Understanding Statistics: Definition and Classification

Definition and Classifications of Statistics

• Statistics can be defined in two ways: plural and singular senses
• Plural sense (layman's definition): Statistics are an aggregate or collection of numerical facts
• Singular sense (formal definition): Statistics involves collecting, organizing, presenting, analyzing, and interpreting numerical data to aid effective decision-making
• Statistics is divided into two main areas based on data use: descriptive and inferential

Descriptive Statistics

• Concerned with summary calculations, graphs, charts, and tables

Inferential Statistics

• Used to generalize from a sample to a population
• Example: Estimating the average income of all families in Ethiopia using data from a few hundred families
• Statistical data often arises from a sample

Stages in Statistical Investigation

• There are five stages in any statistical investigation:
• Collection of data: Measuring, gathering, and assembling raw data
• Data collection methods include surveys, with common methods being telephone surveys, mailed questionnaires, and personal interviews
• Organization of data: Summarizing data in a meaningful way, such as in table form
• Presentation of data: Re-organizing, classifying, compiling, and summarizing data for meaningful presentation
• Analysis of data: Extracting relevant information from summarized data using elementary mathematical operations
• Inference of data: Interpreting and further observing statistical measures through data analysis to form conclusions and inferences
• Probability theory based statistical techniques are required at the inference stage

Definitions

• Statistical Population: A collection of all possible observations of a specified characteristic of interest
• Sample: A subset of the population, selected using a sampling technique to represent the population
• Sampling: The process or method of sample selection from the population
• Sample size: The number of elements or observations included in the sample
• Census: Complete enumeration or observation of all elements in the population
• Parameter: A characteristic or measure obtained from a population
• Statistic: A characteristic or measure obtained from a sample
• Variable: An item of interest that can take on many different numerical values

Types of Variables or Data

• Qualitative Variables: Nonnumeric variables that cannot be measured (e.g., gender, religious affiliation, state of birth)
• Quantitative Variables: Numerical variables that can be measured
  • Discrete variables: Can assume only certain values with gaps between them (e.g., number of bedrooms in a house)
  • Continuous variables: Can assume any value within a specific range (e.g., air pressure in a tire)

Applications, Uses, and Limitations of Statistics

• Statistics is applied in almost all fields of human endeavor
• It is used for obtaining numerical facts in daily life (e.g., prices), in processes like drug invention or pollution assessment, and especially in industries for quality control
• The main function of statistics is to expand knowledge of complex phenomena
• Statistics presents facts precisely, reduces data, measures variations, enables comparison, estimates population characteristics, tests hypotheses, studies variable relationships, and forecasts future events
• Limitations of statistics:
  • Deals only with quantitative information and aggregate facts, not individual data items
  • Statistical data are approximate, not mathematically exact
  • Statistics can be misused, requiring expert handling

Scales of Measurement

• Proper knowledge of data nature is essential for applying correct statistical methods
• Measurement scale: Property of value assigned to data based on order, distance, and fixed zero properties.
• Measurement, in mathematical terms, is a functional mapping from a set of objects to a set of real numbers

Goal of Measurement Systems

• To structure the rule for assigning numbers to objects so that the relationship between objects is preserved in the assigned numbers

Order Property

• Exists when an object with more of an attribute is given a bigger number by the rule system
• For all i, j, if O₁ > Oⱼ, then M(O₁) > M(Oⱼ)

Distance Property

• Concerned with the relationship of differences between objects
• The unit of measurement means the same thing throughout the scale of numbers
• For all i, j, k, l, if Oᵢ - Oⱼ ≥ Oₖ - Oₗ, then M(Oᵢ) - M(Oⱼ) ≥ M(Oₖ) - M(Oₗ)

Fixed Zero Property

• A measurement system possesses a rational zero if an object with none of the attribute is assigned the number zero
• Fixed Zero exists if M(O₀) = 0 (where O₀ is an object with none of the attribute in question)
• Property of fixed zero is needed for ratios between numbers to be meaningful

Scale Types

• Measurement: Assigning numbers to objects or events systematically
• Four common levels: nominal, ordinal, interval, and ratio, each with different properties

Nominal Scales

• Possess none of the order, distance, and fixed zero properties
• Classifies data into mutually exclusive and all-inclusive categories without order or ranking
• No arithmetic and relational operations can be applied
• Examples: political party preference, sex, marital status, country code, regional differentiation

Ordinal Scales

• Possess the property of order but not distance
• The property of fixed zero is not important if the property of distance is not satisfied
• Classifies data into categories that can be ranked, but differences between ranks do not exist
• Arithmetic operations are not applicable, but relational operations are
• Ordering is the sole property
• Examples: letter grades, rating scales, military status

Interval Scales

• Possess order and distance properties, but not fixed zero
• Classifies data that can be ranked, and differences are meaningful
• No meaningful zero, so ratios are meaningless
• All arithmetic operations except division are applicable
• Relational operations are also possible
• Examples: IQ, temperature in °F

Ratio Scales

• Possess all three properties: order, distance, and fixed zero
• Fixed zero allows meaningful interpretation of ratios
• Example: the ratio of Bekele's height to Martha's height is 1.32
• Level of measurement where data can be ranked, differences are meaningful, and there is a true zero
• True ratios exist between the different units of measure
• All arithmetic and relational operations are applicable
• Examples: weight, height, number of students, age