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Statistics Basics for Trading

Difficulty beginner

Descriptive Statistics

Central Tendency

Measure Formula When to Use
Mean x̄ = Σxᵢ/n Normally distributed data
Median Middle value of sorted data Skewed distributions, outliers
Mode Most frequent value Categorical data

Dispersion

Measure Formula Interpretation
Range max - min Total spread
Variance σ² = Σ(xᵢ - x̄)²/n Average squared deviation
Standard Deviation σ = √σ² Average deviation from mean
MAD Σ xᵢ - x̄
IQR Q3 - Q1 Middle 50% spread

Skewness and Kurtosis

Skewness — Asymmetry of distribution 

Skewness = E[(X - μ)³] / σ³

Positive skew: Long right tail (occasional large gains)
Negative skew: Long left tail (occasional large losses)
Zero skew: Symmetric

where: μ mean · σ standard deviation · numerator = third central moment · division by σ³ makes the measure scale-free. does: measures distributional asymmetry. Equity returns typically print negative skew — crashes are sharper than rallies — which Sharpe ratio ignores.

Kurtosis — Tail thickness 

Kurtosis = E[(X - μ)⁴] / σ⁴

Excess Kurtosis = Kurtosis - 3 (excess over normal)

Positive excess: Fat tails (extreme events more likely)
Zero excess: Normal distribution
Negative excess: Thin tails

where: μ mean · σ standard deviation · fourth central moment normalized by σ⁴ · subtracting 3 centres the metric on the normal distribution. does: quantifies tail thickness relative to a normal. Real return series routinely show excess kurtosis of 5–20 — the reason VaR-from-normal massively understates tail risk.

Probability Distributions

Normal Distribution

f(x) = (1 / σ√(2π)) × e^(-(x-μ)²/(2σ²))

68% of data within ±1σ
95% of data within ±2σ
99.7% of data within ±3σ

where: μ mean · σ standard deviation · f(x) density at x · e Euler's number · the prefactor 1/(σ√(2π)) is the normalizing constant making total area = 1. does: the gaussian bell curve. The default null model for returns despite known fat-tail violations — every parametric VaR, Black-Scholes, and z-test ultimately rests on this density.

Trading Application: Modeling returns (approximation), VaR calculations, options pricing.

Log-Normal Distribution

Asset prices (not returns) are typically log-normally distributed: 

If ln(S) ~ N(μ, σ²), then S ~ LogNormal(μ, σ²)

Prices cannot go negative
Returns can be negative

where: S price level · ln(S) log-price · N(μ, σ²) normal distribution with given mean/variance · LogNormal distribution of exp of a normal. does: the standard price-process assumption: log-prices are normal, so prices are positive and right-skewed. Underlies geometric Brownian motion and Black-Scholes.

Student's t-Distribution

Heavier tails than normal distribution
Degrees of freedom (ν) control tail thickness
As ν → ∞, converges to normal

where: ν degrees of freedom (lower ν = fatter tails) · in finance ν is typically estimated in the 3–6 range for daily returns. does: a fat-tailed alternative to the normal. Used for return modelling, robust regression, and small-sample t-tests where the variance is unknown.

Trading Application: Better models for asset returns (fat tails observed in practice).

Fat Tails in Financial Data

Financial returns consistently exhibit: - Negative skewness (crashes > rallies) - Excess kurtosis (fat tails) - Volatility clustering (GARCH effects)

Sampling and Estimation

Sample vs. Population

Sample Population
Mean x̄ μ
Variance s² = Σ(xᵢ - x̄)²/(n-1) σ² = Σ(xᵢ - μ)²/N
Std Dev s σ

Note the Bessel correction: n-1 for sample variance.

Standard Error

SE = σ / √n

Measures precision of sample mean as estimate of population mean
Larger samples → smaller SE → more precise estimates

where: SE standard error of the sample mean · σ population (or sample) standard deviation · n sample size. does: standard deviation of the sampling distribution of the mean. Halving SE requires 4× the trades — the brutal arithmetic of statistical significance in finance.

Confidence Intervals

CI = x̄ ± z × SE

95% CI: z = 1.96
99% CI: z = 2.576
90% CI: z = 1.645

where: x̄ sample mean · z standard-normal quantile for chosen confidence level · SE standard error of the mean. does: range of plausible values for the true mean. Used to report performance with honest uncertainty bands instead of a point estimate.

Hypothesis Testing

Framework

  1. Null Hypothesis (H₀) — Default assumption (e.g., strategy has no edge)
  2. Alternative Hypothesis (H₁) — What you want to prove (e.g., strategy has positive edge)
  3. Significance Level (α) — Probability of Type I error (typically 0.05)
  4. Test Statistic — Calculated from data
  5. P-value — Probability of observing result if H₀ is true
  6. Decision — Reject H₀ if p-value < α

Common Tests in Trading

Test Purpose Example
t-test Compare means Strategy returns vs. zero
F-test Compare variances Volatility before vs. after
Chi-square Test independence Signal vs. outcome
Mann-Whitney Non-parametric comparison Robust alternative to t-test

Type I vs. Type II Errors

H₀ True H₀ False
Reject H₀ Type I Error (α) Correct (Power)
Fail to Reject Correct Type II Error (β)

In Trading Context: - Type I: Concluding strategy works when it doesn't (false positive) - Type II: Concluding strategy doesn't work when it does (false negative)

Covariance and Correlation

Covariance

Cov(X,Y) = Σ(xᵢ - x̄)(yᵢ - ȳ) / (n-1)

Positive: Variables move together
Negative: Variables move oppositely
Zero: No linear relationship

where: xᵢ, yᵢ paired observations · x̄, ȳ sample means · n-1 Bessel-corrected denominator. does: scale-dependent measure of joint variation. Raw covariance is hard to interpret across assets — divide by σₓσᵧ for the unitless correlation.

Correlation

ρ(X,Y) = Cov(X,Y) / (σₓ × σᵧ)

Range: -1 to +1
+1: Perfect positive correlation
0: No linear correlation
-1: Perfect negative correlation

where: Cov(X,Y) covariance · σₓ, σᵧ standard deviations of X and Y · ρ Pearson correlation coefficient. does: standardised joint-movement metric. The atomic building block of portfolio risk, pair trading, and diversification analysis — but only captures linear dependence.

Correlation ≠ Causation

Just because two assets are correlated does not mean one causes the other. Spurious correlations are common in finance.

Practical Applications

In Strategy Development

  1. Distribution Analysis — Check if returns are normal (they're not)
  2. Statistical Significance — Is your edge real or noise?
  3. Correlation Analysis — Portfolio diversification
  4. Rolling Statistics — Detect regime changes
  5. Hypothesis Testing — Validate strategy assumptions

Common Pitfalls

Pitfall Problem Solution
Assuming normality Returns have fat tails Use robust statistics
Ignoring autocorrelation Returns not independent Use HAC standard errors
Small sample sizes Unreliable estimates Require sufficient data
Look-ahead bias Using future information Proper walk-forward testing
Multiple testing Finding false patterns Apply Bonferroni correction

Key Formulas Reference

Mean: x̄ = Σxᵢ/n
Variance: σ² = Σ(xᵢ - x̄)²/(n-1)
Std Dev: σ = √σ²
Z-score: z = (x - μ) / σ
Correlation: ρ = Cov(X,Y) / (σₓ × σᵧ)
Standard Error: SE = σ / √n
95% CI: x̄ ± 1.96 × SE

Next Steps