Understanding the Normal Distribution in Biostatistics

In a clinical setting, the mean might represent the average value of a laboratory measurement or a physiological characteristic in a target population.

The Standard Deviation Determines the Spread

The parameter \(\sigma\) is the standard deviation. It measures the typical spread of observations around the mean.

A smaller standard deviation produces a narrower, taller curve. A larger standard deviation produces a wider, flatter curve.

The 68–95–99.7 Rule

For a normal distribution, a particularly useful approximation is the empirical rule. It describes how much probability lies within one, two, and three standard deviations of the mean.

For example, suppose systolic blood pressure is approximately normally distributed with mean 120 mmHg and standard deviation 10 mmHg. Then approximately 68% of measurements fall between:

Approximately 95% fall between:

And approximately 99.7% fall between 90 and 150 mmHg.

Standardization: Turning a Measurement into a z-Score

A raw measurement is difficult to interpret without knowing where it lies relative to the population distribution. A z-score solves this problem by expressing a value in standard-deviation units from the mean.

For an observation \(x\):

The numerator \(x-\mu\) tells us how far the observation is from the mean. Dividing by \(\sigma\) converts that distance into standard-deviation units.

A useful feature of the z-score is that it puts measurements from different normal distributions onto a common standardized scale.

Worked z-Score Example

Suppose a biomarker has:

• Mean \(\mu=50\)
• Standard deviation \(\sigma=8\)

A patient has a measured value of \(x=66\). The z-score is:

The patient's value is therefore 2 standard deviations above the mean. Under a normal model, that is a relatively high value, but it is not extraordinarily rare.

The Standard Normal Distribution

When \(\mu=0\) and \(\sigma=1\), the normal distribution is called the standard normal distribution.

The standard normal distribution is useful because any normal observation can be converted into a z-score. Probability calculations can then be performed using the same reference distribution.

Finding Probabilities from z-Scores

Suppose \(Z\) follows a standard normal distribution. The cumulative distribution function is:

For example, a commonly used result is:

By symmetry:

And therefore:

This is the statistical basis for the familiar 95% normal-theory interval associated with the critical value 1.96.

Left-Tail, Right-Tail, and Two-Tail Probabilities

The wording of a probability question determines which area under the normal curve is required.

For a continuous distribution, the probability of an exact single point is zero:

This does not mean the point has no practical meaning. It means probability is assigned to intervals or regions under a continuous density.

Percentiles and Reference Values

A percentile tells us where a value falls relative to the distribution. The 50th percentile is the median, which equals the mean for a normal distribution.

If a laboratory measurement is approximately normal, percentiles can be translated back to the original measurement scale using:

For example, the approximate 97.5th percentile is:

Normal Distribution and Clinical Reference Ranges

A reference range is often constructed using a central proportion of a reference population. When the underlying distribution is reasonably normal, a central 95% interval is approximately:

For example, if a reference population has mean hemoglobin of 14.0 g/dL and standard deviation 1.0 g/dL, the approximate central 95% interval under a normal model would be:

giving approximately 12.04 to 15.96 g/dL.

Why Normality Matters in Biostatistics

The normal distribution appears in many statistical procedures. Its importance is not simply that biological data "look bell-shaped." Normal theory provides a mathematical framework for probability calculations and inference.

• Confidence intervals for means often use normal or approximately normal theory.
• Many hypothesis tests rely on normality assumptions or asymptotic approximations.
• Linear regression models commonly assume normally distributed errors for exact small-sample inference.
• ANOVA is based on assumptions concerning model errors and variance structure.
• The Central Limit Theorem explains why sample means can be approximately normal even when individual observations are not.

The Central Limit Theorem Connection

One of the most important ideas in biostatistics is that the normal distribution can arise from averaging, not only from individual observations.

Under suitable conditions, as sample size increases, the sampling distribution of the sample mean becomes approximately normal.

Equivalently, the standard deviation of the sampling distribution of the mean, called the standard error, is:

This is a crucial distinction: the standard deviation \(\sigma\) describes individual observations, whereas the standard error describes the variability of sample means across repeated samples.

