Calculation of mean, median mode and incidence prevalence
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Mean, Median, Mode - and Incidence & Prevalence
PART 1: MEASURES OF CENTRAL TENDENCY
Central tendency in a normal (Gaussian) distribution is characterized by three measures: mean, median, and mode. - Swanson's Family Medicine Review
1. MEAN
The mean is the sum of all values divided by the number of observations.
Formula:
Mean (x̄) = (Sum of all values) / (Number of observations) = Σx / n
Example:
Dataset: 4, 6, 7, 8, 10, 11, 14
- Sum = 60
- n = 7
- Mean = 60 / 7 = 8.57
Geometric mean = the nth root of the product of n numbers. It is used when data follow a log-normal distribution (e.g., antibody titres), as it minimizes the effect of extreme values. - Henry's Clinical Diagnosis and Management by Laboratory Methods
When to use: In a normally distributed (symmetric) dataset, the mean is the best summary statistic. - Rockwood and Green's Fractures in Adults
2. MEDIAN
The median is the value point where the number of observations above equals the number below - i.e., the middle value when data are sorted in order (also called the 50th percentile).
How to calculate:
- Arrange values in ascending order.
- If n is odd: median = the middle value.
- If n is even: median = average of the two middle values.
Example (odd):
Dataset sorted: 4, 6, 7, 8, 10, 11, 14 (n=7)
- Median = 8 (4th value)
Example (even):
Dataset sorted: 4, 6, 7, 8, 10, 11 (n=6)
- Median = (7+8)/2 = 7.5
When to use: The median is preferred when data are skewed or not normally distributed (e.g., income, hospital length of stay, ordinal pain scores). - Rockwood and Green's Fractures in Adults
3. MODE
The mode is the most frequently occurring value in a dataset.
Example:
Dataset: 4, 6, 7, 7, 8, 10, 11, 11, 11, 14
- Mode = 11 (appears 3 times)
A dataset can be:
- Unimodal - one mode
- Bimodal - two modes
- Multimodal - more than two modes
- No mode - all values occur equally
Important note: Standard deviation (SD) is NOT a measure of central tendency - it is a measure of dispersion/spread. - Swanson's Family Medicine Review
Relationship in Normal vs. Skewed Distributions
| Distribution | Mean vs. Median vs. Mode |
|---|---|
| Normal (Gaussian) | Mean = Median = Mode |
| Positively skewed (right) | Mode < Median < Mean |
| Negatively skewed (left) | Mean < Median < Mode |
In a normal distribution: 68.26% of values fall within ±1 SD from the mean; 95.44% within ±2 SD; 99.72% within ±3 SD. - Swanson's Family Medicine Review
PART 2: INCIDENCE AND PREVALENCE
These are the two fundamental measures of disease frequency in epidemiology. - Park's Textbook of Preventive and Social Medicine
1. INCIDENCE RATE
Definition: The number of NEW cases occurring in a defined population during a specified period of time.
Formula:
Incidence Rate = (Number of new cases of a disease during a given time period / Population at risk during that period) × 1000
Example:
- 500 new cases of illness in a population of 30,000 in one year
- Incidence = (500 / 30,000) × 1000 = 16.7 per 1,000 per year
Note: Incidence rate must always include the unit of time (e.g., "per 1,000 per year"). Writing just "16.7 per 1,000" is incomplete. - Park's Textbook of Preventive and Social Medicine
Key features of incidence:
- Counts only new (incident) cases
- Measured over a defined time period
- Denominator = population at risk (disease-free at start)
- Generally used for acute conditions
- Not influenced by disease duration
Incidence Rate (person-time denominator):
Incidence Rate = New cases / Total person-time at risk
This is used in cohort studies where different individuals are followed for different lengths of time.
Special incidence rates:
- Attack rate = (New cases during epidemic / Total population at risk) × 100 - used during outbreaks
- Secondary attack rate = number of exposed persons developing disease within one incubation period after exposure to a primary case
2. PREVALENCE
Definition: "The total number of all individuals who have an attribute or disease at a particular time (or during a particular period) divided by the population at risk of having the attribute or disease at this point in time." - Park's Textbook of Preventive and Social Medicine
Prevalence includes all current cases - both old and new.
(a) Point Prevalence
Point Prevalence = (All current cases at a given point in time / Estimated population at the same point in time) × 100
This is what is meant when "prevalence" is used without further qualification.
(b) Period Prevalence
Period Prevalence = (All existing cases during a defined time interval / Estimated mid-interval population at risk) × 100
Includes cases that started before the period but extended into it, plus new cases arising during the period.
Incidence vs. Prevalence - Visual Illustration
The diagram below from Park's Textbook illustrates the difference clearly. The two vertical lines represent Jan 1 and Dec 31 of a year. Each horizontal bar is a case (open circle = start of illness):
From this diagram:
- Incidence (new cases beginning during the year) = Cases 3, 4, 5, and 8
- Point prevalence (Jan 1) = Cases 1, 2, and 7 (existed on that exact date)
- Point prevalence (Dec 31) = Cases 1, 3, 5, and 8
- Period prevalence (Jan-Dec) = Cases 1, 2, 3, 4, 5, 7, and 8 (all that existed at any time during the year)
3. RELATIONSHIP BETWEEN INCIDENCE AND PREVALENCE
Prevalence depends on two factors: incidence and duration of illness. For a stable population:
P = I × D (Prevalence = Incidence × Mean Duration)
Example:
- Incidence = 10 cases per 1,000 population per year
- Mean duration of disease = 5 years
- Prevalence = 10 × 5 = 50 per 1,000 population
Conversely: Incidence = P / D and Duration = P / I
Key implications of P = I × D: - Park's Textbook of Preventive and Social Medicine
| Situation | Effect on Prevalence |
|---|---|
| High incidence + long duration (e.g., TB, lupus) | High prevalence relative to incidence |
| Acute, short-duration disease (e.g., food poisoning) | Prevalence ≈ incidence (very low prevalence) |
| Rapidly fatal disease | Negligible prevalence |
| Better treatment shortens duration | Prevalence falls (even if incidence unchanged) |
| Treatment prevents death but not cure | Prevalence paradoxically rises |
As Rheumatology (Elsevier, 2022) notes, diseases like systemic lupus have relatively low incidence but long duration - so prevalence is much higher than incidence. This is also why prevalent cases may be a healthier subset ("survivor bias" or depletion of susceptibles).
Quick Comparison Table
| Feature | Incidence | Prevalence |
|---|---|---|
| Cases counted | New cases only | All existing cases (old + new) |
| Time element | Rate over a defined period | Snapshot at a point or period |
| Denominator | Population at risk (disease-free) | Total population |
| Expressed as | Rate per 1,000 per year | Ratio / % |
| Best used for | Aetiology, causation, acute disease | Disease burden, chronic disease planning |
| Analogy | A film (continuous record) | A photograph (instantaneous record) |
Sources: Park's Textbook of Preventive and Social Medicine; Rockwood and Green's Fractures in Adults (10th ed.); Henry's Clinical Diagnosis and Management by Laboratory Methods; Swanson's Family Medicine Review; Rheumatology (Elsevier, 2022)
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