Introduction to Data: Variables, Relationships, and Study Design in Statistics

Numerical Variables

Variables that can be classified as continuous or discrete.

Continuous Variables

Variables that can take on an infinite number of values, including fractional values.

Discrete Variables

Variables that can only take on whole numbers.

Categorical Variables

Qualitative groupings of data that cannot be analyzed using arithmetic.

Ordinal Variables

Categories that follow a natural progression.

Associated Variables

Variables that show some relationship with one another.

Positive Association

When one variable increases, the other also increases.

Negative Association

When one variable increases, the other decreases.

Independent Variables

Variables that have no relationship with one another.

Explanatory Variable

The variable suspected of affecting another variable.

Response Variable

The variable that may be affected by the explanatory variable.

Observational Studies

Studies where the researcher has no interaction with subjects and gathers data without interference.

Experimental Studies

Studies where researchers test relationships between variables and often involve treatment assignments.

Confounding Variables

Variables that are connected to both the explanatory and response variable.

Sources of Bias

Factors that can introduce bias in data collection, such as selection bias and nonresponse bias.

Simple Random Sampling

Each subject in the population is equally likely to be selected.

Stratified Sampling

Dividing the population into homogeneous strata and randomly sampling from each stratum.

Cluster Sampling

Dividing the population into clusters and sampling all cases within selected clusters.

Multistage Sampling

Dividing the population into clusters, randomly sampling clusters, and then sampling within those clusters.

Control in Experimental Design

Controlling differences between groups to reduce confounding variables.

Randomization in Experimental Design

Randomly assigning subjects to account for uncontrollable differences.

Replication in Experimental Design

Repeating a study or using a larger sample size to strengthen results.

Blocking in Experimental Design

Grouping subjects and variables into blocks to control for additional variables.

Single Blinding

Patients are not informed about their treatment.

Double Blinding

Both patients and doctors/nurses are unaware of treatment assignments.

Generalizability of Study Results

A study's results can be generalized when random sampling has been applied for the target population.

Causation Inference

A causation can be inferred when random assignment has been applied.

Random Sample and Assignment

Yes, Yes: Generalize to the population and infer causation.

Random Sampling

Random sampling allows for generalizability because it is indicative of the population at large which was sampled.

Random Assignment

Random assignment allows for making causal conclusions because it eliminates the introduction of confounding variables from specific populations.

Cluster Sampling Efficiency

Cluster sampling is more efficient when it is too difficult and costly to attempt random or stratified sampling.

Blinding

Blinding helps eliminate the placebo effect and other biases by ensuring unbiased data collection.

Scatterplots

Scatterplots show a case-by-case view of two variables, indicating positive or negative relationships.

Describing Numerical Distribution

When describing a numerical variable, mention its shape, center, spread, and unusual observations.

Shape of Numerical Variable

The shape can be right or left skewed or symmetric.

Measures of Center

Common measures of center include mean, median, and mode.

Measures of Spread

Common measures of spread include standard deviation, range, and interquartile range.

Variance

Variance is roughly the average squared distance from the mean.

Robust Statistics

Robust statistics are measures that are not heavily affected by skewness and extreme outliers.

Log Transformation

Log transformations can make the distribution of data more symmetric and easier to model.

Frequency Tables

Frequency tables examine all aspects of one categorical variable.

Contingency Tables

Contingency tables summarize data for two categorical variables.

Hypothesis Testing

Hypothesis testing involves repeating experiments to determine if results are due to random chance.

Null Hypothesis

The null hypothesis states that there is no association between the variables being investigated.

Alternative Hypothesis

The alternative hypothesis states that there is an association between the variables being investigated.

Jurors and Hypothesis Testing

Even if jurors are unconvinced of guilt, it does not mean they believe the defendant is innocent.

Box Plots

Box plots summarize data sets using simple statistics and extreme values.

Histograms

Histograms graphically represent the distribution of numerical data.

Side-by-Side Box Plots

Side-by-side box plots assess the relationship between a numerical and a categorical variable.

Unusual Observations

Unusual observations are classified as outliers after statistical analysis.