Concept Architecture
Concept
Theoretically, Standard Normal Distribution is a continuous probability distribution with a mean of zero and a standard deviation of one. It is a special case of the Normal distribution obtained by standardising a Normally distributed variable and serves as the reference distribution for statistical inference. Because all Normally distributed variables can be transformed into the Standard Normal Distribution through standardisation, it underpins probability calculations, hypothesis testing, confidence intervals, and many statistical procedures used in health economics.
Mathematically, the Standard Normal Distribution is completely defined by its probability density function and cumulative distribution function. Standardisation transforms an observation into a dimensionless z-score by subtracting the population mean and dividing by the population standard deviation. Probabilities and critical values are then obtained from the cumulative distribution of the standardised variable.
In practice, the Standard Normal Distribution is used to calculate probabilities, p-values, critical values, confidence intervals, statistical power, and sample sizes. In health economics it is applied in clinical trial analysis, regression modelling, probabilistic sensitivity analysis, health technology assessment, and economic evaluation whenever Normal approximation methods are appropriate.
Purpose
Used to standardise observations, calculate probabilities and critical values, construct confidence intervals, perform hypothesis testing, estimate statistical power, and support statistical inference in health economic analyses.
Mathematical Formulae
Primary Formula
Probability density function:
�(z) = (1 � �(2�)) ? e^(?z�/2)
where:
- z = standard Normal variable
- ? = 0
- � = 1
Supporting Formulae
Standardisation:
z = (x ? ?) � �
Cumulative distribution function:
�(z) = P(Z � z)
Two-sided probability:
P(|Z| � z) = 2�(z) ? 1
Related Mathematical Methods
- Normal Distribution
- Z-score
- Standard Error
- Confidence Interval
- Hypothesis Testing
- Statistical Power
- Central Limit Theorem
- Sample Size Calculation
Example
A health economist studies annual healthcare costs.
Population mean:
�8,000
Population standard deviation:
�1,500
A patient incurs annual costs of:
�10,250
The standardised score is:
z = (10,250 ? 8,000) � 1,500
= 1.50
Using the Standard Normal Distribution:
P(Z � 1.50) = 0.9332
Therefore, approximately 93.3% of patients are expected to have annual healthcare costs below �10,250 under the assumed Normal distribution.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| STANDARDIZE | =STANDARDIZE(B2,8000,1500) | Convert healthcare costs or outcomes into z-scores. |
| NORM.S.DIST | =NORM.S.DIST(A2,TRUE) | Calculate cumulative probabilities from z-scores. |
| NORM.S.DIST | =NORM.S.DIST(A2,FALSE) | Calculate the standard Normal probability density. |
| NORM.S.INV | =NORM.S.INV(0.975) | Obtain critical values for confidence intervals and hypothesis tests. |
| NORM.DIST | =NORM.DIST(B2,8000,1500,TRUE) | Calculate cumulative probabilities from the original Normal distribution. |
VBA (Optional)
Automate conversion of observations to z-scores and generate probabilities, critical values, and confidence intervals using the Standard Normal Distribution.
Sources
- Casella G, Berger RL. Statistical Inference.
- Rice JA. Mathematical Statistics and Data Analysis.
- Mood AM, Graybill FA, Boes DC. Introduction to the Theory of Statistics.
- DeGroot MH, Schervish MJ. Probability and Statistics.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Drummond MF, et al. Methods for the Economic Evaluation of Health Care Programmes.
- NICE Health Technology Evaluation Manual.
Related Concepts (2)
Library
Publications
1
Bayesian Methods in Health Economics — Gianluca Baio, 1st Edition ed., 2012 (Chapman & Hall / CRC Press)
An overview of Bayesian statistical methods for the analysis of health economic data, covering economic evaluation concepts, statistical cost-effectiveness analysis, Bayesian computation and MCMC, and applied health economic evaluation.
BookView source →
Frequently Asked Questions (6)
What is the standard normal distribution?
A specific normal distribution with a mean of zero and standard deviation of one, used as a reference for standardising other variables.
Source: Casella G, Berger RL. Statistical Inference. 2nd ed. Duxbury; 2002.
What makes the standard normal distribution a common reference?
The standard normal distribution is a normal distribution with a mean of zero and a standard deviation of one, and this fixed form is what makes it a universal reference. Any normally distributed variable can be converted to it by subtracting its mean and dividing by its standard deviation, producing a z-score that says how many standard deviations a value sits from the mean. Because probabilities for this one distribution are tabulated, that conversion lets any normal quantity be handled with a single set of tables. A fixed yardstick for normal variables is its role. Kirkwood and Sterne (2003) describe this.
Source: Kirkwood & Sterne 2003
Why is the standard normal distribution useful?
The standard normal distribution is useful because it provides a common reference to which any normal variable can be standardised, allowing probabilities and percentiles to be looked up or computed once and applied to any normal distribution, and it underlies many statistical tests and confidence intervals. So the standard normal distribution is useful for standardising variables and for inference, since converting to z-scores lets values from different normal distributions be compared and probabilities determined from a single reference, which is why it is central to statistics, appearing in the construction of confidence intervals, hypothesis tests, and the interpretation of standardised values.
Source: Casella & Berger 2002
What is a z-score on the standard normal distribution?
A z-score is a standardised value obtained by subtracting the mean from an observation and dividing by the standard deviation, expressing how many standard deviations the observation lies from the mean. A z-score of zero is at the mean, positive values above it and negative below. So a z-score places a value on the standard normal scale, indicating its position relative to the mean in standard deviation units, which allows values from different distributions to be compared and probabilities to be found from the standard normal distribution, making the z-score a key tool for standardisation and for interpreting how unusual a value is.
Source: Casella & Berger 2002
How is a variable standardised to the standard normal?
A variable is standardised to the standard normal by subtracting its mean and dividing by its standard deviation, which transforms it to have a mean of zero and a standard deviation of one, producing z-scores. If the original variable is normally distributed, the standardised values follow the standard normal distribution. So standardising a variable to the standard normal involves centring it at zero and scaling it to unit standard deviation, which converts it to z-scores on the common standard normal scale, allowing probabilities and comparisons to be made using the properties and tables of the standard normal distribution.
Source: Casella & Berger 2002
How is the standard normal distribution used in inference?
The standard normal distribution is used in inference to compute probabilities, critical values, and confidence intervals for normally distributed estimates: test statistics are often expressed as z-scores compared with the standard normal, and confidence intervals use its critical values, such as about 1.96 for ninety-five per cent. So the standard normal distribution is used throughout inference as the reference for standardised statistics, since many estimates are approximately normal, and referring them to the standard normal yields p-values and interval bounds, which is why its critical values and probabilities are fundamental to constructing tests and confidence intervals for normal or approximately normal quantities.
Source: Casella & Berger 2002
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 25 Dec 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/standard-normal
- Term code
- HE-ES-SA-201
Stable URI · Machine-readable · Resolvable · CC BY 4.0