MTH 265: The Nature of Probability and Statistics Notes

What is Statistics?

  • Statistics is the science of conducting studies to:
    • Collect data
    • Organize data
    • Summarize data
    • Analyze data
    • Draw conclusions from data

Variables and Data

  • Variables: Characteristics or attributes that can assume different values.
  • Data: The values that the variables can assume.

Descriptive and Inferential Statistics

  • Descriptive Statistics:
    • Collection of data
    • Organization of data
    • Summarization of data
    • Presentation of data
  • Inferential Statistics:
    • Generalizing from samples to populations
    • Performing estimations
    • Performing hypothesis tests
    • Determining relationships among variables
    • Making predictions

Population and Sample

  • Population: All subjects that are being studied.
  • Sample: A group of subjects selected from a population.

Qualitative and Quantitative Variables

  • Qualitative Variables:
    • Used to identify (can be numbers or non-numeric).
  • Quantitative Variables:
    • Numerical only.
    • Can be ordered or ranked.

Discrete and Continuous Variables

  • Discrete Variables:
    • Assume values that can be counted.
  • Continuous Variables:
    • Assume values that cannot be counted.

Examples of Discrete Variables

  • Children in a family.
  • Number of students in a classroom.
  • Number of calls received by a switchboard.

Examples of Continuous Variables

  • Temperature (infinite number of readings between any two measurements).
  • Time.
  • Length.
  • Typically, any measurement is continuous.

Boundaries of Recorded Values

  • Rounding is sometimes necessary in recorded values.
  • Boundaries are established in these cases.
  • Example: A recorded value of 15 centimeters has boundaries of 14.5 – 15.5 cm.
    • Values are up to 15.5 but not including 15.5.

Levels of Measurement

  • Nominal:
    • Cannot be ordered.
    • Typically names only.
  • Ordinal:
    • Data can be placed into categories.
    • Data can be ranked.
  • Interval:
    • Ranks data.
    • Differences in measurements exist.
    • No meaningful zero.
  • Ratio:
    • Has all characteristics of interval.
    • Has a true zero.
    • True ratios exist.

Examples of Nominal

  • ZIP Code
  • Gender
  • Eye Color
  • Political Affiliation
  • Religious Affiliation
  • Major Field
  • Nationality

Examples of Ordinal

  • Grade (A, B, C, D, F)
  • Judging (1st place, 2nd place, 3rd place)
  • Rating Scale (poor, good, excellent)
  • Ranking of football teams

Examples of Interval

  • SAT score
  • IQ
  • Temperature

Examples of Ratio

  • Height
  • Weight
  • Time
  • Salary
  • Age

Data Collection Techniques

  • Random Sampling:
    • Subjects are selected by random numbers.
  • Systematic Sampling:
    • Subjects are selected by using every kth subject after the first subject is randomly selected from 1 through k.
  • Stratified Sampling:
    • Subjects are selected by dividing up the population into groups (strata), and subjects are randomly selected within groups.
  • Cluster Sampling:
    • The population is divided into groups (clusters), clusters are randomly chosen, and every subject within these clusters is used.
  • Convenience Sampling:
    • Subjects that are convenient are used (e.g., interviewing subjects entering a local mall).

Observational and Experimental Studies

  • Observational Study:
    • The researcher simply observes what is happening.
  • Experimental Study:
    • The researcher controls one of the variables and tries to determine its influence on other variables.
    • Example: An experimental group and control group in a medical experiment.