Describing Distributions with Numbers

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Section 1.2

• Describing Distributions with Numbers

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VOCABULARY

• Center
• Spread
• Variability
• Quartiles
• 5 number summary
• Box plot
• Variance
• Standard deviation
• Resistant

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Describing a distribution

• To describe a distribution, find the shape,
  center, and spread of the distribution!
• Shape use a histogram or stemplot
• Center use mean, median
• Spread use range, SD, or Q1, Q3

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The MEAN

• The mean is the average value of a set of
  observations.
• Find the sum of the individual observations, then
  divided by the number of total observations, n.
• NOT RESISTANT to outliers. One very large
  observation can skew the mean.

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The Median

• The median is the middle value of a set of
  observations.
• MEANS and MEDIANS behave differently!
• The median is resistant to outliers.

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Mean vs. Median

• If the distribution is Symmetric, meanmedian
• In a skewed distribution, the mean is farther out
  in the tail than the median.
• average value mean
• typical value median

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Measuring Spread- Range

• The range can be used to show the spread of
  observations The range is the largest -smallest
  
• Quartiles are used to measure spread as well
• Q1 first quartile
• Q3 third quartile

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Measuring Spread- Quartiles

• Quartiles are used to measure spread as well

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Quartiles Example

• The highway mileage of 20 gasoline cars arrange
  in order are
• 13 15 16 16 17 19 20 22 23 23 23 24 25 25 26 28
  28 29 32
• Find the mean, median, Q1 and Q3
• Minitab, a statistical software program, gives an
  output like this

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Five Number Summary

• The 5 number summary includes the
• Minimum
• Q1
• Median, M
• Q3
• Maximum
• What is the 5-number summary for the highway
  mileage data?

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One cool thing

• The median splits the data in half
• Q1 and Q3 split the halves in half, so
• 50 of the data lies between Q1 and Q3!
• A boxplot is the visual representation of the 5
  number summary

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Box Plots

• BOX PLOT DEFINITION

• Visual of the 5 number summary
• A central box span the quartiles Q1 and Q3.
• A line the box marks the median
• Lines extend from the minimum and maximum.
• Outliers can be left out, but a dot can be placed
  to show them.

• EXAMPLE

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Example Highway Mileage Boxplot

• Given the data from the previous highway mileage
  example, create a boxplot
• 13 15 16 16 17 19 20 22 23 23 23 24 25 25 26 28
  28 29 32

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Comparing Boxplots

• Side by side comparisons allow you to compare
  medians, and the spread of the data

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1.5 x IQR finds Outliers

• The inter-quartile range (IQR) is Q3-Q1.
• If there is a clear outlier, do we include it in
  our summary? Isnt it a bit misleading to have
  really high or low numbers that arent
  representative of the rest of the population?
• Since the quartiles are resistant to the extreme
  values, we use 1.5x IQR to determine if a value
  is suspected to be an outlier.

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Modified Boxplot

• If you find that there are outliers, you may make
  your whiskers shorter, and put a point where
  the suspected outliers are.

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Modified Box Plot Example

• For the 4 yr college data on pg. 45, we can find
  our 5-number summary
• Min 9420
• Q1 14,286
• Med 16,870
• Q3 21,707.50
• Max 34,850

• The University of Richmond (34, 850) is a clear
  outlier
• Check to see if it is
• IQR Q3-Q111,131.50
• Is U. of Richmond 11,131.50 above or below Q1 or
  Q3?
• YES!!!

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Modified Boxplot Example

• Create a modified boxplot comparing Barry Bonds
  HRs to Babe Ruths
• TI-84
• Barry Bonds (L1)
• 16 25 24 19 33 25 34 46 37 33 42 40 37 34 49 73
  46 45 45
• Babe Ruth (L2)
• 54 59 35 41 46 25 47 60 54 46 49 46 41 34 22
• Plot 1 and Plot 2 ON
• Select modified box plots for both Lists
• Press ZOOM to see the plots.

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Measuring the Spread- Standard Deviation

• The standard deviation measures the spread by
  looking at how far away the observations are from
  their mean.

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Perspective on Variance

• Each value deviates from the mean, positively or
  negatively. The sum of the deviations will always
  be zero.
• Squaring the deviations makes them all positive,
  so that observations far from the mean in either
  direction have large, positive, squared
  deviations.
• The variance is the average of the squared
  deviations.

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Standard Deviation

• The SD, is the square root of the variance so
  the SD is the average deviation from the mean, in
  both directions.

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SD Example

• A persons metabolic rate is the rate a body
  cosumes energy. 7 mens rates are below
  (calories/24 hrs)
• 1792 1666 1362 1614 1 1460 1867 1439
• Enter into L1- Run 1-Var Stats.
• Find the SD using the formula

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Questions about SD

• Why square the deviations?
• To make all number positive, since variation can
  be both -, this takes away negative numbers.
• Why do we emphasize SD instead of variances?
• Once you square the variances and get the
  variance, your number is a squared representation
  of the actual deviations from the mean.
• Why divide by n-1?
• by taking away 1, it adds more accuracy to your
  sample, since the sample should represent the
  population, it makes the SD of your sample an
  unbiased estimator of the true population SD.
  Samples can be inaccurate if your n is not large
  enough, and biased answers can be given
• n-1 is known as the degrees of freedom of the
  SD 1 degree of freedom.

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Properties of SD
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• SD is affected by strong outliers/a skewed
  distribution!
• The SD should only be used when the mean is the
  chosen center, and there a not a lot of outliers
  to skew the SD.

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Choosing Measures of Center and Spread
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Example Stocks vs. Treasury Bills return
Which is better?
Mean return 13.2 SD 17.6
Mean Return 5.0 SD 2.9
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Changing units of Measurement
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Miami Heat Salaries

• Player Salary (in millions)
• Shaquille O'Neal 27.70
• Eddie Jones 13.46
• Dwyane Wade 2.86
• Damon Jones 2.50
• Michael Doleac 2.40
• