1 / 12

Normal Distributions and the Empirical Rule

Normal Distributions and the Empirical Rule. Learning Target: I can use percentiles and the Empirical rule to determine relative standing of data on the standard normal curve. 2.2 a Hw: pg 131: 42, 44, 45, 50, 51. Normal distributions: N ( μ, σ). Symmetric , single peaked and bell shaped.

liang
Download Presentation

Normal Distributions and the Empirical Rule

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Normal Distributions and the Empirical Rule Learning Target: I can use percentiles and the Empirical rule to determine relative standing of data on the standard normal curve. 2.2 a Hw: pg 131: 42, 44, 45, 50, 51

  2. Normal distributions: N (μ, σ) • Symmetric, single peaked and bell shaped. • Center of the curve are μ and M. • Standard deviation σ controls the spread of the curve.

  3. Normal distributions: N (μ, σ) • Inflection points: points where change of curvature takes place is located a distance σ on either side of μ.

  4. Normal curves are a good description of some real data: • test scores • biological measurements • also approximate chance outcomes like tossing coins

  5. The Empirical rule (68-95-99.7 rule) In the normal dist. with mean μ and standard deviation σ. • 68% of the observations fall within of the mean. • 95% of the observations fall within of the mean. • 99.7% of the observations fall within of the mean. 1σ 2σ 3σ

  6. Percentiles:we are interested in seeing where an individual falls relative to the other individuals in the distribution. • First quartile – 25th percentile • Median – 50th percentile • Third quartile – 75th percentile

  7. Ex. 1: Percentiles 84% tile Find the percentiles on above graph at 1, 2 and 3σ’s above and below μ. • At 1σ: 1 - .68 = .32 • 0.32 lie shared above and below 1σ so,.32/2 = .16 • At 1σ above the mean; • 1 – 0.16 = 0.84

  8. Percentiles 84% tile 97.5th % 99.85th % • At 2σ above the mean; • 1 – .025 = 0.975 • At 3σ above the mean; • 1 – 0.0015 = 0.9985

  9. 84% tile 16%tile 97.5th % 2.5%tile .15%tile 99.85th % • Use similar method to find percentiles below the mean • At σ below the mean • At 2σ below the mean • At 3σ below the mean

  10. Exercise 2: Men’s Heights The distribution of adult American men is approximately normal with mean 69inches and standard deviation 2.5 inches. Draw the curve and mark points if inflection.

  11. Recall: mean 69 in. and standard deviation 2.5 in. 16%tile 84% tile 2.5%tile 97.5th % 2.5% .15%tile 99.85th % 64 74 61.5 64 66.5 69 71.5 74 76.5 a) What percent of men are taller than 74 inches? 74 is two standard dev. above the mean. 2.5% b) Between what heights do the middle 95% of men fall? 69 5 =

  12. mean 69 in. and standard deviation 2.5 in. 16% 84% 61.5 64 66.5 69 71.5 74 76.5 c) What percent of men are shorter than 66.5 inches? 16% d) A height of 71.5 inches corresponds to what percentile? 1 - .68 = .32 .32/2 = .16 1 - .16 = .84

More Related