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5.2 The Normal Distribution, Revisited

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Normal Distribution Review

The normal distribution N(μ,σ){N(\mu, \sigma)} is central to statistics. Recall:

Z=X−μσ∼N(0,1)Z = \frac{X - \mu}{\sigma} \sim N(0,1)
Empirical Rule: For X∼N(μ,σ){X \sim N(\mu, \sigma)}:
P(μ−σ<X<μ+σ)≈0.68P(\mu - \sigma < X < \mu + \sigma) \approx 0.68
P(μ−2σ<X<μ+2σ)≈0.95P(\mu - 2\sigma < X < \mu + 2\sigma) \approx 0.95
P(μ−3σ<X<μ+3σ)≈0.997P(\mu - 3\sigma < X < \mu + 3\sigma) \approx 0.997

Finding probabilities:

  • P(X<a){P(X < a)}: standardize → P(Z<z){P(Z < z)} → use table/calculator
  • P(X>a)=1−P(X<a){P(X > a) = 1 - P(X < a)}
  • P(a<X<b)=P(X<b)−P(X<a){P(a < X < b) = P(X < b) - P(X < a)}

Finding values from percentiles: Given P(X<a)=p{P(X < a) = p}: find z∗{z^*} from the table, then a=μ+z∗σ{a = \mu + z^* \sigma}

In this unit

8 lessons

  1. 5.1Introducing Statistics: Why Is My Sample Not Like Yours?Preview lesson
  2. 5.2The Normal Distribution, RevisitedThis lesson
  3. 5.3The Central Limit TheoremPreview lesson
  4. 5.4Biased and Unbiased Point EstimatesPreview lesson
  5. 5.5Sampling Distributions for Sample ProportionsPreview lesson
  6. 5.6Sampling Distributions for Differences in Sample ProportionsPreview lesson
  7. 5.7Sampling Distributions for Sample MeansPreview lesson
  8. 5.8Sampling Distributions for Differences in Sample MeansPreview lesson

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