Normal distribution
- Concept
The bell curve: symmetric, with mean, median and mode equal, and fixed shares of the data within 1, 2 and 3 standard deviations.
What it is
The normal (Gaussian) distribution is a symmetric bell-shaped curve defined by its mean and standard deviation. Many biological measurements come close to it, and almost all parametric tests assume it. Moreover, the means of large samples tend to be normally distributed even when the raw data are not (central limit theorem), which is why it is used to build confidence intervals.
Key facts
- z = (x − mean) / SD. z = 0 is the mean; z = 2 is 2 SD above it.
- The standard normal has mean 0 and SD 1.
- Central limit theorem: the mean of many large samples follows an almost normal distribution, whatever the shape of the data.
Worked example
- Data: in 1 000 university students, systolic pressure has a mean of 116 mmHg and an SD of 10 mmHg, with a bell shape.
- ±1 SD: 106 to 126 mmHg holds about 68% (some 680 students).
- ±2 SD: 96 to 136 mmHg holds about 95% (some 950 students).
- ±3 SD: 86 to 146 mmHg holds about 99.7% (nearly all).
- A student with 136 mmHg has z = (136 − 116) / 10 = 2.0.
- Above z = 2 lies about 2.5% (half of the 5% in the two tails): some 25 of the 1 000 students.
- Reading: being outside the central 95% is uncommon in this population, but it does not by itself mean being ill.
Markers
- z
- SD
- 68–95–99.7
In practice
- A healthy person given 20 independent tests with 95% ranges has a high chance of at least one ‘abnormal’ value: 1 − 0.95²⁰ ≈ 64%.
- Child growth charts use z scores to place a child against the reference population.
Common pitfalls
- Assuming normality without looking at a histogram: many clinical variables (length of stay, costs, counts) are skewed.
- Thinking that ‘normal’ in statistics means ‘healthy’.
- Using mean ± 2 SD as a reference range in very skewed data, where the lower limit can even come out negative.
Test yourself
Think of the answer, then tap the question to see it.
What share of the data lies within ±2 standard deviations of the mean in a normal distribution?
About 95% (exactly 95% within ±1.96 SD).
If the mean is 50 and the SD is 5, what is the z score of a value of 40?
z = (40 − 50) / 5 = −2: two standard deviations below the mean.
How do the mean, median and mode relate in a normal distribution?
They are equal, because the curve is symmetric.
Why do some healthy people have results outside the reference range?
Because the range is usually the central 95% of healthy people: the other 5%, also healthy, fall outside by definition.
Connections
Educational content on epidemiological and statistical methods; the examples use made-up numbers to teach the calculation. Not a clinical or legal guide: current Colombian rules are in the official sources.
Sources: Normal Distribution (NIST/SEMATECH e-Handbook of Statistical Methods, National Institute of Standards and Technology, 2012); Principles of Epidemiology | Lesson 2 - Section 7 (Centers for Disease Control and Prevention, 2012).
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Representative model, for educational purposes only.