Chapter 1: The Nature of Probability and Statistics

Descriptive and Inferential Statistics

General Terms

Statistics: Science of collecting, analyzing and drawing results from data

Statistics studies variables: factors that can assume varied values - by collecting data: specific values of the variable.

Example: Studying the models of cars in a lot; “car model” is the variable; the specific car model, such as “Toyota” or “Honda”, is the data.

Population: The entire set of subjects that are being studied

Sample: A selected portion of the population

It is often unfeasible to study the entire population; samples are used instead

Example: All the cars in a lot are the population. The cars in a select section of the lot is a sample.

Census: Study in which data was collected from every subject in the population

Descriptive vs Inferential

Descriptive Statistics: Summary and analysis of collected data

Inferential Statistics: Inferences made based on collected data

Example: If a sample is studied, analyzing and summarizing the sample data is descriptive; applying the sample data to the rest of the population is inferential.

Inferential statistics uses probability: predicting an event’s chances

Example: When inferring from a sample to a population, statistics predicts the probability of whether or not the sample deviated from the population

Statistics Application

Statistics also studies random variables: variables whose value is determined by chance

Statistics also study the relationship between variables

Example: Studying the color distribution among varied car models

Variables and Types of Data

Types of Variables

Variable types:

• Qualitative Variable: Values are distinct categories

  Example: Car model

• Quantitative Variable: Values are distinct quantities

  There are two types of quantitative variables:

  • Discrete Variables: Values are discrete quantitative units

    Example: Number of wheels

  • Continuous Variables: Values are a point within a continuous range

    Example: Speed

Rounded Values

Values are often rounded. A value’s boundaries is the range within which values are rounded to it.

Levels of Measurement

Values have levels by which they are measured:

• Nominal Level: Qualitative distinctions which are not inherently ranked

  Example: Car model

• Ordinal Level: Qualitative distinctions which are inherently ranked

  Example: Rating on a scale of good, gooder, goodest

• Interval Level: Quantitative distinctions with set intervals, but does not tally a quantity

  Properties:

  • Does not contain an absolute zero
  • Quantities cannot be ratioed - ‘4’ is more than ‘2’ but not ‘double’ 2

  Example: Temperature (C or F) -

  • Measures in set interval degrees, but does not tally a quantity - 40 degrees represents a distinct temperature, not a set of 40 instances
  • 0 degrees similarly represents a specific temperature and does not mean “no temperature”
  • 40 degrees is not ‘double’ 20 degrees.

• Ratio Level: Quantitative distinctions that tally instances

  Example: Number of cars

Data Collection and Sampling Techniques

Methods of Data Collection

There are numerous method of collecting data, including:

• Survey
  • Telephone survey
  • Mailed survey
  • Personal interview
• Record checking
• Direct observation

Biased Samples

A biased sample is one that does not accurately represent the population.

Even if a sample is correctly chosen and is non-biased, there will still be inevitable differences between the sample and population, which can lead to errors when inferring population data from the sample data. Such error are sampling errors: errors due to the mere fact that a sample was used.

However, if the sample was incorrectly chosen and is biased, a subsequent error is a non-sampling error.

Sampling Techniques

When choosing a sample, it is important to avoid choosing a biased sample.

Sample choosing techniques:

• Random Sample: All population members have an equal chance of being selected, and are randomly selected
• Systematic Sample: Population members are numbered; every $k$th member of the population is chosen
• Stratified Sample: The population is divided into groups based on a relevant characteristic; members are then chosen at random from each group
• Cluster Sample: The population is divided into clusters; clusters are randomly chosen and every member of the cluster is studied
• Additional methods: Convenience sample, volunteer / self-selected sample

Experimental Design

Types of Experiment

Observational Study: The researcher observes and collects data but does not experimentally manipulate variables

• Cross Sectional Observational Study: Collects data from a particular point in time
• Retrospective Observational Study: Collects data from past records
• Longitudinal Observational Study: Collects data over a period of time

Experimental Study: The researcher experimentally manipulates a variable to see how it affects the outcome

• Independent / Explanatory Variable: Variable being manipulated by the researcher
• Dependent Variable / Outcome Variable: Resultant variable that is not being actively manipulated

• Often uses two groups: Control Group, in which the independent variable is not manipulated, and Treatment Group, which is treated with the manipulated independent variable.

  Example: Studying if a medication cures a disease; the control group is not treated with the medication, the treatment group is. ‘Medication treatment’ is the independent variable, whether or not the disease is cured is the dependent variable.

Quasi-Experimental Study: An experimental study should use groups with randomly assigned members. A quasi-experimental study uses groups with pre-assigned members.

Example: Testing effects of a medication on hospital patients; a quasi-experimental study may use patients in one hospital as the control group and patients in another hospital as the treatment group, as opposed to randomly choosing patients from both hospitals for both groups

Confounding Variables

Confounding / Lurking Variable: Unaccounted-for variable which affects the outcome, causing researchers to incorrectly attribute the outcome variation to an accounted-for variable

Potential confounding variables in experimental studies:

• Hawthorne Effect: Subjects can behave abnormally due to the fact that they’re aware that they’re being studied
• Placebo Effect: Subjects can respond to a treatment due to their perception that the treatment is effective, even if it is in fact not

Experiment Techniques

Techniques to avoid placebo effect:

• Blinding: Subjects are not informed whether or not they’ve received a real treatment
• Double Blinding: Even if the subjects are unaware whether or not they’ve received a real treatment, if the treatment administrator knows which subjects have received the real treatment, that too may influence the outcome. In a double blinded study, the treatment administrators too are not informed which subjects are receiving the real treatment.

Further techniques to avoid confounding variables:

• Blocking: If there is a potentially confounding variable which varies across the subjects, the subjects may be divided into separate blocks along the variable value, and the experiment is repeated in each block.
  • Where blocking is not used, the experiment is called completely randomized design.
• Matched Pair: A specific highly matched pair is split between the control and treatment group
• Replication: The experiment is repeated with varied group members; lowers the chance of unknown confounding variables