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Lecture 23

The Normal Distribution

Summer 2021

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Announcements

  • Tutor sections signups released! More info here
  • HW 08 released, due Friday, 7/23 at 11:59pm PT
  • Project 1 and Midterm Grades Released
    • Regrade Requests due Friday 7/23 for both
  • Zoom link for lecture will be changing starting tomorrow! Use this link instead: https://berkeley.zoom.us/j/92050640021
  • Midterm Walkthrough Session on Saturday at 12pm (noon) PT

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Weekly Goals

  • Today
    • Center and Spread review
    • The bell shaped curve (Normal Distribution)
    • Central Limit Theorem
  • Friday
    • Sample Means
    • Designing Experiments

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Variance &

Standard Deviation

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How Far from the Mean (Average)?

  • Standard deviation (SD) measures how far the data deviate from their average
  • Variance = Mean Square of Deviations from Average of Data�
  • SD = Root Mean Square of Deviations from Average of Data

R ← M ← S ← D ← A ← D

  • SD has the same units as the data

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Salient Feature of Standard Deviation

No matter what the shape of the distribution,�

the bulk of the data are in the range ��Mean ± A Few SDs

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Standard Units

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Standard Units

  • How many SDs above average?
  • z = (value - Mean)/SD
    • Negative z: value below mean
    • Positive z: value above mean
    • z = 0: value equal to Mean
  • When values are in standard units: Mean = 0, SD = 1
  • Gives us a way to compare/understand data no matter what the original units

(Demo)

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Discussion Question

Find whole numbers that are close to:

​

  1. the average age

​

​

  • the SD of the ages

Average in standard units ≅ 0

≅ 27

​

​

​

Average + 1SD in standard units ≅ 1

27 + SD ≅ 33

SD ≅ 33-27 ≅ 6

(Demo)

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Chebyshev's Inequality

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How Big are Most of the Values?

Chebyshev’s Inequality

No matter what the shape of the distribution,

the proportion of values (i.e., fraction of the population) in the range “Mean ± z SDs” is

is at least 1 - 1/z²

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Chebyshev’s Bounds

Range

Proportion

Mean ± 2 SDs

at least 1 - 1/4 (75%)

Mean ± 3 SDs

at least 1 - 1/9 = 8/9 (88.89%)

Mean ± 4 SDs

at least 1 - 1/16 = 15/16 (93.75%)

Mean ± 5 SDs

at least 1 - 1/25 = 24/25 (96%)

No matter what the distribution looks like!

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The SD and the Histogram

  • Usually, it's not easy to estimate the SD by looking at a histogram.

​

  • But if the histogram has a bell shape, then you can.

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The SD and Bell-Shaped Curves

If a histogram is bell-shaped, then

​

  • the average is at the center

​

  • SD = distance between average and point of inflection on either side

(Demo: Maternal Heights)

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Point of Inflection

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The Normal Distribution

(AKA, Bell-Shaped Curve)

(AKA, Gaussian Distribution)

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The Standard Normal Curve

A beautiful formula that we won’t use at all:

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Bell Curve

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Normal Proportions

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How Big are Most of the Values?

No matter what the shape of the distribution,

the bulk of the data are in the range

“Mean ± A Few SDs”

If a histogram is bell-shaped, then

  • Almost all of the data are in the range

“Mean ± 3 SDs”

​

​

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Bounds and Normal Approximations

NOTE: If our random distribution is Normal, then we don’t need to bootstrap.

Our 95% confidence interval is simply: Mean ± 2 SDs.

Percent in Range

All Distributions (Chebyshev’s)

≥ 1-1/z2

Normal Distribution

Mean ± 1 SD

≥ 1-1/1 = 0 (0%)

≈ 68%

Mean ± 2 SDs

≥ 1-1/22 = 3/4 (75%)

≈ 95%

Mean ± 3 SDs

≥ 1-1/32 = 8/9 (88.89%)

≈ 99.73%

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A “Central” Area

Source: Statistics How To

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Central Limit Theorem

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Sample Means (Averages)

  • The Central Limit Theorem describes how the normal distribution (a bell-shaped curve) is connected to random sample averages.
  • We care about sample averages because they estimate population averages.

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Central Limit Theorem

If the sample is

  • large, and
  • drawn at random with replacement,

​

Then, regardless of the distribution of the population,

​

the probability distribution of the sample average

is roughly normal

​

(Demo)