Z-Scores in Real Life

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📚 Part 1: Quick Concepts & Interpretation

1. Which formula gives the z-score for a value x, with mean μ and standard deviation σ?

z = (x + μ) / σ

z = (x − μ) / σ

z = (μ − x) / σ

2. Interpret this single-case result: a student has z = 1.5 on a test.

The student scored 1.5 standard deviations above the mean

The student scored 1.5 points above the mean

The student is 1.5 times the mean

✏️ Part 2: Applied Activity — Local Sports Comparison

Use the dataset below. For each player calculate their z-score relative to their team's match mean and standard deviation. Show working, then answer the interpretation questions.

Northside Kangaroos — team mean μ = 12.0, σ = 3.0
• Asha: scored 18 (player season SD = 4.0)
• Ben: scored 15 (player season SD = 2.5)
• Chloe: scored 9 (player season SD = 3.2)

Eastside Eagles — team mean μ = 20.0, σ = 5.0
• Dylan: scored 28 (player season SD = 6.0)
• Emma: scored 23 (player season SD = 2.0)
• Faisal: scored 16 (player season SD = 4.5)

5. Calculate z-scores for all six players. (Show calculations.)

6. Rank the six players from best to worst performance relative to their team using the z-scores you calculated. Who performed best relative to their team?

7. Athlete consistency: using the player season SDs given, which player is the most consistent performer? How does season variability affect your confidence in using a single high z-score as evidence for selection?

8. Recommendation: If you may recruit two players for an inter-school match balancing high relative performance and reliability, who would you pick? Give an evidence-based justification using z-scores and season SDs.