Transcript Probability Distribution

Uses of Statistics:
1) Descriptive:
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To describe or summarize a collection of data points
The data set in hand = the population of interest
2) Inferential:
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To make decisions or draw conclusions under conditions
of uncertainty and incompleteness
The data set in hand = a sample or an indicator of some
larger population of interest
Use these data to make an “educated guess” (or
calculated guess) about what we would find if we had full
information
The mathematical idea of probability is used to make
educated guesses with calculated degrees of certainty
Probability (the key concept):
1) A mathematical construct:
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An idealized and abstract theory about
hypothetical events
Used to develop mathematical models of these
hypothetical events
BUT it turns out that many physical events occur
in patterns that closely follow these mathematical
models – at least in the long run
– We can use these models to describe and predict the
likelihood of different event outcomes
– We can use these models to make predictions &
decisions about real world outcomes (with a
calculated chance of error or uncertainty)
– Underlies development of “rational decision-making”
Probability (cont.):
2) Definition: Probability of an outcome =
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# of occurrences of specific outcome .
# of all possible outcomes
E.g., flipping a coin and getting “heads” (1/2) or drawing
a “4” card in an ordinary deck of cards (4/52 = 1/13)
A mathematical expectation of what happens “in the
(very) long run”
Note difference between probabilities and frequencies
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Probabilities = calculated and idealized (expected in the ideal)
Frequencies = measured and tallied (observed in actual events)
3) Arithmetic of probabilities
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Can be combined (by adding or multiplying)
Probabilities of all possible outcomes sum to 1.0
Probability Distribution:
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Refers to the distribution of all possible outcomes
by the likelihood of each one occurring (similar to
frequency distribution)
Sum of all these likelihoods = 1.0
Note difference between:
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Discrete outcomes (only specific values are possible)
 A specific number of values each with a specific probability
 The specific probabilities add up to 1.0
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Continuous outcomes (any values are possible)
 An infinite number of possible values each with a near-zero
probability (of being exactly that value)
 Described by a probability density function where “probability
density” = mathematical likelihood of being in that area
Probability Distributions:
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We can describe probability distributions the same
as we describe frequency distributions – e.g.,
central tendency, dispersion, symmetry –
depending on the type of variable
Characteristics of probability distributions are called
parameters and they are referenced by Greek
letters (because they are mathematical
idealizations)
These correspond to measured characteristics of
frequency distributions (which are referenced by
ordinary English letters but with extra marks)
Probability Distributions:
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Many different kinds of probability
distributions are possible (each with its own
unique likelihood function that links values to
their probabilities)
A few of these probability distributions have
proven very useful because:
 They seem to correspond closely to observed
and naturally occurring patterns
 They are mathematically tractable (calculable
and usable)
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Most famous & useful = Normal Distribution
The Normal Distribution:
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A very specific probability distribution with its
own unique likelihood function that that
– Yields a completely symmetric distribution that
always has the same “bell” shape”
– Contains only 2 parameters which exactly
determine the size of every Normal distribution:
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μ (central tendency)
σ (dispersion around the center)
– Very well known with exactly calculated
probabilities (prob. Densities)
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68%  95%  99.7%
(1σ  2σ  3σ)
The Normal Distribution:
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Define a standard normal distribution in
terms of σ – units and centered around the
mean
Convert scores into standard scores:
– Z=X–μ
σ
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Very well known with exactly calculated
probabilities (prob. Densities)
– 68%  95%  99.7%
(1σ  2σ  3σ)