Statistical Thinking – Data Types & Descriptive Statistics (Quick Review)

Introduction

• Focus: Gathering & understanding data; descriptive statistics for quick comparison.

Data Types

• Two broad classes:
  • Numerical / Quantitative
  • Categorical / Qualitative

Numerical Data

• Quantity that can be counted or measured.
  • Discrete
  • Countable, separate values (e.g., number of weekly posts ${0,1,2,\dots}$)
  • Continuous
  • Any value in a range (e.g., time on social media $1.5\,\text{hrs}$)

Categorical Data

• Quality or category labels.
  • Nominal
  • No inherent order (Instagram / Facebook / Snapchat)
  • Ordinal
  • Ordered categories without exact spacing (UI rating: Poor < Fair < Good < Excellent)

Quick Recap Table

• Nominal | Ordinal | Discrete | Continuous

Descriptive Statistics — The 3 M’s

• Mean $\bar{x}=\frac{\sum<em>{i=1}^{n}x</em>i}{n}$
• Median Middle value after sorting.
• Mode Most frequent value/category.

Properties of the Mean

• Add constant $c$ to every data point ⇒ $\bar{x}_{\text{new}} = \bar{x}+c$
• Multiply every data point by $k$ ⇒ $\bar{x}_{\text{new}} = k\,\bar{x}$

Mean vs. Median vs. Mode

• Mean sensitive to outliers (e.g., ${2,3,4,5,6,20}$ ⇒ $\bar{x}=6.67$ ≠ typical).
• Median preferred when extreme values exist.
• Mode preferred when:
  • Data are categorical / non-numeric.
  • Interest is in most common choice.
  • Distribution highly skewed (income example).

Moving (Window) Average

• Compute mean over a sliding window (e.g., 3-point moving average) to smooth time-series fluctuations.

Choosing the Right Measure

• Symmetric, outlier-free numeric data → Mean.
• Skewed or outlier-heavy numeric data → Median.
• Categorical or desire for most common value → Mode.

Key Takeaways

• Distinguish data type first; it guides valid statistics.
• Understand how transformations affect mean.
• Be aware of outliers before selecting a measure of center.
• Moving averages help reveal trends in sequential data.