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Lecture 26

The Normal Curve

DATA 8

Fall 2017

Slides created by John DeNero (denero@berkeley.edu) and Ani Adhikari (adhikari@berkeley.edu)

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Announcements

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Questions for This Week

  • How can we quantify natural concepts like “center” and “variability”?

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  • Why do many of the empirical distributions that we generate come out bell shaped?

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  • How is sample size related to the accuracy of an estimate?

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Standard Deviation (Review)

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How Far from the Average?

  • Standard deviation (SD) measures roughly how far the data are from their average

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  • SD = root mean square of deviations from average

5 4 3 2 1

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  • SD has the same units as the data

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Why Use the SD?

  • The first reason:

No matter what the shape of the distribution,

the bulk of the data are in the range “average ± a few SDs”

There are two main reasons.

  • The second reason:

Coming up later in this lecture ...

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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 “average ± a few SDs”

Chebyshev’s Inequality

No matter what the shape of the distribution,

the proportion of values in the range “average ± z SDs” is

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at least 1 - 1/z²

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Chebyshev’s Bounds

Range

Proportion

average ± 2 SDs

at least 1 - 1/4 (75%)

average ± 3 SDs

at least 1 - 1/9 (88.888…%)

average ± 4 SDs

at least 1 - 1/16 (93.75%)

average ± 5 SDs

at least 1 - 1/25 (96%)

No matter what the distribution looks like

(Demo)

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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 average
    • Positive z: value above average
    • z = 0: value equal to average
  • When values are in standard units: average = 0, SD = 1
  • Chebyshev: At least 96% of the values of z are between -5 and 5

(Demo)

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Discussion Question

Find whole numbers that are close to:

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  1. the average age

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  • the SD of the ages

(Demo)

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The SD and the Histogram

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

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  • 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

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  • the average is at the center

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  • the SD is the distance between the average and the points of inflection on either side

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Attendance

bit.ly/attend8

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The Normal 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 “average ± a few SDs”

If a histogram is bell-shaped, then

  • Almost all of the data are in the range

“average ± 3 SDs”

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Bounds and Normal Approximations

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A “Central” Area

(Demo)

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Central Limit Theorem

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Second Reason for Using the SD

If the sample is

  • large, and
  • drawn at random with replacement,

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Then, regardless of the distribution of the population,

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the probability distribution of the sample sum

(or of the sample average) is roughly bell-shaped

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(Demo)