Question

In a class of 60 students, 40 speak one language, 15 speak two languages, and 5 speak three languages. How many standard deviations above the average is a trilingual student?

Answer 1

Answer 2

Calculate the Mean (Average) Number of Languages Spoken:

• Total students = 60

• Total languages spoken = (40 × 1) + (15 × 2) + (5 × 3) = 40 + 30 + 15 = 85

• Mean (μ) = Total languages / Total students = 85 / 60 ≈ 1.417

1. Calculate the Variance (σ²):

• Monolingual students: 40 × (1 – 1.417)² ≈ 40 × 0.174 ≈ 6.96

• Bilingual students: 15 × (2 – 1.417)² ≈ 15 × 0.340 ≈ 5.10

• Trilingual students: 5 × (3 – 1.417)² ≈ 5 × 2.507 ≈ 12.54

• Squared deviations from the mean:

• Sum of squared deviations = 6.96 + 5.10 + 12.54 = 24.60

• Variance (σ²) = Sum of squared deviations / Total students = 24.60 / 60 ≈ 0.41

2. Standard Deviation (σ):

• σ = √Variance = √0.41 ≈ 0.64

3. Calculate How Many SDs Above the Mean a Trilingual Student Is:

• A trilingual student speaks 3 languages.

• Difference from mean = 3 – 1.417 ≈ 1.583

• Standard deviations above mean = 1.583 / 0.64 ≈ 1.45

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Conclusion:

A student who speaks 3 languages is 1.45 standard deviations above the class average.