Chapter 2 — Descriptive Statistics¶

1. Statistics Fundamentals¶

Term	Definition
Statistics	The discipline of collecting, organizing, analyzing, interpreting, and presenting data
Population	The complete set of instances of interest (e.g., all students in a school)
Sample	A subset of the population collected by a defined procedure
Sampling	Selecting and analyzing a subset to estimate population values in a quantified way

Induction vs. Deduction¶

Type	Direction	Description
Induction (inference)	Sample → Population	Generalizing from sample to population; larger sample = closer estimate
Deduction	Population → Sample	Reasoning about a sample drawn from a known population (probability is deduction)

Descriptive Statistics¶

Methods for summarizing and visualizing samples to help humans understand data.

Scope	Name	Definition
Single attribute	Univariate	Frequency tables, statistics, plots for one column
Two attributes	Bivariate	Relationships between pairs of columns
3+ attributes	Multivariate	Relationships across many columns

2. Univariate Analysis — Frequencies¶

Frequency Type	Definition	Formula
Absolute	Count of occurrences of a value	count(v)
Relative	Fraction of total occurrences	count(v) / n
Absolute cumulative	Count of values ≤ v	sum of abs. freqs up to v
Relative cumulative	Fraction of values ≤ v	sum of rel. freqs up to v

The relative frequency column defines an empirical frequency distribution.
The relative cumulative frequency column defines an empirical CDF.

A discrete attribute has a probability mass function (PMF)
A continuous attribute has a probability density function (PDF)

3. Univariate Analysis — Location Statistics¶

Location statistics return a central value from a set. Also called measures of central tendency.

Statistic	Definition	Scale validity
Minimum	Smallest value	Ordinal, Quantitative
Maximum	Largest value	Ordinal, Quantitative
Mode	Most frequent value	All scales (Nominal, Ordinal, Quantitative)
Median	Middle value in sorted order	Ordinal, Quantitative
Mean	Sum / n	Quantitative (eventually Ordinal for numeric scales)

Median formula (sorted dataset of n values): - n odd → median = x[(n+1)/2] - n even → median = (x[n/2] + x[n/2 + 1]) / 2

Robustness: Median and mode are more robust than mean when data is skewed or has extreme values.

Quartiles¶

Quartiles split sorted data into four equal parts. There are multiple methods; the position method is the most general:

Count n observations and sort them.
For Q_k (k = 1, 2, 3): compute n × (k/4).
If the result is an integer → Q_k = mean of values at positions n×(k/4) and n×(k/4)+1
If not an integer → round up; Q_k = value at that position

Quartile	Also known as	Interpretation
Q1	Lower quartile	Larger than 25% of values
Q2	Median	Larger than 50% of values
Q3	Upper quartile	Larger than 75% of values

4. Univariate Analysis — Dispersion Statistics¶

Dispersion statistics measure how spread out the values are.

Statistic	Formula	Notes
Amplitude	max − min	Sensitive to outliers
Interquartile Range (IQR)	Q3 − Q1	Robust to outliers
MAD (Mean Absolute Deviation)	Σ\|xᵢ − x̄\| / (n−1)	Average absolute distance from mean (sample)
Standard Deviation	√(Σ(xᵢ − x̄)² / (n−1))	Typical distance from mean (sample)
Variance	s² = Σ(xᵢ − x̄)² / (n−1)	Square of std dev

Population vs. Sample formulas:

	Population	Sample
Mean	μ	x̄
Std Dev	σ = √(Σ(xᵢ−μ)²/n)	s = √(Σ(xᵢ−x̄)²/(n−1))
MAD	Σ\|xᵢ−μ\|/n	Σ\|xᵢ−x̄\|/(n−1)

Dividing by n−1 (Bessel's correction) gives an unbiased estimate of the population variance from a sample.

5. Univariate Analysis — Visualization¶

Chart	Best for	Notes
Pie chart	Nominal scales	Parts of a whole
Bar chart	Qualitative or quantitative with few values	Easy comparison
Line chart	Time / ordered data	Emphasizes trend
Area chart	Comparing distributions or time series	Filled region under curve
Histogram	Quantitative (continuous) distributions	Groups values into bins (cells)
Box plot	Showing quartiles + spread	Min, Q1, median, Q3, Max
Stacked bar	Frequency of one attribute split by another	Shows sub-group composition

Histogram bin count rule of thumb: use ≈ √n bins (where n = number of data points). The best value is always problem-dependent.

