# Lecture 10: The Normal Distribution

## Statistics / Biostatistics

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## Learning Objectives

By the end of this lecture, students will be able to:

1. Describe the properties of the normal (Gaussian) distribution
2. Standardize values using the z-score transformation
3. Use the standard normal table to find probabilities and percentiles
4. Apply the normal distribution to clinical data
5. Assess normality using graphical and statistical methods

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## Lecture Content

### I. The Normal Distribution

The normal distribution, also called the Gaussian distribution after Carl Friedrich Gauss, is the most important continuous probability distribution in statistics. Many biological and clinical measurements are approximately normally distributed, including height, blood pressure, serum cholesterol, and IQ scores. Its probability density function (PDF) is f(x) = [1 / (sigma * sqrt(2 * pi))] * exp[-(x - mu)^2 / (2 * sigma^2)], with two parameters: mu (the mean) and sigma (the standard deviation). The notation X ~ N(mu, sigma^2) indicates that X follows a normal distribution with mean mu and variance sigma^2.

### II. Properties of the Normal Distribution

The normal distribution is bell-shaped and symmetric about the mean. The mean, median, and mode are all equal to mu. The distribution is completely determined by its two parameters: mu determines the center (location) while sigma determines the spread (scale). The curve is asymptotic to the x-axis, meaning the tails extend to infinity in both directions but never quite reach zero. The total area under the curve equals 1, and the inflection points occur at mu - sigma and mu + sigma.

The empirical rule applies: approximately 68.27% of values fall within mu +/- 1*sigma, 95.45% within mu +/- 2*sigma, and 99.73% within mu +/- 3*sigma.

<image>A figure showing three normal distribution curves overlaid on the same axis. Curve 1: N(0, 1) -- standard normal, tall and narrow. Curve 2: N(0, 4) -- same center but wider and shorter. Curve 3: N(3, 1) -- shifted to the right but same spread as Curve 1. Each curve is labeled with its mu and sigma values. Annotations show how mu shifts the curve horizontally and sigma changes the width. The area under each curve is noted as equal to 1.</image>

### III. The Standard Normal Distribution

The standard normal distribution is the special case where mu = 0 and sigma = 1, denoted Z ~ N(0, 1). Any normal variable X can be converted to Z using the z-score (standardization): z = (x - mu) / sigma. The z-score tells you how many standard deviations a value x is above or below the mean. A z-score of 0 means x is at the mean, z = 1.5 means x is 1.5 standard deviations above the mean, and z = -2.0 means x is 2 standard deviations below the mean. Standard normal tables (z-tables) provide cumulative probabilities P(Z <= z).

### IV. Finding Probabilities Using the Z-Table

To find P(Z <= z), read directly from the table. For P(Z > z), use 1 - P(Z <= z). For P(a <= Z <= b), compute P(Z <= b) - P(Z <= a).

As an example, suppose adult male systolic blood pressure follows N(120, 15^2). To find P(SBP > 140), first calculate z = (140 - 120) / 15 = 1.33. From the table, P(Z <= 1.33) = 0.9082, so P(SBP > 140) = 1 - 0.9082 = 0.0918, or about 9.2%.

To find the systolic blood pressure value at the 90th percentile, look up the z-value where P(Z <= z) = 0.90, which gives z = 1.28. Then x = mu + z * sigma = 120 + 1.28 * 15 = 139.2 mmHg.

### V. Applications of the Normal Distribution in Medicine

The normal distribution underpins many clinical applications. Reference ranges are typically defined as the mean +/- 2 standard deviations, capturing the central 95% of values. For example, a normal fasting glucose range is derived from the population distribution. Growth charts use percentiles derived from normal distribution models. In pediatrics, z-scores for height-for-age and weight-for-age express a child's measurements relative to a reference population. Quality control in laboratories monitors measurements for drift using normal distribution-based rules. Pharmacokinetic parameters are often modeled as normally distributed when developing drug dosing guidelines.

<image>A standard normal distribution curve with shaded regions. The entire curve is divided into labeled regions: the central area between z = -1.96 and z = +1.96 is shaded lightly and labeled "95% of values (reference range)." The left tail beyond z = -1.96 and the right tail beyond z = +1.96 are shaded darker and each labeled "2.5%." Below the curve, a number line translates z-scores to a clinical example (e.g., hemoglobin values with mu = 14 g/dL and sigma = 1.5 g/dL), showing the corresponding clinical reference range of approximately 11-17 g/dL.</image>

### VI. Normal Approximation to Other Distributions

The normal distribution can approximate other distributions under appropriate conditions. For the **normal approximation to the binomial**, when np >= 5 and n(1-p) >= 5, use X ~ N(np, np(1-p)) with a continuity correction of +/- 0.5. For example, if X ~ Binomial(100, 0.4), approximate it as N(40, 24), giving P(X <= 35) approximately equal to P(Z <= (35.5 - 40) / sqrt(24)) = P(Z <= -0.918) = 0.179. For the **normal approximation to the Poisson**, when lambda >= 20, use X ~ N(lambda, lambda). The Central Limit Theorem, covered in Lecture 11, provides the theoretical foundation for these approximations.

### VII. Assessing Normality

Assessing whether data are approximately normal is important because many statistical methods assume normality. Graphical methods include examining a histogram for bell-shaped symmetry, constructing a normal Q-Q plot (quantile-quantile plot) where points should fall approximately on a straight line if data are normal (curvature suggests skewness or heavy tails), and checking a boxplot for symmetry with the median centered in the box and equal whiskers.

Statistical tests for normality include the Shapiro-Wilk test (good for small to moderate samples with n < 5000), the Kolmogorov-Smirnov test (which compares the sample distribution to a theoretical normal), and the Anderson-Darling test. A caution applies to large samples, where these tests may reject normality for trivially small deviations that have no practical consequence. In practice, graphical assessment combined with knowledge of the variable is preferred over reliance on formal tests alone.

<image>A Q-Q (quantile-quantile) plot comparison. Panel A: Data from a normal distribution -- points closely follow the diagonal reference line. Panel B: Right-skewed data -- points curve upward away from the line at the upper end. Panel C: Heavy-tailed data -- points deviate from the line at both ends (S-shape). Each panel includes a small inset histogram of the corresponding data for comparison. Annotations describe the interpretation of each pattern.</image>

### VIII. Transformations for Non-Normal Data

When data are not normal, transformations can sometimes achieve approximate normality. The **log transformation** is useful for right-skewed data such as income or biomarker concentrations; if log(X) is normal, then X follows a log-normal distribution. The **square root transformation** is useful for count data, and the **reciprocal transformation** is occasionally applied. After transformation, standard normal-based methods can be used, but results must be back-transformed for clinical interpretation. An alternative approach is to use nonparametric methods that do not assume normality, which will be covered in Lecture 22.

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