Psyc 235:Introduction to

Statistics

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To-Do

• ALEKS: aim for 18 hours spent by the end of this week

• Jan 30th Target Date for Descriptive Statistics

• Watch videos:1. Picturing Distributions2. Describing Distributions3. Normal Distributions

Quiz 1

• NOT GRADED• available starting 8am Thurs Jan

31st, through Friday• can do on ALEKS from home, etc• No access to any other learning or

reviewing materials until either they finish the quizzes or after Friday

• 3.5 hour time limit

Review:(2 Steps Forward and 1 Step Back)

• Distribution For a given variable:

the possible numerical values & the number of times they occur in the data

Many ways to represent visually

Summarizing Distributions

• Descriptive Measures of Data Measures of C_nt__l T__d__cy Measures of D__p_rs__n

Central Tendency

• Mean, Median, Mode Mean vs Median & outliers

(Bill Gates example) skewed distributions

Standard Deviation

• Conceptually: about how far, generally, each datum is

from the mean 2 formulas??

Population vs Sample

• In Psychology: Population: hypothetical, unobservable

not just all humans who ARE, but all humans who COULD BE.

must estimate mean, standard deviation, from:

Sample is the only thing we ever have

Descriptive -> Inferential?

• How can we make inferences about a population if we just have data from a sample?

• How can we evaluate how good our estimate is?

• “Do these sample data really reflect what’s going on in the population, or are they maybe just due to chance?”

PROBABILITY

• The tool that will allow us to bridge the gap from descriptive to inferential

• we’ll start by using simple problems, in which probability can be calculated by merely COUNTING

Flipping a Coin

• Say I flip a coin... OMG Heads!!!! Do you care? Why Not?

• Sample Space: (draw on board) collection of all possible outcomes for a given

phenomenon coin toss: {H,T}

mutually exclusive: either one happens, or the other

Flipping a Coin

• Probability(Heads)?• So.... must the next one be Tails?• No!

Independent trials Random Phenomenon:

can’t predict individual outcome can predict pattern in the LONG RUN

• Probability: relative # times something happens in the long run

2 Coin Flips

• OMG 2 Heads! impressed yet?

• Sample space (draw on board) Prob(2 Heads): 1/4 outcome: single observation

• OMG 2 of same! Prob(2 Heads OR 2 Tails): event: subset of the sample space made of 1

or more possible outcomes

Larger Point

• OMG 30 Heads in a row! NOW maybe you’re finally interested...

• OMG drew 3 yellow cars! interesting? boring? can’t tell!

• Descriptive Stats: measuring & summarizing outcomes

• Inferential Stats: to understand some outcome, must consider it in context of all possible outcomes that could’ve occurred (sample space)

Counting Rules

• Count up the possible outcomes that is: define the sample space

• 2 Main ways to do this: Permutations

when order matters

Combinations when order doesn’t matter

Permutation:Ordered Arrangement

• Example used: Horse Race...• MUTANT HORSE RACE!

Permutation:Ordered Arrangement

“HorseFace McBusterWorthy wins 1st place!!”

...in a one-horse race!

# Horses (n) # Winning Places (r) # Outcomes

1 13 13 3

Permutation:Ordered Arrangement

• For n objects, when taking all of them (r=n), there are n! possible permutations.

• 3 horses (n) & 3 winning places (r) --> 3*2*1=6 possible outcomes

• For n objects taken r at a time: n!(n-r)!

• 7 horses & 3 winning places?...

Combination:Unordered Arrangement

• Example used: Combo Plate!

QuickTime™ and a decompressor

are needed to see this picture.

Combination:Unordered Arrangement

• Mexican restaurant’s menu: taco, burrito, enchilada

• How many different 3-item combos can you get?

# Menu Items (n) Combo Size (r) # Outcomes

3 3

Combination:Unordered Arrangement

• Mexican restaurant’s menu: taco, burrito, enchilada tamale, quesadilla, taquito, chimichanga

• How many different 3-item combos can you get?

# Menu Items (n) Combo Size (r) # Outcomes

3 37 3

Combination: Unordered Arrangement

• For n objects, when taking all of them (r=n), there is 1 combination

• For n objects taken r at a time: n!r!(n-r)!

Multiplication Principle(a.k.a. Fundamental Counting

Principle)

• For 2 independent phenomenon, how many different ways are there for them to happen together? # possible joint outcomes?

• Simply multiply the # possible outcomes for the two individual phenomena

• Example: flip coin & roll die• 2*6=12

Multiplication Principle(a.k.a. Fundamental Counting

Principle)

• Can be used with Permutations &/or Combinations

• Ex: Lunch at the Racetrack 7 horses racing 7 items on the cafe menu I see the results of the race (1st, 2nd,

3rd) and order a 3-item combo plate. How many different ways can this happen?

Calculating Probabilities

• Counting rules (Permutation, Combination, Multiplication): Define sample space (# possible

outcomes)• Probability of a specific outcome:

1 sample space

• Probability of an event? event: subset of sample space made of 1

or more possible outcomes

Calculating Probabilities

• Sample Space: 7 Micro Machines (3 yellow, 4 red)

• Outcome: draw the yellow corvette Probability = 1/7

• Event: draw any yellow car there are 3 outcomes that could satisfy this

event: yellow corvetter, yellow pickup, yellow taxi

Probability = 3/7

Probability of Draws w/ Replacement

• Replacement: resetting the sample space each time --> independent phenomena so use multiplication principle

• Ex: 3 draws with replacement Event: drawing a red car all 3 times Probability: 4/7 * 4/7 * 4/7 = 64/343 = 0.187 =18.7%

Probability of Draws w/o Replacement

1. Use counting rules to define sample space

2. Use counting rules to figure out how many possible outcomes satisfy the event

3. divide #2 by #1.

Probability of Draws w/o Replacement

• Ex: Drawing 3 cars w/o replacement Event: drawing 2 red & 1 yellow

(don’t care about order) --> use Combinations

Define Sample space: Count outcomes that satisfy event

treat red & yellow as independent use combinations, then multiplication

principle Divide

Recap

• Today: Probability is the tool we’ll use to make

inferences about a population, from a sample

Counting rules: define sample space for simple phenomena

Intro to calculating probability• Next time:

Probability rules, more about events, Venn diagrams

Remember

• Quiz 1 starting Thursday• Office hours Thursday• Lab• Put your ALEKS hours in!!