Box Plot Anatomy¶

  Max ─┬─ whisker top
       │
  Q3  ─┼─ top of box
       │
  Q2  ─┼─ median line
       │
  Q1  ─┼─ bottom of box
       │
  Min ─┴─ whisker bottom

Skewness¶

Skewness measures the asymmetry of a distribution.

Skew type	Shape	Tail direction	mean vs. median
Symmetric	Bell	Both equal	mean ≈ median
Positive (right) skew	Right tail longer	Mass concentrated on left	mean > median
Negative (left) skew	Left tail longer	Mass concentrated on right	mean < median

A unimodal distribution has exactly one peak (mode).

6. Common Univariate Distributions¶

Uniform Distribution x ~ U(a, b)¶

Values equally distributed in [a, b]; zero probability outside.

PDF: f(x) = 1/(b−a) for a ≤ x ≤ b, else 0
CDF: P(x < x₀) = (x₀−a)/(b−a) for a ≤ x₀ ≤ b

Normal (Gaussian) Distribution x ~ N(μ, σ)¶

Symmetric bell curve, fully defined by mean μ and standard deviation σ.

μ: locates the peak (center)
σ: controls width (spread)
~68% of values within μ ± σ; ~95% within μ ± 2σ; ~99.7% within μ ± 3σ

In a sample we speak of proportions (relative frequencies).
In a population we speak of probabilities.

7. Bivariate Analysis — Two Quantitative Attributes¶

Visualizations¶

Chart	Purpose
Scatter plot	Visualize distribution of paired values; reveals general trends
3D histogram	Show joint frequency of two quantitative attributes

Covariance¶

Measures the degree of linear co-variation between two attributes.

cov(xi, xj) = (1/(n−1)) × Σ (xki − x̄i)(xkj − x̄j)

Sign	Meaning
Positive	Both attributes increase/decrease together
Negative	One increases as the other decreases
Near zero	No linear relationship

Limitation: covariance value depends on the scale of the attributes (not unit-free).

Pearson's Correlation Coefficient (r)¶

Scale-independent version of covariance. Always in [−1, 1].

r(xi, xj) = cov(xi, xj) / (si × sj)

Value	Interpretation
+1	Perfect positive linear relationship
0	No linear relationship
−1	Perfect negative linear relationship

Spearman's Rank Correlation (ρ)¶

Pearson's r applied to the ranks of the values instead of the values themselves.

Use when data is ordinal, non-linear, or has outliers
Less sensitive to outliers than Pearson (outlier gets capped at its rank value)
ρ = 1 if the relationship is any monotonically increasing function (not just linear)

How to compute: 1. Sort each attribute and assign ranks 1, 2, …, n. For ties, use the average rank. 2. Apply Pearson's formula to the rank columns.

Pearson vs Spearman: When data is roughly elliptical with no outliers, both give similar results. Spearman is preferred for ordinal data or when outliers are present.

8. Bivariate Analysis — Qualitative + Quantitative¶

Use grouped box plots: one box plot per category value, all on the same axis.

X-axis: categories of the qualitative attribute
Y-axis: values of the quantitative attribute
Reveals whether the quantitative distribution differs across groups

9. Bivariate Analysis — Two Qualitative Attributes¶

Contingency Table¶

Matrix of joint absolute (or relative) frequencies. - Rows: values of attribute 1 - Columns: values of attribute 2 - Row totals on right; column totals on bottom; grand total bottom-right

Mosaic Plot¶

Visual equivalent of a contingency table — areas are proportional to relative joint frequencies.

10. Bivariate Analysis — Two Ordinal Attributes¶

Use the same techniques as the qualitative case plus:

Spearman's rank correlation (preferred over Pearson for ordinal data)
Scatter plots with jitter — add a small random offset to each point to prevent overlapping dots at the same (rank, rank) position

Quick-Reference: Choosing the Right Method¶

Attribute types	Best statistic	Best visualization
Nominal (1)	Mode, frequency table	Pie, bar chart
Ordinal (1)	Median, quartiles	Bar chart, box plot
Quantitative (1)	Mean, std dev, quartiles	Histogram, box plot
Quant + Quant	Pearson r, covariance	Scatter plot
Qual + Quant	Grouped stats per category	Grouped box plots
Qual + Qual	Contingency table	Mosaic plot
Ordinal + Ordinal	Spearman ρ	Jitter scatter plot