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  • Chapter 1: Introduction to data

    OpenIntro Statistics, 2nd Edition

  • Case study

    1 Case study

    2 Data basics

    3 Overview of data collection principles

    4 Observational studies and sampling strategies

    5 Experiments

    6 Examining numerical data

    7 Considering categorical data

    8 Case study: Gender discrimination

    OpenIntro Statistics, 2nd Edition

    Chp 1: Intro. to data

  • Case study

    Treating Chronic Fatigue Syndrome

    Objective: Evaluate the effectiveness of cognitive-behaviortherapy for chronic fatigue syndrome.

    Participant pool: 142 patients who were recruited from referralsby primary care physicians and consultants to a hospital clinicspecializing in chronic fatigue syndrome.

    Actual participants: Only 60 of the 142 referred patients enteredthe study. Some were excluded because they didnt meet thediagnostic criteria, some had other health issues, and somerefused to be a part of the study.

    Deale et. al. Cognitive behavior therapy for chronic fatigue syndrome: A randomized controlled trial. The American Journal of

    Psychiatry 154.3 (1997).

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 2 / 94

  • Case study

    Study design

    Patients randomly assigned to treatment and control groups, 30patients in each group:

    Treatment: Cognitive behavior therapy collaborative, educative,and with a behavioral emphasis. Patients were shown on howactivity could be increased steadily and safely withoutexacerbating symptoms.Control: Relaxation No advice was given about how activitycould be increased. Instead progressive muscle relaxation,visualization, and rapid relaxation skills were taught.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 3 / 94

  • Case study

    Results

    The table below shows the distribution of patients with goodoutcomes at 6-month follow-up. Note that 7 patients dropped out ofthe study: 3 from the treatment and 4 from the control group.

    Good outcomeYes No Total

    Treatment 19 8 27GroupControl 5 21 26Total 24 29 53

    Proportion with good outcomes in treatment group:

    19/27 0.70 70%Proportion with good outcomes in control group:

    5/26 0.19 19%

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 4 / 94

  • Case study

    Results

    The table below shows the distribution of patients with goodoutcomes at 6-month follow-up. Note that 7 patients dropped out ofthe study: 3 from the treatment and 4 from the control group.

    Good outcomeYes No Total

    Treatment 19 8 27GroupControl 5 21 26Total 24 29 53

    Proportion with good outcomes in treatment group:

    19/27 0.70 70%

    Proportion with good outcomes in control group:

    5/26 0.19 19%

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 4 / 94

  • Case study

    Results

    The table below shows the distribution of patients with goodoutcomes at 6-month follow-up. Note that 7 patients dropped out ofthe study: 3 from the treatment and 4 from the control group.

    Good outcomeYes No Total

    Treatment 19 8 27GroupControl 5 21 26Total 24 29 53

    Proportion with good outcomes in treatment group:

    19/27 0.70 70%Proportion with good outcomes in control group:

    5/26 0.19 19%OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 4 / 94

  • Case study

    Understanding the results

    Do the data show a real difference between the groups?

    Suppose you flip a coin 100 times. While the chance a coin landsheads in any given coin flip is 50%, we probably wont observeexactly 50 heads. This type of fluctuation is part of almost anytype of data generating process.

    The observed difference between the two groups (70 - 19 =51%) may be real, or may be due to natural variation.

    Since the difference is quite large, it is more believable that thedifference is real.

    We need statistical tools to determine if the difference is so largethat we should reject the notion that it was due to chance.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 5 / 94

  • Case study

    Understanding the results

    Do the data show a real difference between the groups?

    Suppose you flip a coin 100 times. While the chance a coin landsheads in any given coin flip is 50%, we probably wont observeexactly 50 heads. This type of fluctuation is part of almost anytype of data generating process.

    The observed difference between the two groups (70 - 19 =51%) may be real, or may be due to natural variation.

    Since the difference is quite large, it is more believable that thedifference is real.

    We need statistical tools to determine if the difference is so largethat we should reject the notion that it was due to chance.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 5 / 94

  • Case study

    Generalizing the results

    Are the results of this study generalizable to all patients with chronicfatigue syndrome?

    These patients had specific characteristics and volunteered to be apart of this study, therefore they may not be representative of allpatients with chronic fatigue syndrome. While we cannot immediatelygeneralize the results to all patients, this first study is encouraging.The method works for patients with some narrow set ofcharacteristics, and that gives hope that it will work, at least to somedegree, with other patients.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 6 / 94

  • Case study

    Generalizing the results

    Are the results of this study generalizable to all patients with chronicfatigue syndrome?

    These patients had specific characteristics and volunteered to be apart of this study, therefore they may not be representative of allpatients with chronic fatigue syndrome. While we cannot immediatelygeneralize the results to all patients, this first study is encouraging.The method works for patients with some narrow set ofcharacteristics, and that gives hope that it will work, at least to somedegree, with other patients.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 6 / 94

  • Data basics

    1 Case study

    2 Data basicsObservations and variablesTypes of variablesRelationships among variablesAssociated and independent variables

    3 Overview of data collection principles

    4 Observational studies and sampling strategies

    5 Experiments

    6 Examining numerical data

    7 Considering categorical data

    8 Case study: Gender discrimination

    OpenIntro Statistics, 2nd Edition

    Chp 1: Intro. to data

  • Data basics Observations and variables

    Data matrix

    Data collected on students in a statistics class on a variety ofvariables:

    variable

    Stu. gender intro extra dread1 male extravert 32 female extravert 23 female introvert 4 4 female extravert 2 observation...

    ......

    ......

    86 male extravert 3

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 7 / 94

  • Data basics Types of variables

    Types of variables

    all variables

    numerical categorical

    continuous discrete regularcategorical ordinal

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 8 / 94

  • Data basics Types of variables

    Types of variables (cont.)

    gender sleep bedtime countries dread

    1 male 5 12-2 13 32 female 7 10-12 7 23 female 5.5 12-2 1 44 female 7 12-2 25 female 3 12-2 1 36 female 3 12-2 9 4

    gender:

    sleep:

    bedtime:

    countries:

    dread:

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 9 / 94

  • Data basics Types of variables

    Types of variables (cont.)

    gender sleep bedtime countries dread

    1 male 5 12-2 13 32 female 7 10-12 7 23 female 5.5 12-2 1 44 female 7 12-2 25 female 3 12-2 1 36 female 3 12-2 9 4

    gender: categorical

    sleep:

    bedtime:

    countries:

    dread:

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 9 / 94

  • Data basics Types of variables

    Types of variables (cont.)

    gender sleep bedtime countries dread

    1 male 5 12-2 13 32 female 7 10-12 7 23 female 5.5 12-2 1 44 female 7 12-2 25 female 3 12-2 1 36 female 3 12-2 9 4

    gender: categorical

    sleep:

    bedtime:

    countries:

    dread:

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 9 / 94

  • Data basics Types of variables

    Types of variables (cont.)

    gender sleep bedtime countries dread

    1 male 5 12-2 13 32 female 7 10-12 7 23 female 5.5 12-2 1 44 female 7 12-2 25 female 3 12-2 1 36 female 3 12-2 9 4

    gender: categorical

    sleep: numerical, continuous

    bedtime:

    countries:

    dread:

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 9 / 94

  • Data basics Types of variables

    Types of variables (cont.)

    gender sleep bedtime countries dread

    1 male 5 12-2 13 32 female 7 10-12 7 23 female 5.5 12-2 1 44 female 7 12-2 25 female 3 12-2 1 36 female 3 12-2 9 4

    gender: categorical

    sleep: numerical, continuous

    bedtime:

    countries:

    dread:

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 9 / 94

  • Data basics Types of variables

    Types of variables (cont.)

    gender sleep bedtime countries dread

    1 male 5 12-2 13 32 female 7 10-12 7 23 female 5.5 12-2 1 44 female 7 12-2 25 female 3 12-2 1 36 female 3 12-2 9 4

    gender: categorical

    sleep: numerical, continuous

    bedtime: categorical, ordinal

    countries:

    dread:

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 9 / 94

  • Data basics Types of variables

    Types of variables (cont.)

