Inverse Normal Calculation

Module 3 - Section 4: Inverse Normal Calculation

Introduction to Inverse Normal Calculations

  • In certain scenarios, an area is provided, and the goal is to find the corresponding value on the measurement scale.

  • This value is such that the cumulative area up to this value equals the given area.

Z-table for Standard Normal Distribution

  • The Z-table provides critical values for standard normal distribution (Z-values) based on the second decimal place.

Sample Z-table Excerpt

  • Z-values and corresponding cumulative areas (up to 4 decimal places):

    • Row 0.0:

    • Area = 0.5000 at z = 0.00

    • Area = 0.5040 at z = 0.01

    • Area = 0.5080 at z = 0.02

    • Row 0.1:

    • Area = 0.5398 at z = 0.10

    • Area = 0.5438 at z = 0.11

    • Area = 0.5478 at z = 0.12

    • Continue for rows 0.2 through 3.9 displaying various cumulative areas.

  • The areas represent the probability that a standard normal variable is less than the corresponding Z-value.

Example Problems

Finding the Smallest 2%

  • Problem: Identify the z-value that captures the smallest 2%.

  • Mathematically represented as:
    P(z < z^*) = 0.02

  • This means we need to find the Z-value where the area to the left of it is 0.02.

Finding the Largest 5%

  • Problem: Identify the z-value such that the probability of being greater than it is 5%.

  • Mathematically represented as:
    P(z > z^*) = 0.05

  • This entails finding the Z-value where the area to the right of it is 0.05.

Finding Middle 95%

  • Problem: Calculate z-values for the middle 95% of the distribution.

  • Mathematically expressed as:
    P(−z∗≤z≤z∗)=0.95P(-z^* ≤ z ≤ z^*) = 0.95

  • In this case, two Z-values are found, which mark the boundaries of the central 95% of the distribution.

Example with Non-standard Distribution

  • Scenario: For a normally distributed variable where mean (µ) = 100 and standard deviation (σ) = 5, determine the value corresponding to the smallest 30%.

  • Calculation Steps:

    • Identify the Z-value corresponding to 0.30 from the standard normal distribution.

    • Transform back to the original scale using:
      y∗=extµ+z∗imesextσy^* = ext{µ} + z^* imes ext{σ}

Summary of Backward Normal Calculations

  1. Statement of the Problem: Determine the proportion of interest in the context provided.

  2. Locate the Z-value: Use the Z-table to find the closest entry to the proportion and identify the corresponding Z-value (z*).

  3. Unstandardize: Convert the Z-value back to the original measurement scale using the inversion formula:
    y∗=extµ+z∗imesextσy^* = ext{µ} + z^* imes ext{σ}

Application Example: Length of Human Pregnancy

  • Mean: 266 days

  • Standard Deviation: 16 days

  • Calculations Required:

    1. Probability of Lasting Longer than 280 Days: Find the corresponding Z-value and calculate probability.

    2. Longest 10% Shortest Pregnancies: Find the Z-value for the lowest 10% and convert using the same unstandardization formula.

Conclusion

  • Understanding inverse normal calculations is essential for statistics involving normally distributed data, allowing one to derive important probabilities and cut-offs relevant in various analyses.

Acknowledgements

  • Thank you for engaging with this content!