L5 - Reference Ranges for the Normal Distribution – Detailed Study Notes

Normal Distribution: Reference Ranges

• Symmetric, bell-shaped probability distribution.

• Key parameters:

  • \mu (mean)

  • \sigma (standard\ deviation)

• Visual convention in course:

  • Plot \mu at the apex of the curve.

  • Mark \mu \pm 1\sigma,\ \mu \pm 2\sigma,\ \mu \pm 3\sigma on the horizontal axis (both left and right).

Empirical ("68–95–99.7") Rule

• 68\% of observations fall in [\mu-1\sigma,\ \mu+1\sigma].

• 95\% fall in [\mu-2\sigma,\ \mu+2\sigma].

• 99.7\% fall in [\mu-3\sigma,\ \mu+3\sigma].

• Because of symmetry:

  • 50\% are below \mu; 50\% are above \mu.

  • Region-by-region breakdown (per tail & side):

  • 34\% between \mu and \mu\pm1\sigma (each side).

  • 13.5\% between 1\sigma and 2\sigma (each side).

  • 2.35\% between 2\sigma and 3\sigma (each side).

  • 0.15\% beyond 3\sigma (each tail).

Interpreting Percentages as Probabilities

• E.g. "Chance a random observation lies within \pm1\sigma of the mean = 0.68".

• Useful for back-of-envelope answers without consulting a normal table.

Worked Example: Intracranial Pressure (ICP) in TBI Patients

• Dataset: n = 80 traumatic brain-injury patients.

• Assumed normal with

  • \mu = 20\ \text{mmHg}

  • \sigma = 5\ \text{mmHg}

Step-by-Step Problem-Solving Template

1. DRAW the bell curve.

2. LABEL \mu and each integer multiple of \sigma:

   • \mu = 20

   • \mu \pm 1\sigma = 15,\ 25

   • \mu \pm 2\sigma = 10,\ 30

   • \mu \pm 3\sigma = 5,\ 35

3. SHADE the region asked about.

4. READ the proportion directly from the 68-95-99.7 rule (or normal table).

5. CONVERT region limits back to raw units if needed.

Q1: Median ICP

• For a perfect normal, \text{median} = \mu.

• Therefore \tilde{x} = 20\ \text{mmHg}.

Q2: Probability 15–25 mmHg ("high but no treatment")

• Interval corresponds to [\mu-1\sigma,\ \mu+1\sigma].

• Probability = 68\% = 0.68.

Q3: Limits Capturing 95 % of Patients

• Need [\mu-2\sigma,\ \mu+2\sigma] because 95\% rule.

• Calculate:

  • Lower = 20 - 2(5) = 10\ \text{mmHg}

  • Upper = 20 + 2(5) = 30\ \text{mmHg}

• Interpretation: "95\% of similar TBI patients will show ICP between 10 and 30\ \text{mmHg}."

Importance of the Diagram

• Prevents skipping logic steps.

• Visual check for symmetry and correct region selection.

• Reduces algebraic mistakes especially under exam pressure.

Broader Connections & Practical Relevance

• Normal reference ranges underpin clinical decision thresholds (e.g., ICP management).

• Same methodology appears in quality control ("Six-Sigma"), finance (VaR approximations), and psychometrics (IQ scores).

• Ethical note: Over-reliance on normality assumptions can misclassify outliers, leading to over-treatment or under-treatment. Always validate assumptions (explored in next lecture).

Numerical Summary (Example)

• Median = 20\ \text{mmHg}.

• 68\% between 15 and 25\ \text{mmHg}.

• 95\% between 10 and 30\ \text{mmHg}.

Follow-up lecture: handling situations where data are non-normal, or where empirical rule fails (e.g., skewed biomarkers).