    gender sleep bedtime countries dread

    1 male 5 12-2 13 32 female 7 10-12 7 23 female 5.5 12-2 1 44 female 7 12-2 25 female 3 12-2 1 36 female 3 12-2 9 4

    gender: categorical

    sleep: numerical, continuous

    bedtime: categorical, ordinal

    countries:

    dread:

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 9 / 94

  • Data basics Types of variables

    Types of variables (cont.)

    gender sleep bedtime countries dread

    1 male 5 12-2 13 32 female 7 10-12 7 23 female 5.5 12-2 1 44 female 7 12-2 25 female 3 12-2 1 36 female 3 12-2 9 4

    gender: categorical

    sleep: numerical, continuous

    bedtime: categorical, ordinal

    countries: numerical, discrete

    dread:

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 9 / 94

  • Data basics Types of variables

    Types of variables (cont.)

    gender sleep bedtime countries dread

    1 male 5 12-2 13 32 female 7 10-12 7 23 female 5.5 12-2 1 44 female 7 12-2 25 female 3 12-2 1 36 female 3 12-2 9 4

    gender: categorical

    sleep: numerical, continuous

    bedtime: categorical, ordinal

    countries: numerical, discrete

    dread:

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 9 / 94

  • Data basics Types of variables

    Types of variables (cont.)

    gender sleep bedtime countries dread

    1 male 5 12-2 13 32 female 7 10-12 7 23 female 5.5 12-2 1 44 female 7 12-2 25 female 3 12-2 1 36 female 3 12-2 9 4

    gender: categorical

    sleep: numerical, continuous

    bedtime: categorical, ordinal

    countries: numerical, discrete

    dread: categorical, ordinal - could also be used as numerical

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 9 / 94

  • Data basics Types of variables

    Practice

    What type of variable is a telephone area code?

    (a) numerical, continuous

    (b) numerical, discrete

    (c) categorical

    (d) categorical, ordinal

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 10 / 94

  • Data basics Types of variables

    Practice

    What type of variable is a telephone area code?

    (a) numerical, continuous

    (b) numerical, discrete

    (c) categorical

    (d) categorical, ordinal

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 10 / 94

  • Data basics Relationships among variables

    Relationships among variables

    Does there appear to be a relationship between number of alcoholicdrinks consumed per week and age at first alcohol consumption?

    0 10 20 30 40 50 60 70

    3.0

    3.5

    4.0

    Hours of study / week

    GPA

    Can you spot anything unusual about any of the data points?

    There is one student with GPA > 4.0, this is likely a data error.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 11 / 94

  • Data basics Relationships among variables

    Relationships among variables

    Does there appear to be a relationship between number of alcoholicdrinks consumed per week and age at first alcohol consumption?

    0 10 20 30 40 50 60 70

    3.0

    3.5

    4.0

    Hours of study / week

    GPA

    Can you spot anything unusual about any of the data points?

    There is one student with GPA > 4.0, this is likely a data error.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 11 / 94

  • Data basics Relationships among variables

    Relationships among variables

    Does there appear to be a relationship between number of alcoholicdrinks consumed per week and age at first alcohol consumption?

    0 10 20 30 40 50 60 70

    3.0

    3.5

    4.0

    Hours of study / week

    GPA

    Can you spot anything unusual about any of the data points?

    There is one student with GPA > 4.0, this is likely a data error.OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 11 / 94

  • Data basics Associated and independent variables

    Practice

    Based on the scatterplot on theright, which of the following state-ments is correct about the headand skull lengths of possums?

    l

    l

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    85 90 95 10050

    55

    60

    65

    head length (mm)

    skul

    l wid

    th (m

    m)

    (a) There is no relationship between head length and skull width, i.e.the variables are independent.

    (b) Head length and skull width are positively associated.

    (c) Skull width and head length are negatively associated.

    (d) A longer head causes the skull to be wider.

    (e) A wider skull causes the head to be longer.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 12 / 94

  • Data basics Associated and independent variables

    Practice

    Based on the scatterplot on theright, which of the following state-ments is correct about the headand skull lengths of possums?

    l

    l

    l

    ll

    l

    ll

    l

    ll

    l

    l

    ll

    l

    l

    l l

    l

    ll

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    l

    l

    l

    l

    l

    l

    l

    ll

    l ll

    l

    85 90 95 10050

    55

    60

    65

    head length (mm)

    skul

    l wid

    th (m

    m)

    (a) There is no relationship between head length and skull width, i.e.the variables are independent.

    (b) Head length and skull width are positively associated.

    (c) Skull width and head length are negatively associated.

    (d) A longer head causes the skull to be wider.

    (e) A wider skull causes the head to be longer.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 12 / 94

  • Data basics Associated and independent variables

    Associated vs. independent

    When two variables show some connection with one another,they are called associated variables.

    Associated variables can also be called dependent variables andvice-versa.

    If two variables are not associated, i.e. there is no evidentconnection between the two, then they are said to beindependent.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 13 / 94

  • Overview of data collection principles

    1 Case study

    2 Data basics

    3 Overview of data collection principlesPopulations and samplesAnecdotal evidenceSampling from a populationExplanatory and response variablesObservational studies and experiments

    4 Observational studies and sampling strategies

    5 Experiments

    6 Examining numerical data

    7 Considering categorical data

    8 Case study: Gender discrimination

    OpenIntro Statistics, 2nd Edition

    Chp 1: Intro. to data

  • Overview of data collection principles Populations and samples

    Populations and samples

    http:// well.blogs.nytimes.com/ 2012/ 08/ 29/

    finding-your-ideal-running-form

    Research question: Can peoplebecome better, more efficientrunners on their own, merely byrunning?

    Population of interest: All people

    Sample: Group of adult women who recently joined a running groupPopulation to which results can be generalized: Adult women, if thedata are randomly sampled

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 14 / 94

  • Overview of data collection principles Populations and samples

    Populations and samples

    http:// well.blogs.nytimes.com/ 2012/ 08/ 29/

    finding-your-ideal-running-form

    Research question: Can peoplebecome better, more efficientrunners on their own, merely byrunning?Population of interest:

    All people

    Sample: Group of adult women who recently joined a running groupPopulation to which results can be generalized: Adult women, if thedata are randomly sampled

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 14 / 94

  • Overview of data collection principles Populations and samples

    Populations and samples

    http:// well.blogs.nytimes.com/ 2012/ 08/ 29/

    finding-your-ideal-running-form

    Research question: Can peoplebecome better, more efficientrunners on their own, merely byrunning?Population of interest: All people

    Sample: Group of adult women who recently joined a running groupPopulation to which results can be generalized: Adult women, if thedata are randomly sampled

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 14 / 94

  • Overview of data collection principles Populations and samples

    Populations and samples

    http:// well.blogs.nytimes.com/ 2012/ 08/ 29/

    finding-your-ideal-running-form

    Research question: Can peoplebecome better, more efficientrunners on their own, merely byrunning?Population of interest: All people

    Sample: Group of adult women who recently joined a running group

    Population to which results can be generalized: Adult women, if thedata are randomly sampled

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 14 / 94

  • Overview of data collection principles Populations and samples

    Populations and samples

    http:// well.blogs.nytimes.com/ 2012/ 08/ 29/

    finding-your-ideal-running-form

    Research question: Can peoplebecome better, more efficientrunners on their own, merely byrunning?Population of interest: All people

    Sample: Group of adult women who recently joined a running groupPopulation to which results can be generalized:

    Adult women, if thedata are randomly sampled

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 14 / 94

  • Overview of data collection principles Populations and samples

    Populations and samples

    http:// well.blogs.nytimes.com/ 2012/ 08/ 29/

    finding-your-ideal-running-form

    Research question: Can peoplebecome better, more efficientrunners on their own, merely byrunning?Population of interest: All people

    Sample: Group of adult women who recently joined a running groupPopulation to which results can be generalized: Adult women, if thedata are randomly sampled

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 14 / 94

  • Overview of data collection principles Anecdotal evidence

    Anecdotal evidence and early smoking research

    Anti-smoking research started in the 1930s and 1940s whencigarette smoking became increasingly popular. While somesmokers seemed to be sensitive to cigarette smoke, others werecompletely unaffected.

    Anti-smoking research was faced with resistance based onanecdotal evidence such as My uncle smokes three packs a dayand hes in perfectly good health, evidence based on a limitedsample size that might not be representative of the population.

