Statistics 101 Statistics 101

Class 6Class 6

OverviewOverview• Sample and populationsSample and populations• Why a SampleWhy a Sample• Types of samplesTypes of samples• Revisiting our Deck of Cards Example Revisiting our Deck of Cards Example • Another ExampleAnother Example• The THREE Distributions The THREE Distributions • Relationships between the THREE Relationships between the THREE

DistributionsDistributions• SummarySummary

GRADE

0 20 40 60 80 100

05

1015

•Ben is a 4th grader in an underperforming school

•In one case, Ben’s math exam score is 10 points above the mean in his school

•BUT, Ben’s exam score is 10 points below the mean for students in his grade in the country

•It is useful to interpret Ben’s performance relative to average performance.

Ben’s class

Ben’s grade

across the country

Mean of class = 40

Mean of students across

country = 60

Sample vs PopulationSample vs Population

Ben’s class

Ben’s grade

across the country

Mean of class = 40

Ben’s class

Sample and PopulationSample and Population

• Population Population parameters and parameters and sample statisticssample statistics

Why a Sample?Why a Sample?

• We want to learn about a certain populationWe want to learn about a certain population

• The population we are interested in is BIGThe population we are interested in is BIG

• If we take a sample from that populationIf we take a sample from that population

we can learn things about the population we can learn things about the population from the samplefrom the sample

• Inferential statistics is all about trying to Inferential statistics is all about trying to make an inference from a sample to a make an inference from a sample to a populationpopulation

Types of SamplesTypes of Samples

• Random samplesRandom samples

• Systematic samplesSystematic samples

• Haphazard samplesHaphazard samples

• Convenience samplesConvenience samples

• Biased samplesBiased samples

Let’s Brainstorm about Let’s Brainstorm about Selecting a Sample for a Selecting a Sample for a

QuestionQuestionQuestion: What percent of homeless Question: What percent of homeless

people in the United States suffer people in the United States suffer from mental illnessfrom mental illness

Remember our deck of Remember our deck of cards?cards?• Population – 52Population – 52• Mean – 340/52 equals about 6.5Mean – 340/52 equals about 6.5• Let’s calculate the variance up at the Let’s calculate the variance up at the

boardboard• Ok- well that was an EASY population Ok- well that was an EASY population

to deal withto deal with• Lets take a sample – deal a hand of Lets take a sample – deal a hand of

solitaire on the computersolitaire on the computer

Who did their homework?Who did their homework?

• What’s the variation associated with What’s the variation associated with the population of cards?the population of cards?

• What’s the variation associated with What’s the variation associated with the sample solitaire hand that I dealt the sample solitaire hand that I dealt last class?last class?

• Did anyone draw ten solitaire hands Did anyone draw ten solitaire hands as requested? What were the ten as requested? What were the ten means of those hands?means of those hands?

A more general example A more general example

4

Population of scores

= 10.00 and = 6.05

0

14

9

15

20

Sample of 5 scores drawn randomly from the population

M = 11.6 and SD = 6.78

Add cards to deck and sample again

4

0

14

9

15

20

Take the mean of each sample and set it aside

11.6

11

9.2

12.4

11.8

6.8

12

10.2

13.2

9.4

The distribution of these sample means can be used to quantify sampling error

Three Important Three Important DistributionsDistributions• Distribution of the populationDistribution of the population

• Distribution of YOUR sampleDistribution of YOUR sample

• Distribution of the means of many samples Distribution of the means of many samples drawn from the population (sampling drawn from the population (sampling distribution)distribution)

• IF you keep this straight – you are GOLDEN! IF you keep this straight – you are GOLDEN! If you keep confusing these – you are in If you keep confusing these – you are in TROUBLETROUBLE

Relationships between the THREE KEY Distributions – The Central Limit Theorem

• Sample has n observations, M and SDSample has n observations, M and SD• Population has N observation, mu and sigmaPopulation has N observation, mu and sigma• Sample distribution can have ANY SHAPE Sample distribution can have ANY SHAPE

WHATSOEVERWHATSOEVER• Sampling distribution- the distribution of the Sampling distribution- the distribution of the

means of many samples - is ALWAYS NORMALmeans of many samples - is ALWAYS NORMAL• A good estimate of the mean (mu) of your A good estimate of the mean (mu) of your

population is the mean the sampling distributionpopulation is the mean the sampling distribution• Standard deviation of sampling distribution is Standard deviation of sampling distribution is

called the standard error and is = SD/ncalled the standard error and is = SD/n1/21/2

• There is a 95% chance that the mean of the There is a 95% chance that the mean of the POPULATION (denoted mu) is contained within POPULATION (denoted mu) is contained within the interval of the M of sample plus or minus 1.96 the interval of the M of sample plus or minus 1.96 * standard error* standard error

SummarySummary• Sample and populationsSample and populations• Why a SampleWhy a Sample• Types of samplesTypes of samples• Revisiting our Deck of Cards Example Revisiting our Deck of Cards Example • Another ExampleAnother Example• The THREE Distributions The THREE Distributions • Relationships between the THREE Relationships between the THREE

DistributionsDistributions• SummarySummary