Lecture Notes on Activity in Class 4E: Empirical Rule and Z-Scores

Introduction

• Focus on empirical rule and z-scores.
• Shift from politics to clinical trials (toxicity studies on mice).
• Study of spleen and liver toxicity in mice exposed to chemicals.

Key Concepts

Empirical Rule

• Also known as the 68-95-99.7 rule.
• Applied to bell-shaped (normal) distributions.
• Describes the spread of data:
  • 68% of data within 1 standard deviation (SD) from the mean.
  • 95% of data within 2 SDs from the mean.
  • 99.7% of data within 3 SDs from the mean.
• Not all data follow a normal distribution; caution needed.

Z-Scores (Standard Scores)

• Defined as (observation - mean) / standard deviation.
• Measures how many standard deviations an observation is from the mean.
• Positive z-score: observation above the mean.
• Negative z-score: observation below the mean.

Clinical Trials Background

• Conducted to evaluate drug safety and efficacy.
• Human trials vs. animal trials (focus on mice for control).
• Discussion on why people volunteer for trials (curiosity, desperation, money).

Study on Mice

• Mice used in clinical trials to understand chemical toxicity.
• Focus on liver and spleen weights of 26-week-old female mice for consistency.
• Liver and spleen are filter organs often affected by toxins.

Liver and Spleen Data

Liver

• Mean: 0.999 grams
• Standard Deviation: 0.087 grams
• Normal distribution is assumed.
• Calculated ranges:
  • 68%: 0.912 to 1.086 grams
  • 95%: 0.825 to 1.173 grams
  • 99.7%: 0.738 to 1.26 grams

Spleen

• Mean: 0.086 grams
• Standard Deviation: 0.007 grams
• Normal distribution is assumed.
• Calculated ranges:
  • 68%: 0.079 to 0.093 grams
  • 95%: 0.072 to 0.1 grams
  • 99.7%: 0.065 to 0.107 grams

Interpretation of Unusual Results

• Unusual observations are generally those beyond 2 standard deviations.
• Liver weights less than 0.825 or greater than 1.173 are unusual.
• Spleen weights less than 0.072 or greater than 0.1 are unusual.

Z-Score Calculation and Interpretation

• Example given for a mouse liver weight of 1.07 grams:

  • Z-score: Approximately 0.816
  • Indicates liver is less than 1 SD above the mean (typical).

• Example for a mouse spleen weight of 0.104 grams:

  • Z-score: Approximately 2.571
  • Indicates spleen is more than 2.5 SDs above the mean (unusual and indicates high toxicity).

Conclusion

• Z-scores and the empirical rule are effective tools for determining the usualness of an observation.
• They aid in comparing different distributions (e.g., liver vs. spleen).
• Unusual sizes in organs (like spleen) can indicate higher toxicity levels.

Wrap-Up

• Practice activity suggested to reinforce learning.
• Suggestion to take breaks between study sessions for effective learning.