    It was concluded that smoking is a complex human behavior, byits nature difficult to study, confounded by human variability.

    In time researchers were able to examine larger samples ofcases (smokers), and trends showing that smoking has negativehealth impacts became much clearer.

    Brandt, The Cigarette Century (2009), Basic Books.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 15 / 94

  • Overview of data collection principles Sampling from a population

    Census

    Wouldnt it be better to just include everyone and sample theentire population?

    This is called a census.

    There are problems with taking a census:

    It can be difficult to complete a census: there always seem to besome individuals who are hard to locate or hard to measure. Andthese difficult-to-find people may have certain characteristics thatdistinguish them from the rest of the population.Populations rarely stand still. Even if you could take a census, thepopulation changes constantly, so its never possible to get aperfect measure.Taking a census may be more complex than sampling.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 16 / 94

  • Overview of data collection principles Sampling from a population

    Census

    Wouldnt it be better to just include everyone and sample theentire population?

    This is called a census.

    There are problems with taking a census:

    It can be difficult to complete a census: there always seem to besome individuals who are hard to locate or hard to measure. Andthese difficult-to-find people may have certain characteristics thatdistinguish them from the rest of the population.Populations rarely stand still. Even if you could take a census, thepopulation changes constantly, so its never possible to get aperfect measure.Taking a census may be more complex than sampling.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 16 / 94

  • Overview of data collection principles Sampling from a population

    http:// www.npr.org/ templates/ story/ story.php?storyId=125380052

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 17 / 94

  • Overview of data collection principles Sampling from a population

    Exploratory analysis to inference

    Sampling is natural.

    Think about sampling something you are cooking - you taste(examine) a small part of what youre cooking to get an ideaabout the dish as a whole.When you taste a spoonful of soup and decide the spoonful youtasted isnt salty enough, thats exploratory analysis.If you generalize and conclude that your entire soup needs salt,thats an inference.For your inference to be valid, the spoonful you tasted (thesample) needs to be representative of the entire pot (thepopulation).

    If your spoonful comes only from the surface and the salt iscollected at the bottom of the pot, what you tasted is probably notrepresentative of the whole pot.If you first stir the soup thoroughly before you taste, your spoonfulwill more likely be representative of the whole pot.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 18 / 94

  • Overview of data collection principles Sampling from a population

    Exploratory analysis to inference

    Sampling is natural.Think about sampling something you are cooking - you taste(examine) a small part of what youre cooking to get an ideaabout the dish as a whole.

    When you taste a spoonful of soup and decide the spoonful youtasted isnt salty enough, thats exploratory analysis.If you generalize and conclude that your entire soup needs salt,thats an inference.For your inference to be valid, the spoonful you tasted (thesample) needs to be representative of the entire pot (thepopulation).

    If your spoonful comes only from the surface and the salt iscollected at the bottom of the pot, what you tasted is probably notrepresentative of the whole pot.If you first stir the soup thoroughly before you taste, your spoonfulwill more likely be representative of the whole pot.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 18 / 94

  • Overview of data collection principles Sampling from a population

    Exploratory analysis to inference

    Sampling is natural.Think about sampling something you are cooking - you taste(examine) a small part of what youre cooking to get an ideaabout the dish as a whole.When you taste a spoonful of soup and decide the spoonful youtasted isnt salty enough, thats exploratory analysis.

    If you generalize and conclude that your entire soup needs salt,thats an inference.For your inference to be valid, the spoonful you tasted (thesample) needs to be representative of the entire pot (thepopulation).

    If your spoonful comes only from the surface and the salt iscollected at the bottom of the pot, what you tasted is probably notrepresentative of the whole pot.If you first stir the soup thoroughly before you taste, your spoonfulwill more likely be representative of the whole pot.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 18 / 94

  • Overview of data collection principles Sampling from a population

    Exploratory analysis to inference

    Sampling is natural.Think about sampling something you are cooking - you taste(examine) a small part of what youre cooking to get an ideaabout the dish as a whole.When you taste a spoonful of soup and decide the spoonful youtasted isnt salty enough, thats exploratory analysis.If you generalize and conclude that your entire soup needs salt,thats an inference.

    For your inference to be valid, the spoonful you tasted (thesample) needs to be representative of the entire pot (thepopulation).

    If your spoonful comes only from the surface and the salt iscollected at the bottom of the pot, what you tasted is probably notrepresentative of the whole pot.If you first stir the soup thoroughly before you taste, your spoonfulwill more likely be representative of the whole pot.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 18 / 94

  • Overview of data collection principles Sampling from a population

    Exploratory analysis to inference

    Sampling is natural.Think about sampling something you are cooking - you taste(examine) a small part of what youre cooking to get an ideaabout the dish as a whole.When you taste a spoonful of soup and decide the spoonful youtasted isnt salty enough, thats exploratory analysis.If you generalize and conclude that your entire soup needs salt,thats an inference.For your inference to be valid, the spoonful you tasted (thesample) needs to be representative of the entire pot (thepopulation).

    If your spoonful comes only from the surface and the salt iscollected at the bottom of the pot, what you tasted is probably notrepresentative of the whole pot.If you first stir the soup thoroughly before you taste, your spoonfulwill more likely be representative of the whole pot.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 18 / 94

  • Overview of data collection principles Sampling from a population

    Sampling bias

    Non-response: If only a small fraction of the randomly sampledpeople choose to respond to a survey, the sample may no longerbe representative of the population.

    Voluntary response: Occurs when the sample consists of peoplewho volunteer to respond because they have strong opinions onthe issue. Such a sample will also not be representative of thepopulation.

    cnn.com, Jan 14, 2012

    Convenience sample: Individuals who are easily accessible aremore likely to be included in the sample.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 19 / 94

  • Overview of data collection principles Sampling from a population

    Sampling bias

    Non-response: If only a small fraction of the randomly sampledpeople choose to respond to a survey, the sample may no longerbe representative of the population.Voluntary response: Occurs when the sample consists of peoplewho volunteer to respond because they have strong opinions onthe issue. Such a sample will also not be representative of thepopulation.

    cnn.com, Jan 14, 2012

    Convenience sample: Individuals who are easily accessible aremore likely to be included in the sample.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 19 / 94

  • Overview of data collection principles Sampling from a population

    Sampling bias

    Non-response: If only a small fraction of the randomly sampledpeople choose to respond to a survey, the sample may no longerbe representative of the population.Voluntary response: Occurs when the sample consists of peoplewho volunteer to respond because they have strong opinions onthe issue. Such a sample will also not be representative of thepopulation.

    cnn.com, Jan 14, 2012

    Convenience sample: Individuals who are easily accessible aremore likely to be included in the sample.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 19 / 94

  • Overview of data collection principles Sampling from a population

    Sampling bias

    Non-response: If only a small fraction of the randomly sampledpeople choose to respond to a survey, the sample may no longerbe representative of the population.Voluntary response: Occurs when the sample consists of peoplewho volunteer to respond because they have strong opinions onthe issue. Such a sample will also not be representative of thepopulation.

    cnn.com, Jan 14, 2012

    Convenience sample: Individuals who are easily accessible aremore likely to be included in the sample.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 19 / 94

  • Overview of data collection principles Sampling from a population

    Sampling bias

    Non-response: If only a small fraction of the randomly sampledpeople choose to respond to a survey, the sample may no longerbe representative of the population.Voluntary response: Occurs when the sample consists of peoplewho volunteer to respond because they have strong opinions onthe issue. Such a sample will also not be representative of thepopulation.

    cnn.com, Jan 14, 2012

    Convenience sample: Individuals who are easily accessible aremore likely to be included in the sample.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 19 / 94

  • Overview of data collection principles Sampling from a population

    Sampling bias example: Landon vs. FDR

    A historical example of a biased sample yielding misleading results:

    In 1936, Landonsought theRepublicanpresidentialnomination opposingthe re-election ofFDR.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 20 / 94

  • Overview of data collection principles Sampling from a population

    The Literary Digest Poll

    The Literary Digest polled about 10 millionAmericans, and got responses from about2.4 million.

    The poll showed that Landon would likelybe the overwhelming winner and FDRwould get only 43% of the votes.

