Understanding Z-Scores

Introduction to Z-Scores

• Z-scores are used to interpret individual data points in relation to the rest of the data.
• They help understand how a particular score compares to the average of a dataset.

Example: Fictitious Attractiveness Measure

• Imagine a scale rating attractiveness.
• Three individuals scored as follows:
  • Shifty: 35
  • Mickey: 65
  • Antonio: 90
• Initial scores tell us relative attractiveness among the three but not how they compare to others.

Importance of Mean and Deviation Scores

• Mean: Average score is necessary to understand scores in context.
  • Example: Average score on the test is 60.
• Determine who is above or below the mean:
  • Shifty: 35 (25 points below the mean)
  • Mickey: 65 (5 points above the mean)
  • Antonio: 90 (30 points above the mean)

Understanding Standard Deviation

• Standard Deviation: Measures average amount of deviation among scores.
• Necessary to understand how large individual deviations are.
  • In the example, standard deviation is 10.

Computing Z-Scores

• Formula: Subtract the mean from the raw score, then divide by the standard deviation.
• Z-Scores for example:
  • Shifty: z = (35-60)/10 = -2.50
  • Mickey: z = (65-60)/10 = 0.50
  • Antonio: z = (90-60)/10 = 3.00

Interpretation of Z-Scores with Normal Distribution

• Z-scores show where each score falls on a normal curve.
  • Shifty is far below the mean.
  • Mickey is near the mean.
  • Antonio is far above the mean.
• Provides a clear picture of relative standing in the dataset.

Conclusion

• Z-scores are a standardized way to compare individual scores against a dataset.
• They provide insight into whether a score is typical or atypical relative to others.