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Lecture 26

Center and Spread

DATA 8

Spring 2020

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Weekly Goals

  • Today
    • Describing distributions: “center” and “spread”
    • How big are most of the values?
  • Wednesday
    • The bell shaped curve and its relation to large random samples
  • Friday
    • The variability in a random sample average
    • Choosing the size of a random sample

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Confidence Intervals For Testing

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Using a CI for Testing

What if we want to do a hypothesis test, but we can’t simulate under the null?

  • Null hypothesis: Population average = x
  • Alternative hypothesis: Population average ≠ x
  • Cutoff for P-value: p%
  • Method:
    • Construct a (100-p)% confidence interval for the population average
    • If x is not in the interval, reject the null
    • If x is in the interval, can’t reject the null

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Center and Spread

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Questions

  • 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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Average

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The Average (or Mean)

Data: 2, 3, 3, 9 Average = (2+3+3+9)/4 = 4.25

  • Need not be a value in the collection
  • Need not be an integer even if the data are integers
  • Somewhere between min and max, but not necessarily halfway in between
  • Same units as the data
  • Smoothing operator: collect all the contributions in one big pot, then split evenly

(Demo)

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Discussion Question

Are the medians of these two distributions the same or different? Are the means the same or different? If you say “different,” then say which one is bigger.

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Comparing Mean and Median

  • Mean: Balance point of the histogram
  • Median: Half-way point of data; half the area of histogram is on either side of median
  • If the distribution is symmetric about a value, then that value is both the average and the median.
  • If the histogram is skewed, then the mean is pulled away from the median in the direction of the tail.

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Standard Deviation

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Defining Variability

Plan A: “biggest value - smallest value”

  • Doesn’t tell us much about the shape of the distribution

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Plan B:

  • Measure variability around the mean
  • Need to figure out a way to quantify this

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(Demo)

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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 next time.

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Chebyshev's Inequality

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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 - average)/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

(Demo)

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Point of Inflection

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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

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Central Limit Theorem

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Sample 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,

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Then, regardless of the distribution of the population,

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the probability distribution of the sample sum

(or the sample average) is roughly normal

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(Demo)