    Election result: FDR won, with 62% of thevotes.

    The magazine was completely discredited because of the poll,and was soon discontinued.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 21 / 94

  • Overview of data collection principles Sampling from a population

    The Literary Digest Poll what went wrong?

    The magazine had surveyed

    its own readers,registered automobile owners, andregistered telephone users.

    These groups had incomes well above the national average ofthe day (remember, this is Great Depression era) which resultedin lists of voters far more likely to support Republicans than atruly typical voter of the time, i.e. the sample was notrepresentative of the American population at the time.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 22 / 94

  • Overview of data collection principles Sampling from a population

    Large samples are preferable, but...

    The Literary Digest election poll was based on a sample size of2.4 million, which is huge, but since the sample was biased, thesample did not yield an accurate prediction.

    Back to the soup analogy: If the soup is not well stirred, it doesntmatter how large a spoon you have, it will still not taste right. Ifthe soup is well stirred, a small spoon will suffice to test the soup.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 23 / 94

  • Overview of data collection principles Sampling from a population

    Practice

    A school district is considering whether it will no longer allow high schoolstudents to park at school after two recent accidents where students wereseverely injured. As a first step, they survey parents by mail, asking themwhether or not the parents would object to this policy change. Of 6,000 sur-veys that go out, 1,200 are returned. Of these 1,200 surveys that were com-pleted, 960 agreed with the policy change and 240 disagreed. Which of thefollowing statements are true?

    I. Some of the mailings may have never reached the parents.

    II. The school district has strong support from parents to move forwardwith the policy approval.

    III. It is possible that majority of the parents of high school studentsdisagree with the policy change.

    IV. The survey results are unlikely to be biased because all parents weremailed a survey.

    (a) Only I (b) I and II (c) I and III (d) III and IV (e) Only IV

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 24 / 94

  • Overview of data collection principles Sampling from a population

    Practice

    A school district is considering whether it will no longer allow high schoolstudents to park at school after two recent accidents where students wereseverely injured. As a first step, they survey parents by mail, asking themwhether or not the parents would object to this policy change. Of 6,000 sur-veys that go out, 1,200 are returned. Of these 1,200 surveys that were com-pleted, 960 agreed with the policy change and 240 disagreed. Which of thefollowing statements are true?

    I. Some of the mailings may have never reached the parents.

    II. The school district has strong support from parents to move forwardwith the policy approval.

    III. It is possible that majority of the parents of high school studentsdisagree with the policy change.

    IV. The survey results are unlikely to be biased because all parents weremailed a survey.

    (a) Only I (b) I and II (c) I and III (d) III and IV (e) Only IV

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 24 / 94

  • Overview of data collection principles Explanatory and response variables

    Explanatory and response variables

    To identify the explanatory variable in a pair of variables, identifywhich of the two is suspected of affecting the other:

    explanatory variablemight affectresponse variable

    Labeling variables as explanatory and response does notguarantee the relationship between the two is actually causal,even if there is an association identified between the twovariables. We use these labels only to keep track of whichvariable we suspect affects the other.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 25 / 94

  • Overview of data collection principles Observational studies and experiments

    Observational studies and experiments

    Observational study: Researchers collect data in a way that doesnot directly interfere with how the data arise, i.e. they merelyobserve, and can only establish an association between theexplanatory and response variables.

    Experiment: Researchers randomly assign subjects to varioustreatments in order to establish causal connections between theexplanatory and response variables.If youre going to walk away with one thing from this class, let itbe correlation does not imply causation.

    http:// xkcd.com/ 552/

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 26 / 94

  • Overview of data collection principles Observational studies and experiments

    Observational studies and experiments

    Observational study: Researchers collect data in a way that doesnot directly interfere with how the data arise, i.e. they merelyobserve, and can only establish an association between theexplanatory and response variables.Experiment: Researchers randomly assign subjects to varioustreatments in order to establish causal connections between theexplanatory and response variables.

    If youre going to walk away with one thing from this class, let itbe correlation does not imply causation.

    http:// xkcd.com/ 552/

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 26 / 94

  • Overview of data collection principles Observational studies and experiments

    Observational studies and experiments

    Observational study: Researchers collect data in a way that doesnot directly interfere with how the data arise, i.e. they merelyobserve, and can only establish an association between theexplanatory and response variables.Experiment: Researchers randomly assign subjects to varioustreatments in order to establish causal connections between theexplanatory and response variables.If youre going to walk away with one thing from this class, let itbe correlation does not imply causation.

    http:// xkcd.com/ 552/

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 26 / 94

  • Observational studies and sampling strategies

    1 Case study

    2 Data basics

    3 Overview of data collection principles

    4 Observational studies and sampling strategiesConfoundingSampling strategies

    5 Experiments

    6 Examining numerical data

    7 Considering categorical data

    8 Case study: Gender discrimination

    OpenIntro Statistics, 2nd Edition

    Chp 1: Intro. to data

  • Observational studies and sampling strategies Confounding

    http:// www.peertrainer.com/ LoungeCommunityThread.aspx?ForumID=1&ThreadID=3118

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 27 / 94

  • Observational studies and sampling strategies Confounding

    What type of study is this, observational study or an experiment? Girlswho regularly ate breakfast, particularly one that includes cereal, were slimmer than

    those who skipped the morning meal, according to a study that tracked nearly 2,400

    girls for 10 years. [...] As part of the survey, the girls were asked once a year what

    they had eaten during the previous three days.

    This is an observational study since the researchers merely observedthe behavior of the girls (subjects) as opposed to imposing treatmentson them.

    What is the conclusion of the study?

    There is an association between girls eating breakfast and beingslimmer.

    Who sponsored the study?

    General Mills.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 28 / 94

  • Observational studies and sampling strategies Confounding

    What type of study is this, observational study or an experiment? Girlswho regularly ate breakfast, particularly one that includes cereal, were slimmer than

    those who skipped the morning meal, according to a study that tracked nearly 2,400

    girls for 10 years. [...] As part of the survey, the girls were asked once a year what

    they had eaten during the previous three days.

    This is an observational study since the researchers merely observedthe behavior of the girls (subjects) as opposed to imposing treatmentson them.What is the conclusion of the study?

    There is an association between girls eating breakfast and beingslimmer.

    Who sponsored the study?

    General Mills.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 28 / 94

  • Observational studies and sampling strategies Confounding

    What type of study is this, observational study or an experiment? Girlswho regularly ate breakfast, particularly one that includes cereal, were slimmer than

    those who skipped the morning meal, according to a study that tracked nearly 2,400

    girls for 10 years. [...] As part of the survey, the girls were asked once a year what

    they had eaten during the previous three days.

    This is an observational study since the researchers merely observedthe behavior of the girls (subjects) as opposed to imposing treatmentson them.What is the conclusion of the study?

    There is an association between girls eating breakfast and beingslimmer.Who sponsored the study?

    General Mills.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 28 / 94

  • Observational studies and sampling strategies Confounding

    What type of study is this, observational study or an experiment? Girlswho regularly ate breakfast, particularly one that includes cereal, were slimmer than

    those who skipped the morning meal, according to a study that tracked nearly 2,400

    girls for 10 years. [...] As part of the survey, the girls were asked once a year what

    they had eaten during the previous three days.

    This is an observational study since the researchers merely observedthe behavior of the girls (subjects) as opposed to imposing treatmentson them.What is the conclusion of the study?

    There is an association between girls eating breakfast and beingslimmer.Who sponsored the study?

    General Mills.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 28 / 94

  • Observational studies and sampling strategies Confounding

    3 possible explanations

    1. Eating breakfast causes girls to be thinner.

    2. Being thin causes girls to eat breakfast.

    3. A third variable is responsible for both. What could it be?An extraneous variable that affects both the explanatory and theresponse variable and that make it seem like there is arelationship between the two are called confounding variables.

    Images from: http:// www.appforhealth.com/ wp-content/ uploads/ 2011/ 08/ ipn-cerealfrijo-300x135.jpg,

    http:// www.dreamstime.com/ stock-photography-too-thin-woman-anorexia-model- image2814892.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 29 / 94

  • Observational studies and sampling strategies Confounding

    3 possible explanations

    1. Eating breakfast causes girls to be thinner.

    2. Being thin causes girls to eat breakfast.

    3. A third variable is responsible for both. What could it be?An extraneous variable that affects both the explanatory and theresponse variable and that make it seem like there is arelationship between the two are called confounding variables.

