some sources of measurement errors?
• Instrument uncertainty (caliper vs. ruler)
• Use half the smallest division.
• Measurement error (using an instrument
  incorrectly)
• Measure your height - not hold ruler level.
• Variations in the size of the object (spaghetti
  is bumpy)
• Statistical uncertainty

L 9 0.5 cm L 8.5 0.3 cm L 11.8 0.1
cm
42
If no error is given, assume half the last
significant figure.

• That's why you don't write 25.367941 mm.

43
How do you account for errors in calculations?

• The way you combine errors depends on the math
  function
• added or subtracted
• multiplied or divide
• other functions
• The sum of two lengths is Leq L1 L2. What
  is error in Leq?
• The area is of a room is A L x W. What is
  error in A?
• A simple error calculation gives the largest
  probable error.

44
Sum or difference

• What is the error if you add or subtract numbers?
• The absolute error is the sum of the absolute
  errors.

45
What is the error in length of molding to put
around a room?

• L1 5.0cm ? 0.5cm and L2 6.0cm ? 0.3cm.
• The perimeter is
• The error (upper bound) is

46
Errors can be large when you subtract similar
values.

• Weight of container 30 5 g
• Weight of container plus nuts 35 5 g
• Weight of nuts?

47
What is the error in the area of a room?

• L 5.0cm ? 0.5cm and W 6.0cm ? 0.3cm.
• What is the relative error?
• What is the absolute error?

Board Derivation
48
Product or quotient

• What is error if you multiply or divide?
• The relative error is the sum of the relative
  errors.

49
Multiply by constant

• What if you multiply a variable x by a constant
  B?
• The error is the constant times the absolute
  error.

50
What is the error in the circumference of a
circle?

• C 2 p R
• For R 2.15 0.08 cm
• ?C 2 p (0.08 cm)
• 0.50 cm

51
Powers and exponents

• What if you square or cube a number?
• The relative error is the exponent times the
  relative error.

52
What is the error in the volume of a sphere?

• V 4/3 p R3
• For R 2.15 0.08 cm
• V 41.6 cm3
• ?V/V 3 (0.08 cm/2.15 cm)
• 0.11
• ?V 0.11 41.6 cm3
• 4.6 cm3

53
What is the error in the volume of a sphere?
54
Lab Calculus of Errors Explanation
55
How much error did you have in your remote
measurement result?