σ = √[ Σᵢ(xᵢ − μ)² / N ]

Step-by-step:
  1. Calculate mean: μ = Σxᵢ / N
  2. Subtract mean from each value
  3. Square each difference
  4. Sum the squares
  5. Divide by N
  6. Take square root

Example: Data = {2, 4, 4, 4, 5, 5, 7, 9}
  μ = (2+4+4+4+5+5+7+9)/8 = 40/8 = 5
  Differences: -3,-1,-1,-1,0,0,2,4
  Squares: 9,1,1,1,0,0,4,16 → sum=32
  σ² = 32/8 = 4
  σ = √4 = 2

Use σ only when you have data for the ENTIRE population — all test scores, all factory outputs, etc. This is rare in practice.

s = √[ Σᵢ(xᵢ − x̄)² / (n − 1) ]

Bessel's Correction: divide by (n−1)
  • Compensates for using x̄ instead of μ
  • Makes s² an unbiased estimator of σ²
  • Note: s itself is slightly biased

Example: Same data, but as sample
  x̄ = 5 (same)
  Sum of squares = 32 (same)
  s² = 32/(8−1) = 32/7 = 4.571
  s = √4.571 = 2.138

As n → ∞, s → σ
  n=10: n−1=9  (10% correction)
  n=100: n−1=99 (1% correction)
  n=1000: negligible difference

Almost always use sample SD in practice. Dividing by n−1 corrects the bias from estimating the mean from the same sample.

Variance:
  σ² = Σ(xᵢ − μ)² / N      (population)
  s² = Σ(xᵢ − x̄)² / (n−1)  (sample)

Variance properties:
  Var(X+Y) = Var(X) + Var(Y)
    (if X,Y independent)
  Var(cX) = c² × Var(X)
  Var(X+c) = Var(X)

Coefficient of Variation (CV):
  CV = (s / x̄) × 100%

Example:
  Test A: mean=80, SD=10 → CV=12.5%
  Test B: mean=50, SD=10 → CV=20%

Same SD, but B is more variable
relative to its mean.

CV allows comparing variability between datasets with different units or scales. A stock with CV=30% is more volatile than one with CV=10% regardless of price.

z = (x − μ) / σ   (population)
z = (x − x̄) / s   (sample)

Interpretation:
  z = 0:   at the mean
  z = 1:   1 SD above mean
  z = −2:  2 SD below mean

Example: SAT score = 1350
  Mean (μ) = 1060, SD (σ) = 195
  z = (1350 − 1060) / 195 = 1.49
  → 93.2 percentile

Standardization:
  Converts any normal distribution
  to N(0,1) — standard normal
  Enables comparison across scales

Outlier detection:
  |z| > 2: unusual (5% chance)
  |z| > 3: rare (0.3% chance)

Z-scores transform raw data into a 'how many SDs from the mean?' scale, enabling comparison between SAT, GPA, height, weight — any normally distributed variable.