Community medicine chapters
Normal distribution and probability
The shape and properties of the normal (Gaussian) curve, the 68-95-99.7 rule, z-scores, how normal ranges are set, and the basic laws of probability used in medicine.
- The normal curve is bell-shaped and symmetrical; mean = median = mode, and the total area under it is 1 (100%).
- About 68% of values lie within mean ± 1 SD, 95% within ± 2 SD (exactly ±1.96), and 99.7% within ± 3 SD.
- Probability runs from 0 (impossible) to 1 (certain); add probabilities for mutually exclusive "either/or" events and multiply them for independent "both/and" events.
On this page
Overview
Many biological measurements (height, haemoglobin, blood pressure in a population) follow a normal distribution: most values cluster around the average and become rarer towards the extremes.
The normal curve lets us define normal ranges, calculate z-scores, and underpins most statistical tests.
The normal (Gaussian) curve
Properties
- Bell-shaped and symmetrical about the mean.
- Mean = median = mode, all at the centre.
- The tails approach the horizontal axis but never touch it.
- The total area under the curve = 1 (100% of observations).
- Its shape is fully described by two values: the mean (position) and the standard deviation (spread).
- Skewness = 0.
The 68-95-99.7 rule
| Range | Share of values |
|---|---|
| Mean ± 1 SD | About 68% |
| Mean ± 2 SD (exactly ± 1.96 SD) | About 95% |
| Mean ± 3 SD (exactly ± 2.58 SD gives 99%) | About 99.7% |
95% of values fall within mean ± 1.96 SD. Only about 5% lie outside: 2.5% in each tail.
Example: systolic BP in adults has a mean of 120 and SD of 10 mmHg. About 95% of adults will have a systolic BP between 100 and 140.
The standard normal curve and z-scores
Any normal distribution can be converted to the standard normal distribution, which has a mean of 0 and SD of 1.
The z-score (standard normal deviate) shows how many SDs a value lies from the mean:
z = (value − mean) ÷ SD
- z = 0: exactly at the mean.
- z = +2: two SDs above the mean.
- z = −2: two SDs below the mean (about the 2.5th percentile).
z-scores let us compare values measured on different scales, and they are used in growth charts: weight-for-height below −2 SD indicates wasting, and below −3 SD severe wasting (see acute malnutrition).
Systolic BP: mean = 120, SD = 10 mmHg
z for a BP of 135 = (135 − 120) ÷ 10 = +1.5 (1.5 SD above the mean)
95% range = mean ± 1.96 SD
= 120 ± (1.96 × 10) = 120 ± 19.6
= 100.4 to 139.6 mmHg
z = (x − mean) ÷ SD. A z-score of −2 means the value is 2 SD below the mean.
Normal (reference) ranges
Laboratory "normal ranges" are usually set as the central 95% of values in healthy people: mean ± 2 SD.
By definition, about 5% of healthy people fall outside the normal range. A value outside the range is not always abnormal, and doing many tests on one healthy person makes at least one "abnormal" result likely.
Distributions that are not normal
- Positively skewed: a long tail to the right (incubation periods, income, length of stay). Mean > median > mode.
- Negatively skewed: long tail to the left (age at death in rich countries). Mean < median < mode.
- Bimodal: two peaks (age of onset of some diseases).
For skewed data, the median and interquartile range describe the data better, and non-parametric tests are used (see statistical tests).
Probability
Probability (P) is the chance that an event will occur.
- It ranges from 0 (impossible) to 1 (certain).
- P = number of ways the event can happen ÷ total number of possible outcomes.
- P(event does not happen) = 1 − P(event).
Rules of probability
| Rule | Used when | Formula | Example |
|---|---|---|---|
| Addition rule | Events are mutually exclusive (cannot happen together): "A or B" | P(A or B) = P(A) + P(B) | A baby's blood group is A or B: add the two probabilities |
| Multiplication rule | Events are independent (one does not affect the other): "A and B" | P(A and B) = P(A) × P(B) | Two babies both being boys: 0.5 × 0.5 = 0.25 |
P(boy) = 0.5 for each birth (independent)
Two children both boys = 0.5 × 0.5 = 0.25
Three children all boys = 0.5 × 0.5 × 0.5 = 0.125
At least one girl in three = 1 − 0.125 = 0.875
"Or" with mutually exclusive events → add. "And" with independent events → multiply.
Probability distributions
- Normal distribution: continuous data.
- Binomial distribution: the number of "successes" in a fixed number of trials with two outcomes (for example the number of boys in families of four).
- Poisson distribution: counts of rare events in a fixed time or space (for example cases of a rare disease per month).
How this links to statistical tests
Inferential tests compare a result with what would be expected by chance under the normal distribution. A difference more than about 1.96 standard errors from the null value has a probability of less than 5% (p < 0.05) of arising by chance. See hypothesis testing.
Expect: properties of the normal curve (mean = median = mode, symmetry, area = 1), the percentages within 1, 2 and 3 SD, 1.96 for 95%, the z-score formula, how normal ranges are set and why 5% of healthy people fall outside, and the addition and multiplication rules of probability.
Quick revision
- Normal curve: bell-shaped, symmetrical, mean = median = mode, area = 1.
- ±1 SD 68%; ±2 SD 95% (±1.96 exactly); ±3 SD 99.7%.
- z = (value − mean) ÷ SD; standard normal: mean 0, SD 1.
- Normal range: mean ± 2 SD; 5% of healthy people fall outside.
- Probability 0 to 1; "or" (mutually exclusive) add; "and" (independent) multiply.
- Binomial: two outcomes. Poisson: rare events.
Written by The House Job from Excel Community Medicine and Park's Textbook of Preventive and Social Medicine (named in the UHS PGET syllabus).
FAQ
What are the properties of a normal distribution?
It is bell-shaped and symmetrical, the mean, median and mode are equal, and about 68%, 95% and 99.7% of values lie within one, two and three standard deviations of the mean.
What is a z-score?
The number of standard deviations a value lies above or below the mean: (value − mean) ÷ SD.
How is a normal laboratory range usually defined?
As the mean plus or minus two standard deviations, covering the central 95% of healthy people.
When do you multiply probabilities?
When you want the chance of two independent events both happening.