    Images from: http:// www.appforhealth.com/ wp-content/ uploads/ 2011/ 08/ ipn-cerealfrijo-300x135.jpg,

    http:// www.dreamstime.com/ stock-photography-too-thin-woman-anorexia-model- image2814892.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 29 / 94

  • Observational studies and sampling strategies Confounding

    3 possible explanations

    1. Eating breakfast causes girls to be thinner.

    2. Being thin causes girls to eat breakfast.

    3. A third variable is responsible for both. What could it be?An extraneous variable that affects both the explanatory and theresponse variable and that make it seem like there is arelationship between the two are called confounding variables.

    Images from: http:// www.appforhealth.com/ wp-content/ uploads/ 2011/ 08/ ipn-cerealfrijo-300x135.jpg,

    http:// www.dreamstime.com/ stock-photography-too-thin-woman-anorexia-model- image2814892.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 29 / 94

  • Observational studies and sampling strategies Confounding

    3 possible explanations

    1. Eating breakfast causes girls to be thinner.

    2. Being thin causes girls to eat breakfast.

    3. A third variable is responsible for both. What could it be?An extraneous variable that affects both the explanatory and theresponse variable and that make it seem like there is arelationship between the two are called confounding variables.

    Images from: http:// www.appforhealth.com/ wp-content/ uploads/ 2011/ 08/ ipn-cerealfrijo-300x135.jpg,

    http:// www.dreamstime.com/ stock-photography-too-thin-woman-anorexia-model- image2814892.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 29 / 94

  • Observational studies and sampling strategies Confounding

    Prospective vs. retrospective studies

    A prospective study identifies individuals and collects informationas events unfold.

    Example: The Nurses Health Study has been recruitingregistered nurses and then collecting data from them usingquestionnaires since 1976.

    Retrospective studies collect data after events have taken place.Example: Researchers reviewing past events in medical records.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 30 / 94

  • Observational studies and sampling strategies Sampling strategies

    Obtaining good samples

    Almost all statistical methods are based on the notion of impliedrandomness.

    If observational data are not collected in a random frameworkfrom a population, these statistical methods the estimates anderrors associated with the estimates are not reliable.

    Most commonly used random sampling techniques are simple,stratified, and cluster sampling.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 31 / 94

  • Observational studies and sampling strategies Sampling strategies

    Simple random sample

    Randomly select cases from the population, where there is no impliedconnection between the points that are selected.

    Index

    Index

    Stratum 1

    Stratum 2

    Stratum 3

    Stratum 4

    Stratum 5

    Stratum 6

    Cluster 1

    Cluster 2

    Cluster 3

    Cluster 4

    Cluster 5

    Cluster 6

    Cluster 7

    Cluster 8

    Cluster 9

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 32 / 94

  • Observational studies and sampling strategies Sampling strategies

    Stratified sample

    Strata are made up of similar observations. We take a simple randomsample from each stratum.

    Index

    Index

    Stratum 1

    Stratum 2

    Stratum 3

    Stratum 4

    Stratum 5

    Stratum 6

    Cluster 1

    Cluster 2

    Cluster 3

    Cluster 4

    Cluster 5

    Cluster 6

    Cluster 7

    Cluster 8

    Cluster 9

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 33 / 94

  • Observational studies and sampling strategies Sampling strategies

    Cluster sample

    Clusters are usually not made up of homogeneous observations, andwe take a simple random sample from a random sample of clusters.Usually preferred for economical reasons.

    Index

    Index

    Stratum 1

    Stratum 2

    Stratum 3

    Stratum 4

    Stratum 5

    Stratum 6

    Cluster 1

    Cluster 2

    Cluster 3

    Cluster 4

    Cluster 5

    Cluster 6

    Cluster 7

    Cluster 8

    Cluster 9

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 34 / 94

  • Observational studies and sampling strategies Sampling strategies

    Practice

    A city council has requested a household survey be conducted in asuburban area of their city. The area is broken into many distinct andunique neighborhoods, some including large homes, some with onlyapartments. Which approach would likely be the least effective?

    (a) Simple random sampling

    (b) Cluster sampling

    (c) Stratified sampling

    (d) Blocked sampling

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 35 / 94

  • Observational studies and sampling strategies Sampling strategies

    Practice

    A city council has requested a household survey be conducted in asuburban area of their city. The area is broken into many distinct andunique neighborhoods, some including large homes, some with onlyapartments. Which approach would likely be the least effective?

    (a) Simple random sampling

    (b) Cluster sampling

    (c) Stratified sampling

    (d) Blocked sampling

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 35 / 94

  • Experiments

    1 Case study

    2 Data basics

    3 Overview of data collection principles

    4 Observational studies and sampling strategies

    5 Experiments

    6 Examining numerical data

    7 Considering categorical data

    8 Case study: Gender discrimination

    OpenIntro Statistics, 2nd Edition

    Chp 1: Intro. to data

  • Experiments

    Principles of experimental design

    1. Control: Compare treatment of interest to a control group.

    2. Randomize: Randomly assign subjects to treatments, andrandomly sample from the population whenever possible.

    3. Replicate: Within a study, replicate by collecting a sufficientlylarge sample. Or replicate the entire study.

    4. Block: If there are variables that are known or suspected to affectthe response variable, first group subjects into blocks based onthese variables, and then randomize cases within each block totreatment groups.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 36 / 94

  • Experiments

    More on blocking

    We would like to design an experiment toinvestigate if energy gels makes you run faster:

    Treatment: energy gelControl: no energy gel

    It is suspected that energy gels might affect proand amateur athletes differently, therefore weblock for pro status:

    Divide the sample to pro and amateurRandomly assign pro athletes to treatment andcontrol groupsRandomly assign amateur athletes totreatment and control groupsPro/amateur status is equally represented inthe resulting treatment and control groups

    Why is this important? Can you think of other variables to block for?

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 37 / 94

  • Experiments

    More on blocking

    We would like to design an experiment toinvestigate if energy gels makes you run faster:

    Treatment: energy gelControl: no energy gel

    It is suspected that energy gels might affect proand amateur athletes differently, therefore weblock for pro status:

    Divide the sample to pro and amateurRandomly assign pro athletes to treatment andcontrol groupsRandomly assign amateur athletes totreatment and control groupsPro/amateur status is equally represented inthe resulting treatment and control groups

    Why is this important? Can you think of other variables to block for?

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 37 / 94

  • Experiments

    More on blocking

    We would like to design an experiment toinvestigate if energy gels makes you run faster:

    Treatment: energy gelControl: no energy gel

    It is suspected that energy gels might affect proand amateur athletes differently, therefore weblock for pro status:

    Divide the sample to pro and amateurRandomly assign pro athletes to treatment andcontrol groupsRandomly assign amateur athletes totreatment and control groupsPro/amateur status is equally represented inthe resulting treatment and control groups

    Why is this important? Can you think of other variables to block for?

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 37 / 94

  • Experiments

    More on blocking

    We would like to design an experiment toinvestigate if energy gels makes you run faster:

    Treatment: energy gelControl: no energy gel

    It is suspected that energy gels might affect proand amateur athletes differently, therefore weblock for pro status:

    Divide the sample to pro and amateurRandomly assign pro athletes to treatment andcontrol groupsRandomly assign amateur athletes totreatment and control groupsPro/amateur status is equally represented inthe resulting treatment and control groups

    Why is this important? Can you think of other variables to block for?

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 37 / 94

  • Experiments

    More on blocking

    We would like to design an experiment toinvestigate if energy gels makes you run faster:

    Treatment: energy gelControl: no energy gel

    It is suspected that energy gels might affect proand amateur athletes differently, therefore weblock for pro status:

    Divide the sample to pro and amateurRandomly assign pro athletes to treatment andcontrol groupsRandomly assign amateur athletes totreatment and control groupsPro/amateur status is equally represented inthe resulting treatment and control groups

    Why is this important? Can you think of other variables to block for?

