Standard Deviation Explained: Why the Bell Curve Is Everywhere

Table of Contents

5. Calculating Z-Scores & Percentiles

Standard deviation measures how spread out data points are from the mean average.

• Population Standard Deviation (σ): σ = √[ Σ(x - μ)² / N ] • 68-95-99.7 Rule: - 68% of data falls within ±1σ of mean - 95% of data falls within ±2σ of mean - 99.7% of data falls within ±3σ of mean

5. Calculating Z-Scores & Percentiles

• Z-Score Formula: Z = (x - μ) / σ • A Z-score of +2.0 means a data point is 2 standard deviations above average (approx 97.7th percentile).

6. Standard Error of the Mean & Central Limit Theorem

The Central Limit Theorem states that as sample size n increases, the sampling distribution of the mean approaches a normal distribution with Standard Error SE = σ / √n.

7. Skewness, Kurtosis & Non-Normal Distributions

Real-world financial distributions often exhibit "heavy tails" (high kurtosis) or right skewness (income distributions).

8. Chebyshev's Inequality for Arbitrary Distributions

For ANY dataset (even non-normal, skewed distributions), Chebyshev's inequality guarantees that at least $1 - 1/k^2$ of the data lies within $k$ standard deviations of the mean ($k > 1$). At least 75% of data lies within $pm 2sigma$, and at least 88.9% lies within $pm 3sigma$.

9. Coefficient of Variation (CV) & Relative Risk

The Coefficient of Variation $CV = (sigma / mu) imes 100%$ measures relative variability, allowing comparison of risk between assets with different price scales (e.g. Bitcoin vs S&P 500).

10. Calculating Sample Variance & Bessel's Correction

• Sample Variance (s²): s² = Σ(x - x̄)² / (n - 1) • Why (n - 1)? Dividing by (n - 1) instead of n corrects sample bias, making s² an unbiased estimator of population variance σ².

11. Standard Deviation in Portfolio Finance (Sharpe Ratio)

Investors use standard deviation $\sigma$ as a measure of portfolio risk. The Sharpe Ratio $SR = (R_p - R_f) / \sigma_p$ measures excess return per unit of risk.

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Frequently Asked Questions (FAQs)

What is the difference between sample and population standard deviation?

Sample standard deviation divides by (n - 1) [Bessel's correction] to remove bias when estimating a larger population.