Worked Example: A Sampling Distribution

Suppose a population has mean 100 and standard deviation 20. If samples of \(n=25\) are repeatedly drawn, the standard error of the sample mean is:

The sampling distribution of the mean is therefore approximately:

A sample mean of 108 has a standardized value:

The important point is that the z-score is calculated using the standard error when standardizing a sample mean, not the population standard deviation of an individual observation.

Normal Distribution in Confidence Intervals

For a normally distributed population with known standard deviation, a two-sided 95% confidence interval for a population mean has the form:

When the population standard deviation is unknown and the sample is relatively small, the t-distribution is generally used instead of the standard normal distribution.

Normal Distribution in Hypothesis Testing

Normal theory also provides the foundation for many test statistics. A generic standardized statistic can often be written as:

A large positive or negative value indicates that the observed estimate is far from the null value relative to its standard error.

For a two-sided test with significance level \(\alpha=0.05\), the standard normal critical values are approximately:

Thus, under the standard normal reference distribution, values beyond approximately \(\pm1.96\) lie in the two tails containing 5% total probability.

Normality Does Not Mean Every Observation Must Be Near the Mean

A common misunderstanding is that data from a normal distribution should cluster tightly around the mean. The standard deviation determines how wide that cluster is. A normal distribution can be very narrow or very broad.

Also, observations beyond two or three standard deviations are not impossible. They are simply less probable.

The exact values are approximately 31.7%, 4.55%, and 0.27%, respectively, under the standard normal distribution.

How to Assess Whether Data Are Approximately Normal

A normality assumption should not be judged solely by looking at whether a histogram "looks like a bell." Several complementary methods are useful.

1. Histogram

A histogram can reveal strong skewness, multiple modes, extreme outliers, or other departures from a normal shape. However, histograms can be unstable with small samples because the appearance depends on the bin width.

2. Q–Q Plot

A normal quantile-quantile plot compares observed quantiles with theoretical normal quantiles. If the data are reasonably compatible with a normal distribution, the points tend to follow an approximately straight pattern.

3. Summary Statistics

Mean, median, standard deviation, skewness, and the presence of extreme observations can help identify departures from symmetry.

4. Formal Normality Tests

Tests such as the Shapiro–Wilk test can be useful in some settings, but a formal test should not be treated as the sole decision criterion. With large samples, small and practically unimportant departures can become statistically significant. With small samples, substantial departures may be difficult to detect.

Normality of the Data vs. Normality of the Model Errors

This distinction is particularly important in clinical statistics.

Many statistical models do not require the raw outcome variable itself to be normally distributed. For example, in linear regression, the relevant normality assumption for classical small-sample inference concerns the model errors, conditional on the predictors, rather than requiring every observed outcome to be normally distributed.

Similarly, many clinical endpoints are analyzed using distributions specifically suited to their measurement scale. Binary outcomes are commonly analyzed with binomial or logistic models, and time-to-event outcomes are often analyzed using survival methods.

When a Normal Distribution Is a Poor Model

The normal distribution may be inappropriate when the variable has strong characteristics that conflict with its assumptions.

• Strong right skew: common for some laboratory and cost variables.
• Strong left skew: possible when observations are bounded above.
• Heavy tails: extreme observations occur more often than a normal model predicts.
• Bounded variables: proportions and percentages cannot extend indefinitely in both directions.
• Count data: integer-valued event counts are not naturally continuous normal variables.
• Binary data: yes/no outcomes are not normally distributed.
• Multimodal data: mixtures of distinct populations may produce multiple peaks.

In these settings, an appropriate transformation or a distribution/model better matched to the endpoint may be preferable.

Normal Distribution vs. t-Distribution

The normal and t-distributions are closely related but should not be treated as interchangeable.

The heavier tails of the t-distribution account for additional uncertainty when the population standard deviation is estimated from the sample.

