Sample Mean:
  x̄ = (x₁ + x₂ + ... + xₙ) / n
  x̄ = Σxᵢ / n

Population Mean:
  μ = Σxᵢ / N

Example: {3, 7, 7, 19, 24}
  x̄ = (3+7+7+19+24)/5
  x̄ = 60/5 = 12

Weighted Mean:
  x̄ᵤ = Σ(wᵢ × xᵢ) / Σwᵢ
  GPA example:
    A(4)×3cr + B(3)×4cr + A(4)×3cr
    = (12+12+12)/10 = 3.6 GPA

Geometric Mean (growth rates):
  GM = (x₁ × x₂ × ... × xₙ)^(1/n)
  Returns: +10%, −5%, +20%
  GM = (1.1×0.95×1.2)^(1/3) = 1.0772
  → 7.72% avg annual return

The arithmetic mean is the 'balance point' of a dataset. It uses ALL values, which makes it sensitive to outliers. The geometric mean is better for percentages and growth rates.

Finding the Median:
  1. Sort data ascending
  2. If n is odd: median = middle value
     position = (n+1)/2
  3. If n is even: median = average of
     two middle values
     positions = n/2 and n/2+1

Example (odd n=5): {3, 7, 7, 19, 24}
  Sorted: 3, 7, [7], 19, 24
  Position: (5+1)/2 = 3rd value
  Median = 7

Example (even n=6): {3, 7, 7, 19, 24, 30}
  Sorted: 3, 7, [7, 19], 24, 30
  Positions: 3rd and 4th
  Median = (7+19)/2 = 13

With outlier: {3, 7, 7, 19, 24, 1000}
  Mean = 176.7 (skewed by 1000!)
  Median = 13 (barely changed)
  → Median is ROBUST to outliers

The median is the 50th percentile — exactly half the data falls below it. It's preferred for skewed data (income, house prices, reaction times) because outliers can't distort it.

Finding the Mode:
  Count frequency of each value
  Mode = value(s) with highest frequency

Unimodal: {1, 2, 2, 3, 4}
  Mode = 2 (appears twice)

Bimodal: {1, 2, 2, 3, 3, 4}
  Modes = 2 and 3 (both appear twice)

Multimodal: {1,1, 2,2, 3,3}
  Three modes: 1, 2, 3

No mode: {1, 2, 3, 4, 5}
  All values appear once
  (some say 'no mode')

Categorical data (mode only option):
  Colors: {red, blue, blue, green, red,
           blue, green}
  Mode = blue (3 occurrences)
  Mean/median don't apply here

For grouped/continuous data:
  Modal class = class with highest freq
  Mode ≈ L + [(f₁−f₀)/(2f₁−f₀−f₂)] × h

Mode is the only central tendency measure for categorical (nominal) data. It's also used in fashion ('à la mode' = in style), reflecting 'what's most common.'

Trimmed Mean (robust):
  Remove top/bottom k% of data
  Calculate mean of remaining
  Olympic scoring: drop highest
  and lowest judge (trimmed mean)

Harmonic Mean (rates):
  HM = n / Σ(1/xᵢ)
  Average speed: 60mph out, 40mph back
  HM = 2/(1/60 + 1/40) = 48 mph
  NOT simple mean of 50 mph!

Geometric Mean (growth):
  GM = (∏xᵢ)^(1/n)
  Investment returns, population growth
  Always ≤ Arithmetic Mean

Midrange:
  (max + min) / 2
  Quick but not robust

Relationship (for positive data):
  Harmonic ≤ Geometric ≤ Arithmetic
  Equal only when all values equal
  Called the AM-GM-HM inequality

Different averages suit different data types. Harmonic mean for rates, geometric mean for growth/compounding, trimmed mean for outlier-resistant averaging. Know which one to use for your context.