Chapter 1 Notes: Basic Ideas (Sampling, Data Types, Design of Experiments, Bias)

Sampling (Section 1.1: Terminology)

• Statistics is the science of planning studies and experiments, obtaining, organizing, presenting, analyzing, interpreting data, and drawing conclusions.

• Population: the entire collection of individuals about which information is sought.

• Sample: a subset of the population containing the individuals that are actually observed.

• Census: the collection of information from every member of the population.

• Sampling: a simple random sample of size $n$ is a sample chosen by a method in which each collection of population items is equally likely to make up the sample.

  • Example: From a class, a group of $5$ people is needed. Everyone’s name is written on a ticket and placed in a raffle drum. Each student has an equal chance to be part of the sample (group).

• Convenience sample: a sample that is not drawn by a well-defined random method.

  • Example: Asking customers waiting in line.

• Problems with samples of convenience: they may differ systematically in some way from the population.

• Systematic sampling: in every $k$-th item, a sample is chosen (e.g., every $k$-th person).

  • The value of $k$ can be randomly chosen (you count with a randomly chosen interval).

• Cluster sampling: the population is divided into groups called clusters (subgroups), then all of the items from some randomly chosen clusters are selected.

  • Examples: Classes at a college split into sections; a dormitory split into floors. Example: Floors 1, 4, and 7 are selected.

• Stratified sampling: the population is divided into strata (subgroups), then a simple random sample is drawn from each stratum.

  • Examples: People categorized as Democrats or Republicans; employees categorized as faculty, staff, maintenance, and other.

• Cluster vs. Stratified sampling (key differences):

  • In cluster sampling, every member of the selected group(s) is part of the sample.
  • In stratified sampling, subjects are selected from each subgroup.
  • The key behind cluster sampling is that all members of a group become part of the sample once the group is selected.
  • The key behind stratified sampling is that members are selected from all groups to be part of the sample.

• Voluntary response samples are often used by the media to engage the audience (e.g., radio call-in).

  • Voluntary response bias: People who volunteer an opinion tend to have stronger opinions than typical; those with negative opinions are often more likely to volunteer.

• A representative sample reflects as closely as possible the relevant characteristics of the population under consideration.

  • Question example: If you were estimating the percentage of ASU students who got their flu shot last year, who would you sample?

• Statistic vs Parameter:

  • A statistic is a measurement describing a characteristic of a sample.
  • A parameter is a measurement describing a characteristic of a population.

• Example 1.1:

  • a) In a survey on driving habits, 33% of respondents reported texting while driving. $ext{statistic}$
  • b) The average score for Super Bowl winners is 30.05 points. $ext{parameter}$
  • c) The Chicago Cubs have won 51.3% of all the games they’ve ever played. $ext{parameter}$

Types of Data (Section 1.2)

• Terminology: Variables are the characteristics of individuals on which we collect information.
  • Collectively, the values of the variable we record/measure for each individual are the data.
• Qualitative vs Quantitative Data
  • Qualitative (categorical/attribute) data are non-numerically valued data; they consist of labels or names.
  • Quantitative data consist of numbers representing counts or measurements (numerical data).
• Types of Qualitative Data
  • Ordinal data have a natural ordering.
  • Nominal data do not have a natural ordering.
• Types of Quantitative Data
  • Discrete data have possible values that can be listed or counted.
  • Continuous data can take on any value in some interval; typically from a measurement with an uncountable number of possible values.
• Example 1.2:
  • a) Height — likely continuous data.
  • b) Number of siblings — discrete data.
  • c) Place of birth — qualitative, nominal data.
  • d) Age — typically continuous data (though it can be treated as discrete in some contexts).

Design of Experiments (Section 1.3)

• Terminology:
  • Experimental units: the individuals who are studied. These can be people, animals, plants, or things.
  • When the experimental units are people, they are sometimes called subjects.
  • Treatments: the procedures applied to each experimental unit.
  • The response variable (also called outcome) is what is measured on each experimental unit.
• Types of Studies:
  • In a randomized experiment, we randomly assign treatments and observe the effects on the experimental units (modification).
  • In an observational study, we observe or measure specific characteristics of the experimental units without assigning any treatments (no modification).
• Example 1.3 (observational vs experimental):
  • a) Observational: high vs low air pollution areas; asks people to report how many colds they had in the past year.
  • b) Experimental: a pharmaceutical company compares a new cold medicine to a currently used medicine by assigning groups to receive the new vs the current medication and recording symptoms.

Bias in Studies (Section 1.4)

• Terminology:
  • Studies conducted with methods that tend to overestimate or underestimate the true value are said to be biased.
  • A study conducted by a procedure that produces the correct result on the average is said to be unbiased.
• Bias: Generally the goal is to minimize bias; highly biased studies yield unreliable conclusions.
• Sources of Bias:
  • Voluntary Response Bias: Opinions of volunteers do not reflect the population; voluntary response surveys are highly biased.
  • Self-Interest Bias: People with an interest in the outcome may bias methods to favor their outcome.
  • Advertisements: (context often implies bias in how information is presented or framed).
  • Social Acceptability Bias: People may be reluctant to admit behavior that reflects negatively on them (e.g., asking about voting or other sensitive behaviors).
  • Leading Question Bias: Wording of questions suggests a particular response (e.g., "What is your opinion on decreasing taxes for middle-class families?" vs. "Do you favor decreasing the heavy tax burden on middle-class families?")
• Non-Response Bias: Surveys with a significant proportion of non-responders are often biased; non-responders' opinions may differ from responders.
• Sampling Bias: Occurs when some members of the population are more likely to be included in the sample than others.
  • Samples of convenience almost always have sampling bias.

Additional Notes on Bias and Sampling (Wrap-up)

• Always consider who is included vs who is left out in a sample to assess potential bias.
• Randomization, representative sampling, and minimizing non-response are key methods to reduce bias in studies.
• Distinguish between observational and experimental designs when evaluating potential biases, as biases can operate differently in each design.