Measures of Dispersion – Range, Variance, Standard Deviation, Interquartile Range 📐

🎯 Learning Objectives

By the end of this lesson, you will be able to:
✔ Define and calculate Range, Variance, Standard Deviation (SD), and Interquartile Range (IQR)
✔ Understand their interpretation in medical data
✔ Know why dispersion is important alongside central tendency

1️⃣ Introduction

Knowing the average (mean, median) is not enough. We also need to know how spread out the data is.
This is called dispersion. Two datasets can have the same mean but very different spreads, which can change interpretation in clinical research.

Common measures of dispersion:
✅ Range
✅ Variance
✅ Standard Deviation (SD)
✅ Interquartile Range (IQR)

2️⃣ Range

Definition:

• The difference between the largest and smallest value in the dataset.

✔ Formula:

✔ Example:
BP readings: 110, 120, 130, 140, 150
Range = 150 – 110 = 40 mmHg

✔ Key Features:

• Simple but affected by outliers
• Gives only the extreme spread, not the distribution

3️⃣ Variance

Definition:

• The average of the squared differences from the mean.

✔ Formula:

✔ Example:
If mean = 100, values = 90, 100, 110
Differences = -10, 0, +10 → Squares = 100, 0, 100
Variance = (100 + 0 + 100)/3 = 66.67

✔ Key Features:

• Always positive
• Units are squared (not very interpretable clinically)

4️⃣ Standard Deviation (SD)

Definition:

• The square root of the variance.
• Indicates how much values typically deviate from the mean.

✔ Formula:

✔ Example:
From above, Variance = 66.67 → SD = √66.67 ≈ 8.16

✔ Key Features:

• Most widely used measure of dispersion
• Used in confidence intervals, normal distribution

✔ Clinical Interpretation:

• In a normal distribution:
  • 68% values lie within ±1 SD
  • 95% values within ±2 SD
  • 99.7% values within ±3 SD

5️⃣ Interquartile Range (IQR)

Definition:

• The range between the 25th percentile (Q1) and 75th percentile (Q3).
• It captures the middle 50% of the data, removing extreme values.

✔ Formula:

✔ Example:
Data: 5, 7, 8, 10, 12, 13, 15
Q1 = 7, Q3 = 13 → IQR = 13 – 7 = 6

✔ Key Features:

• Not affected by outliers
• Best for skewed data

6️⃣ Comparison Table

7️⃣ Clinical Relevance

• Range: Quick spread in BP or lab values
• SD: Used in reporting clinical trial results
• IQR: Used in non-parametric statistics, skewed hospital stay data

8️⃣ Quick Check: Test Your Understanding ✅

Q1: Which measure is most affected by extreme values?
a) Range
b) SD
c) IQR
👉 Answer: a) Range

Q2: If Q1 = 20 and Q3 = 30, what is IQR?
👉 Answer: 10

Q3: Which measure is best for skewed data?
👉 Answer: IQR

✨ Key Takeaways

• Range: Simple but outlier-sensitive
• Variance: Basis for SD, not easily interpretable
• Standard Deviation: Most widely used, key for normal distribution
• IQR: Best for skewed data, ignores extremes