A Clinical Example: Interpreting a Laboratory Value

Suppose a laboratory measurement is modeled as normal with:

• \(\mu=100\)
• \(\sigma=15\)

A patient has \(x=130\). The standardized value is:

The measurement is therefore two standard deviations above the population mean. Under the normal model, approximately 2.5% of observations are above \(z=1.96\), so a value of 130 is in the upper tail.

However, whether 130 is clinically abnormal is a separate question. A statistical tail probability is not the same as a diagnostic or clinical decision rule.

Common Mistakes

1. Assuming every biological variable is normal. Many clinical variables are skewed, bounded, discrete, or otherwise non-normal.
2. Confusing standard deviation with standard error. SD describes individual variability; SE describes estimator variability.
3. Using the 68–95–99.7 rule for non-normal data. The empirical rule is a normal-distribution property.
4. Interpreting a z-score as a diagnosis. A z-score describes statistical position, not clinical significance by itself.
5. Assuming a normality test proves normality. No statistical test establishes that a dataset is perfectly normal.
6. Thinking a histogram must look perfectly bell-shaped. Real samples contain random variation and may show modest departures even when a normal model is reasonable.
7. Checking only the raw outcome in a regression model. The relevant assumption may concern residuals or model errors instead.
8. Using 1.96 automatically for every confidence interval. The appropriate reference distribution depends on the analysis.

Practical Normal-Distribution Workflow

Normal Distribution in R

R provides functions for the normal distribution using the prefixes d, p, q, and r.

Calculate a Tail Probability

pnorm(1.96)
# approximately 0.975

1 - pnorm(1.96)
# approximately 0.025

Calculate a Quantile

qnorm(0.975)
# approximately 1.96

qnorm(0.025)
# approximately -1.96

Work on the Original Measurement Scale

mu <- 100
sigma <- 15

pnorm(130, mean = mu, sd = sigma)

qnorm(0.975, mean = mu, sd = sigma)

The first calculation gives the probability of observing a value at or below 130 under the specified normal model. The second gives the 97.5th percentile on the original measurement scale.

Normal Distribution in SAS

SAS also provides functions for normal probabilities and quantiles. For example, the CDF function can be used to evaluate a cumulative normal probability.

data normal_example;
    mu = 100;
    sigma = 15;
    x = 130;

    z = (x - mu) / sigma;
    p_left = cdf("NORMAL", x, mu, sigma);
    p_upper = 1 - p_left;
run;

proc print data=normal_example;
run;

For standardized values, the same probability can be obtained from the standard normal distribution:

data z_example;
    z = 2;
    p_left = cdf("NORMAL", z);
    p_upper = 1 - p_left;
run;

proc print data=z_example;
run;

Normal Distribution and Clinical Trial Analysis

Normal-theory methods appear frequently in clinical trials, particularly for continuous endpoints. Examples include change from baseline, laboratory measurements, vital signs, and patient-reported outcomes.

For a two-arm comparison of mean change, investigators may compare treatment-group means using a t-test or a model such as ANCOVA. In these settings, the normal distribution often enters through assumptions about model errors or the sampling distribution of estimated treatment effects.

This is why understanding the normal distribution is foundational even when the raw clinical endpoint is not perfectly normal.

The Most Important Distinction: Distribution vs. Sampling Distribution

There are two different questions that are often confused:

The second question is central to statistical inference. Confidence intervals and hypothesis tests depend on the sampling distribution of an estimator, not merely on the distribution of the observed raw values.

A Compact Formula Sheet

Worked Example Summary

Consider a normally distributed clinical measurement with mean 50 and standard deviation 8. A patient has a value of 66.

This example illustrates the central purpose of standardization: the raw value 66 becomes interpretable once it is expressed relative to the center and spread of the distribution.

References

Altman, D.G. (1991). Practical Statistics for Medical Research. Chapman & Hall.
Rosner, B. (2016). Fundamentals of Biostatistics, 8th ed. Cengage Learning.
Daniel, W.W. & Cross, C.L. (2018). Biostatistics: A Foundation for Analysis in the Health Sciences, 11th ed. Wiley.
Kirkwood, B.R. & Sterne, J.A.C. (2003). Essential Medical Statistics, 2nd ed. Blackwell Science.