Uses of Statistics:1) Descriptive:

• To describe or summarize a collection of data points• The data set in hand = the population of interest

2) Inferential:• 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:

• 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 =

# 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

– Probabilities = calculated and idealized (expected in the ideal)– Frequencies = measured and tallied (observed in actual events)

3) Arithmetic of probabilities• Can be combined (by adding or multiplying)• Probabilities of all possible outcomes sum to 1.0

Probability Distribution:• 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:

– 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

– 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:• 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:• 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)

• Most famous & useful = Normal Distribution

The Normal Distribution:• 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:• μ (central tendency)• σ (dispersion around the center)

– Very well known with exactly calculated probabilities (prob. Densities)• 68% 95% 99.7% (1σ 2σ 3σ)

The Normal Distribution:• Define a standard normal distribution in

terms of σ – units and centered around the mean

• Convert scores into standard scores:– Z = X – μ

σ

• Very well known with exactly calculated probabilities (prob. Densities)

– 68% 95% 99.7% (1σ 2σ 3σ)