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Statistics 270 - Lecture 3

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Empirical Rule for Bell-Shaped Distributions. Approximately. 68% of the data lie ... What does the empirical rule tell us about 95% of the data? Is this useful? ... – PowerPoint PPT presentation

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Title: Statistics 270 - Lecture 3


1
Statistics 270 - Lecture 3
2
  • Last class types of quantitative variable,
    histograms, measures of center, percentiles and
    measures of spreadwell, we shall finish these
    today
  • Will have completed Chapter 1
  • Assignment 1 Chapter 1, questions 6, 20b, 26,
    36b-d, 48, 60
  • Some suggested problems
  • Chapter 1 1, 5, 13 or 14 (DO histogram), 19, 26,
    29, 33

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Measures of Spread (cont.)
  • 5 number summary often reported
  • Min, Q1, Q2 (Median), Q3, and Max
  • Summarizes both center and spread
  • What proportion of data lie between Q1 and Q3?

7
Box-Plot
  • Displays 5-number summary graphically
  • Box drawn spanning quartiles
  • Line drawn in box for median
  • Lines extend from box to max. and min values.
  • Some programs draw whiskers only to 1.5IQR above
    and below the quartiles

8
  • Can compare distributions using side-by-side
    box-plots
  • What can you see from the plot?

9
Other Common Measure of Spread Sample Variance
  • Sample variance of n observations
  • Can be viewed as roughly the average squared
    deviation of observations from the sample mean
  • Units are in squared units of data

10
Sample Standard Deviation
  • Sample standard deviation of n observations
  • Can be viewed as roughly the average deviation of
    observations from the sample mean
  • Has same units as data

11
Exercise
  • Compute the sample standard deviation and
    variance for the Muzzle Velocity Example

12
  • Variance and standard deviation are most useful
    when measure of center is
  • As observations become more spread out, s
    increases or decreases?
  • Both measures sensitive to outliers
  • 5 number summary is better than the mean and
    standard deviation for describing (I) skewed
    distributions (ii) distributions with outliers

13
Population and Samples
  • Important to distinguish between the population
    and a sample from the population
  • A sample consisting of the entire population is
    called a
  • What is the difference between the population
    mean and the sample mean?
  • The population variance ( or std. deviation) and
    that of the population
  • Population median and sample median?

14
Empirical Rule for Bell-Shaped Distributions
  • Approximately
  • 68 of the data lie in the interval
  • 95 of the data lie in the interval
  • 95 of the data lie in the interval
  • Can use these to help determine range of typical
    values or to identify potential outliers

15
ExamplePutting this all together
  • A geyser is a hot spring that becomes unstable
    and erupts hot gases into the air. Perhaps the
    most famous of these is Wyoming's Old Faithful
    Geyser.
  • Visitors to Yellowstone park most often visit Old
    Faithful to see it erupt. Consequently, it is of
    great interest to be able to predict the interval
    time of the next eruption.

16
ExamplePutting this all together
  • Consider a sample of 222 interval times between
    eruptions (Weisberg, 1985). The first few lines
    of the available data are
  • Goal Help predict the interval between
    eruptionsConsider a variety of plots that may
    shed some light upon the nature of the intervals
    between eruptions

17
ExamplePutting this all together
  • Goal Help predict the interval between eruptions
  • Consider a histogram to shed some light upon the
    nature of the intervals between eruptions

18
ExamplePutting this all together
19
ExamplePutting this all together
  • What does the box-plot show?
  • Is a box-plot useful at showing the main features
    of these data?
  • What does the empirical rule tell us about 95 of
    the data? Is this useful?
  • We will come back to this in a minute

20
Scatter-Plots
  • Help assess whether there is a relationship
    between 2 continuous variables,
  • Data are paired
  • (x1, y1), (x2, y2), ... (xn, yn)
  • Plot X versus Y
  • If there is no natural pairingprobably not a
    good idea!
  • What sort of relationships might we see?

21
ExamplePutting this all together
  • What does this plot reveal?

22
ExamplePutting this all together
23
ExamplePutting this all together
  • Suppose an eruption of 2.5 minutes had just taken
    place. What would you estimate the length of the
    next interval to be?
  • Suppose an eruption of 3.5 minutes had just taken
    place. What would you estimate the length of the
    next interval to be?
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