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 37 / 94

  • Experiments

    Practice

    A study is designed to test the effect of light level and noise level onexam performance of students. The researcher also believes that lightand noise levels might have different effects on males and females,so wants to make sure both genders are equally represented in eachgroup. Which of the below is correct?

    (a) There are 3 explanatory variables (light, noise, gender) and 1response variable (exam performance)

    (b) There are 2 explanatory variables (light and noise), 1 blockingvariable (gender), and 1 response variable (exam performance)

    (c) There is 1 explanatory variable (gender) and 3 response variables(light, noise, exam performance)

    (d) There are 2 blocking variables (light and noise), 1 explanatoryvariable (gender), and 1 response variable (exam performance)

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 38 / 94

  • Experiments

    Practice

    A study is designed to test the effect of light level and noise level onexam performance of students. The researcher also believes that lightand noise levels might have different effects on males and females,so wants to make sure both genders are equally represented in eachgroup. Which of the below is correct?

    (a) There are 3 explanatory variables (light, noise, gender) and 1response variable (exam performance)

    (b) There are 2 explanatory variables (light and noise), 1 blockingvariable (gender), and 1 response variable (exam performance)

    (c) There is 1 explanatory variable (gender) and 3 response variables(light, noise, exam performance)

    (d) There are 2 blocking variables (light and noise), 1 explanatoryvariable (gender), and 1 response variable (exam performance)

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 38 / 94

  • Experiments

    Difference between blocking and explanatory variables

    Factors are conditions we can impose on the experimental units.

    Blocking variables are characteristics that the experimental unitscome with, that we would like to control for.

    Blocking is like stratifying, except used in experimental settingswhen randomly assigning, as opposed to when sampling.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 39 / 94

  • Experiments

    More experimental design terminology...

    Placebo: fake treatment, often used as the control group formedical studies

    Placebo effect: experimental units showing improvement simplybecause they believe they are receiving a special treatment

    Blinding: when experimental units do not know whether they arein the control or treatment group

    Double-blind: when both the experimental units and theresearchers who interact with the patients do not know who is inthe control and who is in the treatment group

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 40 / 94

  • Experiments

    Practice

    What is the main difference between observational studies and exper-iments?

    (a) Experiments take place in a lab while observational studies donot need to.

    (b) In an observational study we only look at what happened in thepast.

    (c) Most experiments use random assignment while observationalstudies do not.

    (d) Observational studies are completely useless since no causalinference can be made based on their findings.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 41 / 94

  • Experiments

    Practice

    What is the main difference between observational studies and exper-iments?

    (a) Experiments take place in a lab while observational studies donot need to.

    (b) In an observational study we only look at what happened in thepast.

    (c) Most experiments use random assignment while observationalstudies do not.

    (d) Observational studies are completely useless since no causalinference can be made based on their findings.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 41 / 94

  • Experiments

    Random assignment vs. random sampling

    Random assignment

    No random assignment

    Random sampling

    Causal conclusion, generalized to the whole

    population.

    No causal conclusion, correlation statement

    generalized to the whole population.

    Generalizability

    No random sampling

    Causal conclusion, only for the sample.

    No causal conclusion, correlation statement only

    for the sample.No

    generalizability

    Causation Correlation

    ideal experiment

    most experiments

    most observational

    studies

    bad observational

    studies

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 42 / 94

  • Examining numerical data

    1 Case study

    2 Data basics

    3 Overview of data collection principles

    4 Observational studies and sampling strategies

    5 Experiments

    6 Examining numerical dataScatterplots for paired dataDot plots and the meanHistograms and shapeVariance and standard deviationBox plots, quartiles, and the medianRobust statisticsTransforming dataMapping data

    7 Considering categorical data

    8 Case study: Gender discrimination

    OpenIntro Statistics, 2nd Edition

    Chp 1: Intro. to data

  • Examining numerical data Scatterplots for paired data

    Scatterplot

    Scatterplots are useful for visualizing the relationship between twonumerical variables.

    Do life expectancy and total fertility appearto be associated or independent?

    They appear to be linearly and negativelyassociated: as fertility increases, lifeexpectancy decreases.

    Was the relationship the same throughoutthe years, or did it change?

    The relationship changed over the years.

    http:// www.gapminder.org/ world

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 43 / 94

  • Examining numerical data Scatterplots for paired data

    Scatterplot

    Scatterplots are useful for visualizing the relationship between twonumerical variables.

    Do life expectancy and total fertility appearto be associated or independent?

    They appear to be linearly and negativelyassociated: as fertility increases, lifeexpectancy decreases.

    Was the relationship the same throughoutthe years, or did it change?

    The relationship changed over the years.

    http:// www.gapminder.org/ world

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 43 / 94

  • Examining numerical data Scatterplots for paired data

    Scatterplot

    Scatterplots are useful for visualizing the relationship between twonumerical variables.

    Do life expectancy and total fertility appearto be associated or independent?

    They appear to be linearly and negativelyassociated: as fertility increases, lifeexpectancy decreases.

    Was the relationship the same throughoutthe years, or did it change?

    The relationship changed over the years.

    http:// www.gapminder.org/ world

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 43 / 94

  • Examining numerical data Dot plots and the mean

    Dot plots

    Useful for visualizing one numerical variable. Darker colors representareas where there are more observations.

    GPA2.5 3.0 3.5 4.0

    How would you describe the distribution of GPAs in this data set? Makesure to say something about the center, shape, and spread of the dis-tribution.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 44 / 94

  • Examining numerical data Dot plots and the mean

    Dot plots & mean

    GPA2.5 3.0 3.5 4.0

    The mean, also called the average (marked with a triangle in theabove plot), is one way to measure the center of a distribution ofdata.

    The mean GPA is 3.59.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 45 / 94

  • Examining numerical data Dot plots and the mean

    Mean

    The sample mean, denoted as x, can be calculated as

    x =x1 + x2 + + xn

    n,

    where x1, x2, , xn represent the n observed values.The population mean is also computed the same way but isdenoted as . It is often not possible to calculate sincepopulation data are rarely available.

    The sample mean is a sample statistic, and serves as a pointestimate of the population mean. This estimate may not beperfect, but if the sample is good (representative of thepopulation), it is usually a pretty good estimate.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 46 / 94

  • Examining numerical data Dot plots and the mean

    Stacked dot plot

    Higher bars represent areas where there are more observations,makes it a little easier to judge the center and the shape of thedistribution.

    GPA

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    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 47 / 94

  • Examining numerical data Histograms and shape

    Histograms - Extracurricular hours

    Histograms provide a view of the data density. Higher barsrepresent where the data are relatively more common.Histograms are especially convenient for describing the shape ofthe data distribution.The chosen bin width can alter the story the histogram is telling.

    Hours / week spent on extracurricular activities0 10 20 30 40 50 60 70

    0

    50

    100

    150

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 48 / 94

  • Examining numerical data Histograms and shape

    Bin width

    Which one(s) of these histograms are useful? Which reveal too muchabout the data? Which hide too much?

    Hours / week spent on extracurricular activities0 20 40 60 80 100

    0

    50

    100

    150

    200

    Hours / week spent on extracurricular activities0 10 20 30 40 50 60 70

    0

    50

    100

    150

    Hours / week spent on extracurricular activities0 10 20 30 40 50 60 70

    0

    20

    40

    60

    80

    Hours / week spent on extracurricular activities0 10 20 30 40 50 60 70

    0

    10

    20

    30

    40

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 49 / 94

  • Examining numerical data Histograms and shape

    Shape of a distribution: modality

    Does the histogram have a single prominent peak (unimodal), severalprominent peaks (bimodal/multimodal), or no apparent peaks(uniform)?

    0 5 10 15

    05

    1015

    0 5 10 15 20

    05

    1015

    0 5 10 15 20

    05

    1015

    20

    0 5 10 15 20

    02

    46

    810

    14

    Note: In order to determine modality, step back and imagine a smooth curve over the

    histogram imagine that the bars are wooden blocks and you drop a limp spaghetti

    over them, the shape the spaghetti would take could be viewed as a smooth curve.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 50 / 94

  • Examining numerical data Histograms and shape

    Shape of a distribution: skewness

    Is the histogram right skewed, left skewed, or symmetric?

