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
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.