Statistics Fundamentals

Statistics

• Help organize and summarize data
• Aid researchers in drawing general conclusions

Population and Samples

• Population: all individuals of interest
• Sample: a selection of individuals from the population, intended to represent the larger population

Variables and Data

• Variable: a characteristic or condition that changes
• Data: measurements and observations
  • Data set: a collection of measurements or observations
  • Datum: a single measurement or observation
  • Score/raw data: unprocessed data

Parameters and Statistics

• Parameter: a value describing a population (denoted by P)
  • Usually derived from population measurements
• Statistic: a value describing a sample (denoted by S)
  • Usually derived from sample measurements

Descriptive and Inferential Statistics

• Descriptive statistics: organize, simplify, and summarize data
• Inferential statistics: techniques for studying samples and drawing generalizations about populations

Sampling Error

• Naturally occurring discrepancies between a sample statistic and its corresponding population parameter
• The goal of inferential statistics is to distinguish between sampling errors and real differences.

Individual Variables and Relationships Among Variables

• Some studies describe individual variables.
• Many studies explore relationships between two or more variables.

The Correlational Method

• Observe two or more variables to determine if a relationship exists.
• Correlation: numerical scores
• Chi-square test: categorical data
• A correlational study can show relationships but cannot explain cause and effect.

Experimental and Nonexperimental Methods

• Experimental studies manipulate one variable to determine its effect on another.
• Non-experimental studies observe relationships between variables without manipulation.
  • Experimental studies can determine cause-and-effect relationships.
  • Non-experimental studies cannot determine cause-and-effect relationships.

The Experimental Method

• Goal: demonstrate a cause-and-effect relationship
  • Two characteristics:
    • Manipulation
    • Control

Terminology in The Experimental Method

• Independent variable: manipulated by the researcher
• Dependent variable: measured by the researcher to assess the effect of the independent variable
• Control condition: participants do not receive the experimental treatment
• Experimental condition: Participants receive the experimental treatment

Non-Experimental Method: Nonequivalent Groups

• Researchers compare preexisting groups
• Cannot determine cause and effect reliably.

Non-Experimental Method: Pre-Post Studies

• Measure the same variable twice for each participant: once before and once after a treatment.
• Cannot determine cause and effect reliably.

Terminology in Non-experimental Research

• Quasi-independent variable: a variable that the researcher cannot manipulate.
• Measuring variables: involves recording observations to establish relationships.

Constructs and Operational Definitions

• Constructs: theoretical concepts.
• Operational definitions: Procedures for measuring constructs.

Types of Variables

• Discrete variables: have separate categories and no values between.
• Continuous variables: can take on any value within a range.

Continuous Variables

• Real limits: The boundaries of the intervals of scores, that are represented on a continuous number line
• Upper real limit: at the top
• Lower real limit: at the bottom

Measuring Variables

• Scale of measurement describes the nature of the categories required to measure a variable.
• Nominal: Different categories; no numerical relation
• Ordinal: Different categories organized in an ordered sequence.
• Interval: Ordered categories; intervals represent equal differences.
• Ratio: Ordered categories: intervals are equal; there is a true zero point.

Four Types of Measurement Scales

• Nominal scale: Categorizes observations using different names
• Ordinal scale: Categorizes observations in an ordered sequence
• Interval scale: Ordered categories; intervals are equal
• Ratio scale: Ordered categories; intervals are equal; true zero point

Statistical Notation

• Σ (Sigma): Represents a sum of numbers
• X: Represents a variable
• N: Represents a population
• n: Represents a sample

Frequency Distributions

• Organized tabulation of how many individuals fall into categories within a distribution.
• Can be tables or graphs.

Frequency Distribution Tables

• Lists categories; values listed from highest to lowest.
• Lists frequency of each value.
• Sum of frequencies = total number of participants.

Frequency Distribution Graphs

• Graphs for frequency distributions: histograms and polygons.
• Histograms: Bars touch one another, representing numerical scores.
• Polygons: Uses dots to represent scores, connected by a line
• Categorical data: bar graphs

Graphs For Population Distributions

• Show frequencies for each score in the population

Relative Frequencies

• Useful when exact frequency is not known; Proportions are presented instead

Smooth Curve

• Useful when scores are continuous variables
• Indicates the overall shape and distribution is not showing exact frequencies, but an overall probability trend

Bar Graph

• Used when scale is Nominal of Ordinal
• Spaces between bars

Describing Frequency Distributions

• Characteristics: Shape, variability, central tendency.

Shape of Distribution

• Symmetrical: Mirror images.
• Skewed: Scores pile up at one end with gradually decreasing frequency at the other end.
  • Positively skewed: tail extends to the right.
  • Negatively skewed: tail extends to the left.

Stem-and-Leaf Displays

• Efficiently displays frequency distributions
• Shows individual scores

Central Tendency

• Goal: Find a single score that represents a typical, or average value in a distribution
• Mean: The balance point of the distribution (the average).
• Median: The midpoint of the distribution.
• Mode: The most frequently occurring score.

When to use Measures of Central Tendency

• Extreme scores or skewed distributions: Median may be more appropriate.
• Undetermined/unknown values: Median or Mode are more suitable.
• Open-ended distributions: Median or Mode may be more suitable.
• Ordinal data: Median might be more appropriate measure.

Variability

• Measures how much scores differ from one another, how spread out scores are.
• Range: The difference between the largest and smallest scores
• Standard deviation: Average distance from the mean

Standard Deviation and Variance

• Standard deviation: Average distance of scores from the mean
• Variance: Average of the squared deviations from the mean

Sample variability and Degrees of freedom

• df=(n-1), in estimating the population variance from sample data.