    0 2 4 6 8 10

    05

    1015

    0 5 10 15 20 25

    020

    4060

    0 20 40 60 80

    05

    1015

    2025

    30

    Note: Histograms are said to be skewed to the side of the long tail.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 51 / 94

  • Examining numerical data Histograms and shape

    Shape of a distribution: unusual observations

    Are there any unusual observations or potential outliers?

    0 5 10 15 20

    05

    1015

    2025

    30

    20 40 60 80 1000

    1020

    3040

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 52 / 94

  • Examining numerical data Histograms and shape

    Extracurricular activities

    How would you describe the shape of the distribution of hours per weekstudents spend on extracurricular activities?

    Hours / week spent on extracurricular activities0 10 20 30 40 50 60 70

    0

    50

    100

    150

    Unimodal and right skewed, with a potentially unusual observation at60 hours/week.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 53 / 94

  • Examining numerical data Histograms and shape

    Extracurricular activities

    How would you describe the shape of the distribution of hours per weekstudents spend on extracurricular activities?

    Hours / week spent on extracurricular activities0 10 20 30 40 50 60 70

    0

    50

    100

    150

    Unimodal and right skewed, with a potentially unusual observation at60 hours/week.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 53 / 94

  • Examining numerical data Histograms and shape

    Commonly observed shapes of distributions

    modality

    unimodal bimodal multimodaluniform

    skewness

    right skew left skewsymmetric

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 54 / 94

  • Examining numerical data Histograms and shape

    Commonly observed shapes of distributions

    modality

    unimodal

    bimodal multimodaluniform

    skewness

    right skew left skewsymmetric

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 54 / 94

  • Examining numerical data Histograms and shape

    Commonly observed shapes of distributions

    modality

    unimodal bimodal

    multimodaluniform

    skewness

    right skew left skewsymmetric

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 54 / 94

  • Examining numerical data Histograms and shape

    Commonly observed shapes of distributions

    modality

    unimodal bimodal multimodal

    uniform

    skewness

    right skew left skewsymmetric

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 54 / 94

  • Examining numerical data Histograms and shape

    Commonly observed shapes of distributions

    modality

    unimodal bimodal multimodaluniform

    skewness

    right skew left skewsymmetric

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 54 / 94

  • Examining numerical data Histograms and shape

    Commonly observed shapes of distributions

    modality

    unimodal bimodal multimodaluniform

    skewness

    right skew left skewsymmetric

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 54 / 94

  • Examining numerical data Histograms and shape

    Commonly observed shapes of distributions

    modality

    unimodal bimodal multimodaluniform

    skewness

    right skew

    left skewsymmetric

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 54 / 94

  • Examining numerical data Histograms and shape

    Commonly observed shapes of distributions

    modality

    unimodal bimodal multimodaluniform

    skewness

    right skew left skew

    symmetric

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 54 / 94

  • Examining numerical data Histograms and shape

    Commonly observed shapes of distributions

    modality

    unimodal bimodal multimodaluniform

    skewness

    right skew left skewsymmetric

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 54 / 94

  • Examining numerical data Histograms and shape

    Practice

    Which of these variables do you expect to be uniformly distributed?

    (a) weights of adult females

    (b) salaries of a random sample of people from North Carolina

    (c) house prices

    (d) birthdays of classmates (day of the month)

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 55 / 94

  • Examining numerical data Histograms and shape

    Practice

    Which of these variables do you expect to be uniformly distributed?

    (a) weights of adult females

    (b) salaries of a random sample of people from North Carolina

    (c) house prices

    (d) birthdays of classmates (day of the month)

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 55 / 94

  • Examining numerical data Histograms and shape

    Application activity: Shapes of distributions

    Sketch the expected distributions of the following variables:

    number of piercings

    scores on an exam

    IQ scores

    Come up with a concise way (1-2 sentences) to teach someone howto determine the expected distribution of any variable.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 56 / 94

  • Examining numerical data Histograms and shape

    Are you typical?

    http:// www.youtube.com/ watch?v=4B2xOvKFFz4

    How useful are centers alone for conveying the true characteristics ofa distribution?

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 57 / 94

  • Examining numerical data Histograms and shape

    Are you typical?

    http:// www.youtube.com/ watch?v=4B2xOvKFFz4

    How useful are centers alone for conveying the true characteristics ofa distribution?

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 57 / 94

  • Examining numerical data Variance and standard deviation

    Variance

    Variance is roughly the average squared deviation from the mean.

    s2 =n

    i=1(xi x)2n 1

    The sample mean is x = 6.71,and the sample size isn = 217.

    The variance of amount ofsleep students get per nightcan be calculated as:

    Hours of sleep / night2 4 6 8 10 12

    0

    20

    40

    60

    80

    s2 =(5 6.71)2 + (9 6.71)2 + + (7 6.71)2

    217 1 = 4.11 hours2

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 58 / 94

  • Examining numerical data Variance and standard deviation

    Variance

    Variance is roughly the average squared deviation from the mean.

    s2 =n

    i=1(xi x)2n 1

    The sample mean is x = 6.71,and the sample size isn = 217.

    The variance of amount ofsleep students get per nightcan be calculated as:

    Hours of sleep / night2 4 6 8 10 12

    0

    20

    40

    60

    80

    s2 =(5 6.71)2 + (9 6.71)2 + + (7 6.71)2

    217 1 = 4.11 hours2

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 58 / 94

  • Examining numerical data Variance and standard deviation

    Variance

    Variance is roughly the average squared deviation from the mean.

    s2 =n

    i=1(xi x)2n 1

    The sample mean is x = 6.71,and the sample size isn = 217.

    The variance of amount ofsleep students get per nightcan be calculated as:

    Hours of sleep / night2 4 6 8 10 12

    0

    20

    40

    60

    80

    s2 =(5 6.71)2 + (9 6.71)2 + + (7 6.71)2

    217 1 = 4.11 hours2

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 58 / 94

  • Examining numerical data Variance and standard deviation

    Variance (cont.)

    Why do we use the squared deviation in the calculation of variance?

    To get rid of negatives so that observations equally distant fromthe mean are weighed equally.

    To weigh larger deviations more heavily.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 59 / 94

  • Examining numerical data Variance and standard deviation

    Variance (cont.)

    Why do we use the squared deviation in the calculation of variance?

    To get rid of negatives so that observations equally distant fromthe mean are weighed equally.

    To weigh larger deviations more heavily.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 59 / 94

  • Examining numerical data Variance and standard deviation

    Standard deviation

    The standard deviation is the square root of the variance, and has thesame units as the data.s

    s =s2

    The standard deviation ofamount of sleep students getper night can be calculatedas:

    s =4.11 = 2.03 hours

    We can see that all of thedata are within 3 standarddeviations of the mean.

    Hours of sleep / night2 4 6 8 10 12

    0

    20

    40

    60

    80

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 60 / 94

  • Examining numerical data Variance and standard deviation

    Standard deviation

    The standard deviation is the square root of the variance, and has thesame units as the data.s

    s =s2

    The standard deviation ofamount of sleep students getper night can be calculatedas:

    s =4.11 = 2.03 hours

    We can see that all of thedata are within 3 standarddeviations of the mean.

    Hours of sleep / night2 4 6 8 10 12

    0

    20

    40

    60

    80

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 60 / 94

  • Examining numerical data Variance and standard deviation

    Standard deviation

    The standard deviation is the square root of the variance, and has thesame units as the data.s

    s =s2

    The standard deviation ofamount of sleep students getper night can be calculatedas:

    s =4.11 = 2.03 hours

    We can see that all of thedata are within 3 standarddeviations of the mean.

    Hours of sleep / night2 4 6 8 10 12

    0

    20

    40

    60

    80

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 60 / 94

  • Examining numerical data Box plots, quartiles, and the median

    Median

    The median is the value that splits the data in half when orderedin ascending order.

    0, 1, 2, 3, 4

    If there are an even number of observations, then the median isthe average of the two values in the middle.

    0, 1, 2, 3, 4, 5 2 + 32

    = 2.5

    Since the median is the midpoint of the data, 50% of the valuesare below it. Hence, it is also the 50th percentile.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 61 / 94

  • Examining numerical data Box plots, quartiles, and the median

    Q1, Q3, and IQR

    The 25th percentile is also called the first quartile, Q1.

