Standard Normal

Standard Normal Distribution in Health Economic Statistics and Modelling

The standard normal distribution, also called the unit normal or z distribution, is the normal curve centred on zero with unit spread, against which health economists read any estimate that has been expressed in standard error or standard deviation units. A mean cost difference, a log hazard ratio or an incremental net benefit that is approximately normal can be rescaled onto it, so a single table of probabilities serves trial analysis and decision modelling alike. Its cumulative probabilities supply P values and the chance that a technology is cost effective, and its quantiles supply the multipliers 1.645, 1.96 and 2.576 that appear in most reported confidence intervals, while its inverse turns uniform random numbers into normal draws for probabilistic models. Standardising a single observation belongs to the z-score page and the wider family to the normal distribution page.

Density, cumulative probability and the inverse function

The NIST/SEMATECH e-Handbook of Statistical Methods calls the case of a normal distribution with location zero and scale one the standard normal distribution, and notes that the general form of the probability functions can be expressed in terms of this standard form. The density, the cumulative function and its inverse do the work in practice.

$$\phi(z) = \frac{1}{\sqrt{2\pi}}\, e^{-z^{2}/2}, \qquad \Phi(z) = \Pr(Z \le z) = \int_{-\infty}^{z} \phi(t)\, dt, \qquad z_{p} = \Phi^{-1}(p)$$

where $Z$ is a standard normal random variable, $z$ is a value on its scale, $\phi(z)$ is the density at $z$, $t$ is the variable of integration, $\Phi(z)$ is the probability that $Z$ does not exceed $z$, $p$ is a probability strictly between 0 and 1, and $z_{p}$ is the quantile with $\Phi(z_{p}) = p$.

The handbook notes that the cumulative function has no simple closed form and is computed numerically, and the same holds for its inverse, the quantile function. The density peaks at $\phi(0) = 1/\sqrt{2\pi} \approx 0.399$, and a density value is not a probability, since probabilities are areas under the curve. The curve is symmetric about zero, so $\Phi(-z) = 1 - \Phi(z)$. Roughly 68.3% of the probability lies within one unit of zero and 95.0% within 1.96 units.

Critical values that recur in trial and meta-analysis reporting

A two-sided interval at confidence level $\gamma = 1 - \alpha$ uses the quantile that leaves $\alpha/2$ in each tail. Three quantiles cover most published reports.

The Cochrane Handbook (chapter 6) uses these values when it recovers a standard error from a published confidence interval. For a large sample a 95% interval is 3.92 standard errors wide, since 3.92 is twice 1.96, and the divisor becomes 3.29 for a 90% interval and 5.15 for a 99% interval, which are twice 1.645 and twice 2.576 to rounding. For mean differences from continuous outcomes the handbook adds that intervals from small samples, which it puts at fewer than about 60 participants per group, should have been calculated from the t distribution, whose slightly larger multipliers replace these values.

Turning reported results into standard errors and P values

Evidence synthesis for health technology assessment often starts with trials that report an estimate with only an interval or a P value. The Cochrane Handbook describes the simple Wald test approach, in which the estimate is divided by its standard error and the result, Z, is compared with a standard normal distribution.

$$Z = \frac{\hat{\theta}}{SE(\hat{\theta})}, \qquad P = 2\,\Phi(-|Z|), \qquad SE(\hat{\theta}) = \frac{U - L}{3.92}$$

where $\hat{\theta}$ is the estimated effect on a scale where it is approximately normal, $SE(\hat{\theta})$ is its standard error, $P$ is the two-sided P value for the null value zero, and $L$ and $U$ are the lower and upper limits of a 95% confidence interval on the same scale.

For ratio measures such as hazard ratios the handbook specifies that the analysis is done on the natural log scale, so the limits are logged before the divisor is applied and the resulting standard error refers to the log of the ratio. The handbook also cautions that standard errors derived this way may not coincide exactly with the true values when a study used a different test.

Sample size and power

The standard normal also fixes the arithmetic of sample size calculation for a trial that will feed an economic evaluation. Whitley and Ball give a formula for comparing means in two groups of equal size.

$$n = \frac{2}{d^{2}}\, c_{p,\text{power}}$$

where $n$ is the number of participants required in each group, $d$ is the standardised difference, equal to the target difference divided by the standard deviation, and $c_{p,\text{power}}$ is a constant set by the significance level and the statistical power chosen.

