L6 — Standard Normal Distribution, Z-Scores & Probabilities for Intracranial Pressure

Scenario Recap: Intracranial Pressure (ICP)

• Context re-introduced from previous videos

  • ICP assumed normally distributed

  • True (population) parameters supplied

  • Mean \mu = 20\text{ mmHg}

  • Standard deviation \sigma = 5\text{ mmHg}

• Clinical trigger: intensive treatment considered when ICP > 27.5\text{ mmHg}

• Visual reference

  • Bell-shaped curve sketched previously with tick-marks at \mu \pm 1\sigma,\, \mu \pm 2\sigma,\, \mu \pm 3\sigma

  • 27.5\text{ mmHg} lies between \mu + 1\sigma (25) and \mu + 2\sigma (30)

Why Reference Ranges Fail Here

• Usual 68–95–99.7 rule (aka 68\%/95\%/99.7\% rule) only tells us

  • \approx 68\% of values within \pm 1\sigma

  • \approx 95\% within \pm 2\sigma

  • \approx 99.7\% within \pm 3\sigma

• 27.5 does not coincide with a standard integer multiple of \sigma → need more precise method

Enter the Standard Normal Distribution (SND)

• Definition: normal distribution with fixed parameters

  • Mean 0, standard deviation 1

• Purpose

  • Eliminates need to compute integrals for every possible \mu,\sigma

  • Areas (probabilities) pre-tabulated once for all

• Z-score converts any normal value to its SND coordinate

  • Formula
    Z = \frac{X_{obs} - \mu}{\sigma}

  • Meaning: number of standard deviations the observation X_{obs} sits from the mean

• When population parameters unknown

  • Substitute sample mean \bar x and sample SD s if sample size n \ge 30\text{–}40 ("large")

Computing Z for ICP = 27.5\text{ mmHg}

• Plug values
  Z = \frac{27.5 - 20}{5} = 1.5

• Interpretation: 27.5 is 1.5\sigma above the mean

Using Table A1 (Essential Medical Statistics)

• Table layout (version used in lecture)

  • Columns: second decimal digit of Z

  • Entries: right-tail probability P(Z > z)

  • Symmetry fact: P(Z > 0) = 0.5 appears as 0.5000 in row 0.0, column 0.00

• Lookup procedure for Z = 1.50

  • Row 1.5, column 0.00 → area = 0.0668

  • This value already corresponds to desired orange region (right of 1.5 \sigma)

Final Probability Statement

• P(ICP > 27.5\text{ mmHg}) = 0.0668 = 6.68\%

• Clinically: only about 1 in 15 patients exceed treatment threshold under stated assumptions

Diagramming & Alternate Scenarios

• Always draw the distribution first to identify

  • Which table area you can obtain directly (orange/purple/blue examples)

  • Which area you actually need (green/purple regions)

• Common manipulations

  1. Left-of-mean regions

  • Table gives right tail; exploit symmetry

  • Example: need P(0 < Z < z) → compute 0.5 - P(Z > z)

  1. Between two positive z-values (both >0)

  • Let z1 < z2

  • Table provides two right tails

    • A1 = P(Z > z1) (orange)

    • A2 = P(Z > z2) (blue)

  • Desired region (purple) = A1 - A2

  1. Two-sided intervals (not explicitly drawn here)

  • Symmetry simplifies calculation: e.g. P(|Z| < z) = 1 - 2P(Z > z)

• Moral: proper sketching prevents sign & subtraction errors

Broader Lecture 3 Take-Home Points

• Reviewed sample vs population parameters

• Defined Bernoulli and binomial distributions (completeness)

• Explored normal distribution properties

  • Reference ranges

  • 68\%/95\%/99.7\% empirical rule

• Introduced standard normal distribution and Z-tables

• Demonstrated full workflow: diagram → Z-score → table lookup → probability statement

• Preview Week 4

  • Sampling distributions of sample statistics, especially \bar X

  • Central Limit Theorem (CLT) as foundational concept