Statistics 101 Class 6

Overview Sample and populations Sample and populations Why a Sample Why a Sample Types of samples Types of samples Revisiting our Deck of Cards Example Revisiting our Deck of Cards Example Another Example Another Example The THREE Distributions The THREE Distributions Relationships between the THREE Distributions Relationships between the THREE Distributions Summary Summary

Ben is a 4 th 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 Population Ben’s class Ben’s grade across the country Mean of class = 40 Ben’s class

Sample and Population Population parameters and sample statistics Population parameters and sample statistics

Why a Sample? We want to learn about a certain population We want to learn about a certain population The population we are interested in is BIG The population we are interested in is BIG If we take a sample from that population If we take a sample from that population we can learn things about the population from the sample Inferential statistics is all about trying to make an inference from a sample to a population Inferential statistics is all about trying to make an inference from a sample to a population

Types of Samples Random samples Random samples Systematic samples Systematic samples Haphazard samples Haphazard samples Convenience samples Convenience samples Biased samples Biased samples

Let’s Brainstorm about Selecting a Sample for a Question Question: What percent of homeless people in the United States suffer from mental illness

Remember our deck of cards? Population – 52 Population – 52 Mean – 340/52 equals about 6.5 Mean – 340/52 equals about 6.5 Let’s calculate the variance up at the board Let’s calculate the variance up at the board Ok- well that was an EASY population to deal with Ok- well that was an EASY population to deal with Lets take a sample – deal a hand of solitaire on the computer Lets take a sample – deal a hand of solitaire on the computer

Who did their homework? What’s the variation associated with the population of cards? What’s the variation associated with the population of cards? What’s the variation associated with the sample solitaire hand that I dealt last class? What’s the variation associated with the sample solitaire hand that I dealt last class? Did anyone draw ten solitaire hands as requested? What were the ten means of those hands? Did anyone draw ten solitaire hands as requested? What were the ten means of those hands?

A more general example

4 Population of scores  = and  = Sample of 5 scores drawn randomly from the population M = 11.6 and SD = 6.78 Add cards to deck and sample again

Take the mean of each sample and set it aside The distribution of these sample means can be used to quantify sampling error

Three Important Distributions Distribution of the population Distribution of the population Distribution of YOUR sample Distribution of YOUR sample Distribution of the means of many samples drawn from the population (sampling distribution) Distribution of the means of many samples drawn from the population (sampling distribution) IF you keep this straight – you are GOLDEN! If you keep confusing these – you are in TROUBLE IF you keep this straight – you are GOLDEN! If you keep confusing these – you are in TROUBLE

Relationships between the THREE KEY Distributions – The Central Limit Theorem Sample has n observations, M and SD Sample has n observations, M and SD Population has N observation, mu and sigma Population has N observation, mu and sigma Sample distribution can have ANY SHAPE WHATSOEVER Sample distribution can have ANY SHAPE WHATSOEVER Sampling distribution- the distribution of the means of many samples - is ALWAYS NORMAL Sampling distribution- the distribution of the means of many samples - is ALWAYS NORMAL A good estimate of the mean (mu) of your population is the mean the sampling distribution A good estimate of the mean (mu) of your population is the mean the sampling distribution Standard deviation of sampling distribution is called the standard error and is = SD/n 1/2 Standard deviation of sampling distribution is called the standard error and is = SD/n 1/2 There is a 95% chance that the mean of the POPULATION (denoted mu) is contained within the interval of the M of sample plus or minus 1.96 * standard error There is a 95% chance that the mean of the POPULATION (denoted mu) is contained within the interval of the M of sample plus or minus 1.96 * standard error

Summary Sample and populations Sample and populations Why a Sample Why a Sample Types of samples Types of samples Revisiting our Deck of Cards Example Revisiting our Deck of Cards Example Another Example Another Example The THREE Distributions The THREE Distributions Relationships between the THREE Distributions Relationships between the THREE Distributions Summary Summary