Measures of variation in medical statistics

Measures of variation (also called measures of dispersion) describe how spread out or dispersed data values are within a dataset. They help clinicians and researchers understand how much variability exists around measures of central tendency (such as the mean or median).

The most commonly used measures of variation are:

• Range
• Interquartile range
• Variance
• Standard deviation

Range

• Definition: The difference between the largest and smallest values in a dataset.
• Formula: Range = Maximum value - Minimum value
• Use: Gives a quick sense of the total spread of data, but is highly affected by outliers.

Interquartile range (IQR)

• Definition: The range of the middle 50% of the data.
• Use: A more robust measure of variability than the range, because it is not affected by extreme values or outliers.
• Formula: IQR=Q3-Q1

​where 𝑄1 is the first quartile (25th percentile) and 𝑄3 is the third quartile (75th percentile).


Variance

• Definition: The average of the squared differences from the mean.
• Use: Reflects how data values are spread around the mean. The units of variance are squared, which can make interpretation less intuitive.
• Formula (Population): σ² = Σ(xi - μ)² / N

Where:
σ² is the population variance.
Σ is the summation symbol.
xi is each individual data point in the population.
μ is the population mean.
N is the total number of data points in the population.

Standard Deviation (SD)

• Definition: The square root of the variance.
• Formula: standard deviation (σ) = √Variance
• Use: The most commonly used measure of spread in medical research. It expresses the typical distance of values from the mean, in the same units as the data.

Graphical Representation: Box and Whisker Plot
A Box and Whisker Plot (often referred to as simply a Box Plot) is a graphical summary of a dataset that displays its distribution, central tendency, and variability.


What it shows:

• Median (middle value of the data)
• Interquartile range (IQR): The box spans from Q1 to Q3. This range contains the middle 50% of the data values.
• Whiskers: Extend to the smallest and largest values within 1.5 × IQR from the quartiles
• Outliers: Displayed as individual points beyond the whiskers

Why it's useful:

• Provides a visual summary of distribution, spread, and outliers
• Easily compares multiple groups or conditions side by side
• Ideal for illustrating non-normally distributed data