Normal Distribution

Normal Distribution: Concept Architecture

Orientation and learning roadmap

This page develops the parameters and probability calculation for a bell-shaped model, then applies it to an illustrative health-economic decision output. It distinguishes the distributions of individual observations, uncertain estimates, and model inputs before assessing whether the normal assumption is suitable.

Shape and parameters

A normal model is symmetric around its mean and has tails extending in both directions without a hard boundary. Its mean $\mu$ sets the centre, while its positive standard deviation $\sigma$ sets the scale; the variance is $\sigma^2$.

For $X\sim N(\mu,\sigma^2)$, the probability density at $x$ is

$$f(x)=\frac{1}{\sigma\sqrt{2\pi}} \exp\!\left[-\frac{(x-\mu)^2}{2\sigma^2}\right], \qquad \sigma>0.$$

A density value is not itself the probability of an exact continuous value; probabilities are areas over intervals under the curve. Symmetry gives $P(X\leq\mu)=0.5$, and approximately 68.3% of the distribution lies between $\mu-\sigma$ and $\mu+\sigma$. The standard normal has mean zero and standard deviation one.

Convert values to probabilities

Subtracting the mean and dividing by the standard deviation converts an observation to a standard-normal score. If $Z=(X-\mu)/\sigma$, then $Z\sim N(0,1)$ under the model, and its cumulative distribution function $\Phi(z)$ gives $P(Z\leq z)$.

For example, if a hypothetical biological measurement follows $N(100,15^2)$ in its stated units, a value of 130 has $z=(130-100)/15=2$. The model implies $P(X>130)=1-\Phi(2)\approx0.0228$, or about 2.28%, for that population and distribution. That probability depends on the assumed distribution and does not establish that actual clinical measurements have normal tails.

Worked health-economic example: incremental net benefit

Suppose an analyst has justified an approximate normal distribution for incremental net monetary benefit (INMB) at a particular willingness-to-pay threshold, with mean £2,000 and standard deviation £4,000. This example assumes the distribution refers to uncertainty about the decision-relevant INMB, not the distribution of individual patient costs or health outcomes.

$$\begin{aligned} P(\mathrm{INMB}>0) &=P\!\left(Z>\frac{0-2000}{4000}\right)\\ &=P(Z>-0.5)=\Phi(0.5)\approx0.6915. \end{aligned}$$

Under that illustrative normal approximation, the intervention has a roughly 69.15% probability of positive INMB at the stated threshold. The calculation is not a recommendation, and it does not say the expected INMB is positive in exactly 69.15% of patients. If the true INMB uncertainty distribution is skewed, multimodal, or strongly affected by correlated costs and effects, direct probabilistic simulation may give a different result.

Which quantity is normal?

The same notation can refer to several different random quantities, so the analyst must name the target of the distribution. A normal model for an individual's biomarker, an approximate sampling distribution for a treatment-effect estimator, and a probability distribution assigned to an uncertain model input answer different questions.

An average may have an approximately normal sampling distribution in some settings even if individual observations are skewed, subject to sample size and dependence conditions. Conversely, normal uncertainty for one regression coefficient does not imply a normal distribution for a derived probability or cost-effectiveness ratio. State the uncertainty source, units, and correlation with other model quantities.

When the assumption is unsuitable

A normal distribution permits negative and arbitrarily large positive values, so it is generally unsuitable as an unmodified model for a probability, a nonnegative absolute cost, or a strictly bounded quantity. Real healthcare costs often have a right tail and many zeroes; a single symmetric model may hide the budget impact of rare expensive cases.

Health-state utility may be bounded above and can have a point mass at full health, while its lower permitted value depends on the instrument and valuation set. Check the actual support, skew, outliers, subgroup patterns, and whether the fitted model reproduces tails relevant to the decision. Transformations, bounded or mixture distributions, and nonparametric methods can be considered when their assumptions and data support them.

Estimation and model checking

For independent observations from a normal population, the sample mean and sample variance provide familiar estimators of location and spread, but their uncertainty depends on the sampling design. A histogram, quantile plot, residual diagnostics, and comparison of predicted versus observed outcomes can expose important departures from an assumed normal model.

Normality of residuals in a linear model is a claim about conditional errors, not a requirement that all raw outcomes are normally distributed. Large samples do not automatically fix selection bias, informative missingness, or an invalid parameter range. In economic evaluation, perform sensitivity checks for distributional choices that materially affect incremental costs, QALYs, or cost-effectiveness probabilities.

Source basis and further learning

The statistical references support the density, standardisation, and probability calculations, while health-economic methods guidance frames distribution choice as an evidence-based modelling decision. The worked numbers are invented and should not be interpreted as an estimate for a real technology.

• US National Institute of Standards and Technology, Normal Distribution, Engineering Statistics Handbook. This gives the normal density and cumulative function.
• SciPy developers, official scipy.stats.norm documentation. This specifies standardisation, density, and cumulative probabilities in a reproducible implementation.
• ISPOR-SMDM Modeling Good Research Practices Task Force, “Model Parameter Estimation and Uncertainty Analysis”, Value in Health (2012). This guides probabilistic analysis and parameter uncertainty in health-economic models.