Their table gives $c_{p,\text{power}} = 7.9$ for a 0.05 significance level with 80% power and 10.5 with 90% power. These constants are squared sums of two standard normal quantiles, since $(1.960 + 0.842)^{2} = 7.85$ and $(1.960 + 1.282)^{2} = 10.51$, where 0.842 and 1.282 are the 80th and 90th percentiles of the standard normal. Their example with $d = 0.78$ and 80% power gives $n = 2 / 0.78^{2} \times 7.9 = 25.97$, so 26 participants are needed in each arm.

Sampling uncertain parameters in probabilistic sensitivity analysis

In probabilistic sensitivity analysis each uncertain input is drawn repeatedly from a distribution, and the NICE manual (PMG36, section 4.7.10) asks that these distributions represent the available evidence on each parameter and that their use be justified. By the inverse transform principle, if $U$ is uniform on the interval 0 to 1 then $\Phi^{-1}(U)$ is standard normal, which is how a spreadsheet model can generate normal draws.

$$\theta^{(k)} = \exp\!\left(\mu + \sigma\, Z^{(k)}\right), \qquad Z^{(k)} = \Phi^{-1}\!\left(U^{(k)}\right)$$

where $k$ indexes the simulation, $U^{(k)}$ is a uniform random number, $Z^{(k)}$ is the corresponding standard normal draw, $\mu$ and $\sigma$ are the mean and standard error of the parameter on the log scale, and $\theta^{(k)}$ is the sampled ratio.

The ISPOR-SMDM task force report on parameter estimation lists the log-normal for relative risks or hazard ratios, so a sampled ratio from a log-normal distribution is a rescaled and exponentiated standard normal draw. The same report states that for regression coefficients the covariance matrix defines the uncertainty and multivariate normality is appropriate for the linear predictor. Correlated draws are then built by multiplying independent standard normal draws by the lower-triangular factor from a Cholesky decomposition of that matrix and adding the coefficient means.

Worked example with a reported hazard ratio

All numbers in this example are illustrative. A cost-effectiveness model needs a hazard ratio for progression, and the only source is a trial reporting a hazard ratio of 0.75 with a 95% confidence interval of 0.60 to 0.94. The analyst needs a log-scale standard error for a log-normal distribution and a P value to check the report.

1. Move to the log scale. The logs are $\ln 0.75 = -0.2877$, $\ln 0.60 = -0.5108$ and $\ln 0.94 = -0.0619$. The geometric midpoint of the limits, $\sqrt{0.60 \times 0.94} = 0.751$, matches the point estimate, as expected for an interval built on the log scale.

2. Recover the standard error. The log interval is $-0.0619 - (-0.5108) = 0.4490$ wide, so the standard error is $SE = 0.4490 / 3.92 = 0.1145$.

3. Recover the test statistic. The Wald statistic is $Z = -0.2877 / 0.1145 = -2.51$, and the two-sided P value is $P = 2\,\Phi(-2.51) = 0.012$, below the conventional 0.05 significance level.

4. Draw parameter values. A uniform number of 0.30 gives $Z = \Phi^{-1}(0.30) = -0.5244$, so the log hazard ratio is $-0.2877 + 0.1145 \times (-0.5244) = -0.3477$ and the sampled hazard ratio is $e^{-0.3477} = 0.706$. A uniform number of 0.95 gives $Z = 1.6449$, a log hazard ratio of $-0.2877 + 0.1145 \times 1.6449 = -0.0994$ and a hazard ratio of $e^{-0.0994} = 0.905$.

5. Check the distribution against the report. The 2.5th and 97.5th percentiles of the sampled hazard ratio are $e^{-0.2877 - 1.96 \times 0.1145} = e^{-0.5121} = 0.599$ and $e^{-0.2877 + 1.96 \times 0.1145} = e^{-0.0633} = 0.939$, which reproduce the published 0.60 and 0.94. The median of the sampled values is 0.75, while their mean is $e^{-0.2877 + 0.1145^{2}/2} = 0.755$, a little higher because the log-normal is right skewed.

