Statistics 270 - Lecture 3

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Statistics 270 - Lecture 3
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• 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?

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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

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

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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

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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

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Exercise

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

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• 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

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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?

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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

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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.

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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

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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

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ExamplePutting this all together
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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

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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?

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ExamplePutting this all together

• What does this plot reveal?

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ExamplePutting this all together
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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?