    The 50th percentile is also called the median.

    The 75th percentile is also called the third quartile, Q3.

    Between Q1 and Q3 is the middle 50% of the data. Therange these data span is called the interquartile range, or the IQR.

    IQR = Q3 Q1

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 62 / 94

  • Examining numerical data Box plots, quartiles, and the median

    Box plot

    The box in a box plot represents the middle 50% of the data, and thethick line in the box is the median.

    # of

    stu

    dy h

    ours

    / we

    ek

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    70

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 63 / 94

  • Examining numerical data Box plots, quartiles, and the median

    Anatomy of a box plot#

    of s

    tudy

    hou

    rs /

    week

    0

    10

    20

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    40

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    70

    lower whisker

    Q1 (first quartile)medianQ3 (third quartile)

    max whisker reach& upper whisker

    suspected outliers

    lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 64 / 94

  • Examining numerical data Box plots, quartiles, and the median

    Whiskers and outliers

    Whiskers of a box plot can extend up to 1.5 IQR away from thequartiles.

    max upper whisker reach = Q3 + 1.5 IQRmax lower whisker reach = Q1 1.5 IQR

    IQR : 20 10 = 10max upper whisker reach = 20 + 1.5 10 = 35max lower whisker reach = 10 1.5 10 = 5

    A potential outlier is defined as an observation beyond themaximum reach of the whiskers. It is an observation thatappears extreme relative to the rest of the data.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 65 / 94

  • Examining numerical data Box plots, quartiles, and the median

    Whiskers and outliers

    Whiskers of a box plot can extend up to 1.5 IQR away from thequartiles.

    max upper whisker reach = Q3 + 1.5 IQRmax lower whisker reach = Q1 1.5 IQR

    IQR : 20 10 = 10max upper whisker reach = 20 + 1.5 10 = 35max lower whisker reach = 10 1.5 10 = 5

    A potential outlier is defined as an observation beyond themaximum reach of the whiskers. It is an observation thatappears extreme relative to the rest of the data.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 65 / 94

  • Examining numerical data Box plots, quartiles, and the median

    Whiskers and outliers

    Whiskers of a box plot can extend up to 1.5 IQR away from thequartiles.

    max upper whisker reach = Q3 + 1.5 IQRmax lower whisker reach = Q1 1.5 IQR

    IQR : 20 10 = 10max upper whisker reach = 20 + 1.5 10 = 35max lower whisker reach = 10 1.5 10 = 5

    A potential outlier is defined as an observation beyond themaximum reach of the whiskers. It is an observation thatappears extreme relative to the rest of the data.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 65 / 94

  • Examining numerical data Box plots, quartiles, and the median

    Outliers (cont.)

    Why is it important to look for outliers?

    Identify extreme skew in the distribution.

    Identify data collection and entry errors.

    Provide insight into interesting features of the data.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 66 / 94

  • Examining numerical data Box plots, quartiles, and the median

    Outliers (cont.)

    Why is it important to look for outliers?

    Identify extreme skew in the distribution.

    Identify data collection and entry errors.

    Provide insight into interesting features of the data.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 66 / 94

  • Examining numerical data Robust statistics

    Extreme observations

    How would sample statistics such as mean, median, SD, and IQR ofhousehold income be affected if the largest value was replaced with$10 million? What if the smallest value was replaced with $10 million?

    Annual Household Income

    ll lll

    ll ll

    lll

    ll l

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    ll

    lll

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    l lll

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    l

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    l l

    l

    0e+00 2e+05 4e+05 6e+05 8e+05 1e+06

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 67 / 94

  • Examining numerical data Robust statistics

    Robust statistics

    Annual Household Income

    ll lll

    ll ll

    lll

    ll l

    llll

    ll

    lll

    l

    l lll

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    0e+00 2e+05 4e+05 6e+05 8e+05 1e+06

    robust not robustscenario median IQR x soriginal data 190K 200K 245K 226Kmove largest to $10 million 190K 200K 309K 853Kmove smallest to $10 million 200K 200K 316K 854K

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 68 / 94

  • Examining numerical data Robust statistics

    Robust statistics

    Median and IQR are more robust to skewness and outliers than meanand SD. Therefore,

    for skewed distributions it is often more helpful to use medianand IQR to describe the center and spread

    for symmetric distributions it is often more helpful to use themean and SD to describe the center and spread

    If you would like to estimate the typical household income for a student,would you be more interested in the mean or median income?

    Median

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 69 / 94

  • Examining numerical data Robust statistics

    Robust statistics

    Median and IQR are more robust to skewness and outliers than meanand SD. Therefore,

    for skewed distributions it is often more helpful to use medianand IQR to describe the center and spread

    for symmetric distributions it is often more helpful to use themean and SD to describe the center and spread

    If you would like to estimate the typical household income for a student,would you be more interested in the mean or median income?

    Median

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 69 / 94

  • Examining numerical data Robust statistics

    Robust statistics

    Median and IQR are more robust to skewness and outliers than meanand SD. Therefore,

    for skewed distributions it is often more helpful to use medianand IQR to describe the center and spread

    for symmetric distributions it is often more helpful to use themean and SD to describe the center and spread

    If you would like to estimate the typical household income for a student,would you be more interested in the mean or median income?

    Median

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 69 / 94

  • Examining numerical data Robust statistics

    Mean vs. median

    If the distribution is symmetric, center is often defined as themean: mean median

    Symmetric

    meanmedian

    If the distribution is skewed or has extreme outliers, center isoften defined as the median

    Right-skewed: mean > medianLeft-skewed: mean < median

    Rightskewed

    meanmedian

    Leftskewed

    meanmedian

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 70 / 94

  • Examining numerical data Robust statistics

    Practice

    Which is most likely true for the distribution of percentage of time actuallyspent taking notes in class versus on Facebook, Twitter, etc.?

    % of time in class spent taking notes0 20 40 60 80 100

    0

    10

    20

    30

    40

    50

    (a) mean> median

    (b) mean < median

    (c) mean median(d) impossible to tell

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 71 / 94

  • Examining numerical data Robust statistics

    Practice

    Which is most likely true for the distribution of percentage of time actuallyspent taking notes in class versus on Facebook, Twitter, etc.?

    % of time in class spent taking notes0 20 40 60 80 100

    0

    10

    20

    30

    40

    50 median: 80%mean: 76%

    (a) mean> median

    (b) mean < median

    (c) mean median(d) impossible to tell

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 71 / 94

  • Examining numerical data Transforming data

    Extremely skewed data

    When data are extremely skewed, transforming them might makemodeling easier. A common transformation is the log transformation.

    The histograms on the left shows the distribution of number ofbasketball games attended by students. The histogram on the rightshows the distribution of log of number of games attended.

    # of basketball games attended0 10 20 30 40 50 60 70

    0

    50

    100

    150

    # of basketball games attended0 1 2 3 4

    0

    10

    20

    30

    40

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 72 / 94

  • Examining numerical data Transforming data

    Extremely skewed data

    When data are extremely skewed, transforming them might makemodeling easier. A common transformation is the log transformation.

    The histograms on the left shows the distribution of number ofbasketball games attended by students. The histogram on the rightshows the distribution of log of number of games attended.

    # of basketball games attended0 10 20 30 40 50 60 70

    0

    50

    100

    150

    # of basketball games attended0 1 2 3 4

    0

    10

    20

    30

    40

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 72 / 94

  • Examining numerical data Transforming data

    Pros and cons of transformations

    Skewed data are easier to model with when they are transformedbecause outliers tend to become far less prominent after anappropriate transformation.

    # of games 70 50 25 log(# of games) 4.25 3.91 3.22

    However, results of an analysis might be difficult to interpretbecause the log of a measured variable is usually meaningless.

    What other variables would you expect to be extremely skewed?

    Salary, housing prices, etc.

    OpenIntro Statistics, 2nd Edition Chp 1: Intro. to data 73 / 94

  • Examining numerical data Transforming data

    Pros and cons of transformations

    Skewed data are easier to model with when they are transformedbecause outliers tend to become far less prominent after anappropriate transformation.

    # of games 70 50 25 log(# of games) 4.25 3.91 3.22

    However, results of an analysis might be difficult to interpretbecause the log of a