6. Interpretation. The reported interval carries enough information to parameterise the model input once it is read through the standard normal on the correct scale. Applying the 3.92 divisor to the untransformed limits, $(0.94 - 0.60) / 3.92 = 0.087$, would give a symmetric normal distribution for the ratio itself and ignore the asymmetry of the published limits, which sit 0.15 below and 0.19 above the estimate.

Probability of cost-effectiveness under a normal approximation

The NICE manual (PMG36, section 4.7.15) asks that, alongside expected costs, outcomes and ICERs, a submission presents the probability that the treatment is cost effective at maximum acceptable ICERs of GBP 25,000 to GBP 35,000 per QALY gained and the error probability that it is not. In a probabilistic model this is normally the share of simulations with positive incremental net monetary benefit, which traces the cost-effectiveness acceptability curve across thresholds. When the incremental net monetary benefit at a given threshold is close to normal, the standard normal gives the same answer directly.

$$\Pr(\text{INMB} > 0) = \Phi\!\left(\frac{m}{s}\right)$$

where INMB is the incremental net monetary benefit at the stated threshold, $m$ is its expected value and $s$ is its standard deviation across the uncertainty distribution.

The shortcut is only as good as the normality of the INMB distribution, which the normal distribution page examines in more depth.

Excel functions for the standard normal

Spreadsheet models usually rely on four Excel functions documented by Microsoft. The table gives each one's documented behaviour and a typical use.

Where the standard normal reference misleads

Most errors arise when the reference distribution is applied to a quantity that is not close to normal on the scale used. Four problems recur in health economic work.

1. Reading patient-level costs as normal. Quoting a standard normal percentile for one patient's cost assumes symmetric costs, yet Thompson and Barber describe highly skewed distributions with a long right-hand tail as typical of cost data. The approximation suits estimates such as mean costs in large samples, the subject of the central limit theorem page.
2. Using 1.96 for small-sample mean differences. The Cochrane Handbook expects t multipliers for mean differences from groups of fewer than about 60, so a z multiplier understates the width of such an interval.
3. Working on the wrong scale. Ratio measures are approximately normal after logging, and a standard error found on the ratio scale gives a distribution that can stray outside plausible values.
4. Reading the P value as the probability of cost-effectiveness. The P value is a tail area computed assuming no difference, which is a different quantity from the probability that PMG36 asks for.

Sources

• Briggs AH, Weinstein MC, Fenwick EAL, Karnon J, Sculpher MJ, Paltiel AD. Model parameter estimation and uncertainty analysis: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force Working Group-6. Medical Decision Making. 2012;32(5):722-732. https://doi.org/10.1177/0272989X12458348
• Higgins JP, Li T, Deeks JJ (editors). Chapter 6: Choosing effect measures and computing estimates of effect (last updated August 2023). In: Higgins JP, Thomas J, Chandler J, et al. (editors). Cochrane Handbook for Systematic Reviews of Interventions, version 6.5. Cochrane; 2024. Sections 6.3.1, 6.3.2 and 6.5.2.3. https://www.cochrane.org/authors/handbooks-and-manuals/handbook/current/chapter-06
• Microsoft. NORM.S.DIST function; NORM.S.INV function; STANDARDIZE function; RAND function. Microsoft Support; accessed 28 October 2026. https://support.microsoft.com/en-gb/excel/functions/norm-s-dist-function
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). Published 31 January 2022, last updated 13 October 2026. Sections 4.7.10 and 4.7.15. https://www.nice.org.uk/process/pmg36
• NIST/SEMATECH. e-Handbook of Statistical Methods. Section 1.3.6.6.1, Normal distribution. National Institute of Standards and Technology; 2012, accessed 28 October 2026. https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm
• Thompson SG, Barber JA. How should cost data in pragmatic randomised trials be analysed? BMJ. 2000;320(7243):1197-1200. https://doi.org/10.1136/bmj.320.7243.1197
• Whitley E, Ball J. Statistics review 4: Sample size calculations. Critical Care. 2002;6(4):335-341. https://doi.org/10.1186/cc1521