Descriptive Statistics & Data Collection

Why Statistics?

Statistics is the science of turning observations into actionable decisions. In business, research, and consulting, statistics helps answer questions about what is happening, for whom, and how much variation exists. A useful analysis connects a question, a variable, an output, and a decision.

• Research: Identifies patterns and variation.

• Business: Guides action and monitoring.

• Consulting: Provides evidence and interprets meaning.

The Evidence Workflow

Statistical analysis follows a structured workflow:

1. Question: Define what needs to be known.

2. Cases: Identify who or what is observed.

3. Variables: Determine how each concept is measured.

4. Description: Detect patterns in the data.

5. Communication: Draw responsible conclusions.

Defining and Collecting Data

Data Structure

In a dataset, each row represents a case (e.g., a customer transaction), each column represents a variable (e.g., region, satisfaction), and each cell contains a recorded value.

Measurement Levels

The type of variable determines the analysis, not its appearance:

• Nominal: Categories without order (e.g., channel, region). Summarized by counts, percentages, and mode.

• Ordinal: Ordered categories (e.g., satisfaction ratings). Summarized by counts, percentages, and median.

• Quantitative (Scale): Numeric values (e.g., basket value, wait time). Summarized by mean, median, standard deviation (SD), and interquartile range (IQR).

Organizing and Visualizing Variables

Univariate Analysis: Qualitative Variables

Categorical analysis answers questions about frequency and proportion:

• Count: = number of cases in category

• Percentage:

• Mode: Most frequent category

Common chart types include sorted bar charts, 100% stacked bars, horizontal bars, and pie charts (used sparingly).

Univariate Analysis: Quantitative Variables

Quantitative analysis requires measures of center and spread:

• Mean: Arithmetic average, sensitive to outliers.

• Median: Middle value, robust to extremes.

• Mode: Most common value.

• Range: Difference between maximum and minimum.

• IQR: Spread of the middle 50% ().

• Standard Deviation (SD): Typical distance from the mean.

• Coefficient of Variation (CV):

Choice of summary depends on data shape: mean for symmetric data, median for skewed or outlier-prone data.

Population vs. Sample Mean

The mean is calculated differently for populations and samples:

• Population Mean:

• Sample Mean:

Population vs. Sample Standard Deviation

Standard deviation measures spread, with formulas differing for population and sample:

• Population SD:

• Sample SD:

Visual Summaries

• Bar Chart: For categorical variables.

• Histogram: For quantitative variables, shows distribution shape.

• Boxplot: Visualizes the five-number summary (min, Q1, median, Q3, max) and identifies outliers.

Numerical Descriptive Measures

Worked Example: Mean vs. Median

One extreme value can shift the mean but not the median, highlighting the importance of inspecting distribution shape before choosing a summary statistic.

Dispersion Measures

• Range:

• IQR:

• Sample SD:

• Coefficient of Variation:

Bivariate Analysis

Bivariate Roadmap

The types of two variables determine the appropriate analysis:

• Qualitative × Qualitative: Crosstab and conditional percentages.

• Qualitative × Quantitative: Group summaries and boxplots.

• Quantitative × Quantitative: Scatterplot and correlation.

Crosstabs

Crosstabs reveal how one categorical distribution changes across another. Conditional percentages (row, column, total) clarify the direction of comparison.

Group Comparisons

Comparing quantitative variables across groups involves examining means/medians, spread (SD/IQR), sample sizes, and overlap in distributions.

Scatterplots and Correlation

Scatterplots show direction, form, strength, and anomalies in the relationship between two quantitative variables. Pearson correlation () standardizes co-movement:

• Covariance:

• Pearson correlation: ,

Correlation captures linear association but not causality or nonlinear relationships.

Sampling and Inference

Sample vs. Population

Descriptive statistics summarize observed data (sample), while inferential statistics estimate population parameters and quantify uncertainty.

Sources of Bias

• Sampling variability: Different samples yield different estimates.

• Coverage bias: Some population members cannot be selected.

• Nonresponse: Respondents differ from nonrespondents.

• Measurement error: Recorded values may not match intended concepts.

Generalization Checklist

• Target population: Who should the result describe?

• Sampling design: How were cases selected?

• Uncertainty method: How variable is the estimate across samples?

Summary Table: Variable Types and Summaries

Conclusion

Descriptive statistics provide the foundation for understanding data, guiding responsible communication, and preparing for inferential methods such as probability, sampling distributions, confidence intervals, and hypothesis testing.