Measures of dispersion

What are the measures of dispersion? Discover how to calculate and interpret data variability in biostatics using simple formula and examples

Measures of Dispersion 

• Dispersion refers to the extent to which a dataset is spread out, measuring the variability of data points. 
  • High dispersion indicates that the data points are widely spread. 
  • Low dispersion shows that the data points cluster closely around a central value. 

Key Measures of Dispersion: 

1. Range 

   • The difference between the maximum and minimum values. 
   • Simple to calculate but sensitive to outliers. 
   • Formula: 
     • • Range=Maximum value−Minimum value
       • 𝑹𝒂𝒏𝒈𝒆=𝑴𝒂𝒙𝒊𝒎𝒖𝒎 𝒗𝒂𝒍𝒖𝒆−𝑴𝒊𝒏𝒊𝒎𝒖𝒎 𝒗𝒂𝒍𝒖𝒆

2. Interquartile Range (IQR) 

   • The difference between the 75th percentile (Q3) and the 25th percentile (Q1). 
   • Represents the spread of the middle 50% of data. 
   • Less sensitive to outliers. 

3. Variance 

   • The average of the squared differences from the mean. 
   • Gives an idea of how much data points vary from the mean, but in squared units. 
   • Less intuitive due to units being squared. 

4. Standard Deviation 

   • The square root of the variance. 
   • Returns dispersion to the original units of measurement. 
   • Widely used and more interpretable than variance. 

5. Mean Absolute Deviation (MAD) 

   • The average of the absolute differences between each data point and the mean. 
   • A direct and intuitive measure of spread. 

6. Coefficient of Variation (CV) 

   • A standardized measure of dispersion. 
   • Formula: 
     • CV=Standard DeviationMean
   • Useful for comparing datasets with different units or scales. 

Example: Standard Deviation Calculation 

• A pharmaceutical company produces tablets intended to contain 100 mg of an active pharmaceutical ingredient (API).  

• A sample shows the following API concentrations (in mg): 

Data:  

• 98, 102, 99, 101, 100, 98, 103, 97, 102, 100 

Steps: 

1. Mean concentration = 100 mg 
2. Differences from the mean: 
   • −2,2,−1,1,0,−2,3,−3,2,0 
3. Squared differences: 
   • 4,4,1,1,0,4,9,9,4 
4. Variance: 

Variance: 

(4+4+1+1+0+4+9+9+4+0)10=3610=3.6 mg24+4+1+1+0+4+9+9+4+010=3610=3.6 mg2

Interpretation: 

• A standard deviation of 1.9 mg shows the variability in API concentration.  
• A lower standard deviation indicates better consistency, crucial for drug safety and efficacy. 

Range 

• The range is the simplest measure of dispersion. 
• It is calculated as the difference between the maximum and minimum values. 
• Provides a quick sense of spread but is sensitive to outliers. 

1. Range in a Discrete Series 

Discrete data consist of separate, distinct values. 

Example: 

• Number of prescriptions filled per day: 

Day	Prescriptions Filled
Monday	45
Tuesday	50
Wednesday	40
Thursday	55
Friday	48
Saturday	60
Sunday	43

• Calculation: 
  • Maximum value = 60 (Saturday) 
  • Minimum value = 40 (Wednesday) 
  • Range = 60 − 40 = 20 
• Interpretation: 

There’s a difference of 20 prescriptions between the busiest and slowest day. 

1. Range in a Continuous Series 

Continuous data can take any value within a range and are often grouped into intervals. 

Example: 

• Reduction in blood pressure (mmHg): 

Blood Pressure Reduction (mmHg)	Number of Patients
10–19	5
20–29	12
30–39	8
40–49	5
50–59	2

• Calculation: 
  • Minimum interval = 10–19 → use 10 as minimum boundary 
  • Maximum interval = 50–59 → use 59 as maximum boundary 
  • Range = 59 − 10 = 49 mmHg 
• Interpretation: 

The approximate spread in blood pressure reduction is 49 mmHg. 

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