Notes on the Normal Distribution and Standard Deviation

Notation and core ideas

• Sigma (σ) denotes the standard deviation; it measures the variation or spread of a random variable around its mean. It answers: how far, on average, data deviate from the mean.
• The standard deviation is a measure of spread with the same units as the variable X.
• Common shorthand: the normal distribution is often written as X ~ N(μ, σ^2), where
  • μ is the mean (center of the distribution),
  • σ is the standard deviation, and
  • σ^2 is the variance.
• The statement “X is normal” typically means X is distributed normally, i.e., X ~ N(μ, σ^2).
• Observed values are written in lowercase x; random variables are written in uppercase X.
• The phrase “norm is normalized” refers to standardizing a normal variable to a standard normal form.

Normal distribution basics

• Distribution notation: $X \sim \mathcal{N}(μ, σ^2)$ which means X has a normal distribution with mean μ and variance σ^2.
• Mean and variance:
  • Mean: $\,E[X] = μ$
  • Variance: $\mathrm{Var}(X) = σ^2$
• Standard deviation: $\mathrm{SD}(X) = σ$
• The normal distribution is symmetric and bell-shaped; many real-valued phenomena are modeled as approximately normal when they are the sum of many small independent effects (Central Limit Theorem).

Standardization (normalization) to the standard normal

• Standardization transforms X into a standard normal variable Z with mean 0 and variance 1:
  • Z = X \2{\mu}{\sigma} = \frac{X - μ}{σ}
  • $Z \sim \mathcal{N}(0, 1)$
• Purpose: enables comparison across different normal distributions and facilitates use of standard normal tables (z-tables) for probabilities.
• If you know μ and σ, you can convert any X to Z and read probabilities from the standard normal distribution.

Worked example

• Suppose a measurement X has parameters μ = 100 and σ = 15.
  • If X = 115, the standardized value is
  • $Z = \frac{115 - 100}{15} = 1$
  • Interpretation: the observed value 115 is 1 standard deviation above the mean.
• If X is exactly 100, then Z = 0 (at the mean).

Probability density and cumulative concepts (normal case)

• PDF of X when X ~ N(μ, σ^2):
  • $f_X(x) = \frac{1}{σ \,\sqrt{2\pi}} \exp\left(-\frac{(x - μ)^2}{2σ^2}\right)$
• CDF (probability up to x):
  • $F<em>X(x) = P(X \le x) = \int</em>{-\infty}^{x} f_X(t) \, dt$
• Probability within an interval:
  • $P(a \le X \le b) = \int<em>{a}^{b} f</em>X(x) \, dx$
• For many practical purposes, probabilities are computed via standard normal tables or software using the z-score transformation.

Notational pitfalls and clarifications

• Distinguish: uppercase X (random variable) vs lowercase x (a realized value).
• When we write X ~ N(μ, σ^2), we mean the distribution of the random variable X, not a specific observed value.
• The term “normalized” commonly refers to converting to the standard normal form via Z = (X - μ)/σ.

Connections and relevance

• The normal model is a foundational assumption in many statistical methods (confidence intervals, hypothesis tests, regression residuals) due to the Central Limit Theorem and mathematical convenience.
• Real-world relevance: measurement error, natural phenomena with many small additive effects, and standardization for comparability across datasets.

Quick summary of key formulas

• Variance and standard deviation:
  • $\mathrm{Var}(X) = σ^2$
  • $\mathrm{SD}(X) = σ$
• Normal distribution notation:
  • $X \sim \mathcal{N}(μ, σ^2)$
• Standardization:
  • $Z = \frac{X - μ}{σ}, \quad Z \sim \mathcal{N}(0, 1)$
• Probability density function (PDF):
  • $f_X(x) = \frac{1}{σ \sqrt{2\pi}} \exp\left(-\frac{(x - μ)^2}{2σ^2}\right)$
• CDF and interval probability:
  • $F<em>X(x) = P(X \le x) = \int</em>{-\infty}^{x} f_X(t) \, dt$
  • $P(a \le X \le b) = \int<em>{a}^{b} f</em>X(x) \, dx$