The Central Limit Idea

Learn central limit idea, standard scores.

Lesson 2 of 32 parts Practice Flashcards

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Central Limit Idea

Objective: Explain central limit idea, apply the stated method, and check its conditions.

For independent identically distributed observations with finite variance, the standardized sample mean approaches a normal distribution as sample size grows. Its SD is σ/√n. The required size depends on population shape; there is no universal sample-size guarantee for every distribution. The central limit theorem concerns the sampling mean, not raw-data normality.

Formulas

$SD(x̄)=σ/√n$

Worked examples

Try itIndependent observations have population SD=42. For samples of size 49, find SD(x̄).Show answer

Step 1: For independent sampling, SD(x̄)=σ/√n. Step 2: Compute (42)/√(49). Step 3: The SD is 6; approximate normal shape requires additional conditions.

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Central Limit Idea

Objective: Explain central limit idea, apply the stated method, and check its conditions.

Formulas

$SD(x̄)=σ/√n$

Worked examples

Try itIndependent observations have population SD=42. For samples of size 49, find SD(x̄).Show answer

Step 1: For independent sampling, SD(x̄)=σ/√n. Step 2: Compute (42)/√(49). Step 3: The SD is 6; approximate normal shape requires additional conditions.

Standard Scores

Objective: Explain standard scores, apply the stated method, and check its conditions.

A standard score z=(x−μ)/σ measures signed distance from a mean in SD units, with σ>0. Normal probability calculations additionally require a normal model; standardization alone does not make an arbitrary distribution normal. A z-score transformation does not by itself establish normality.

Formulas

$z=(x−μ)/σ$

Worked examples

Try itA synthetic normal model has mean 7 and SD=6. Find z for x=49.Show answer

Step 1: Standardize with z=(x−μ)/σ. Step 2: Compute (49−7)/6. Step 3: z=7.

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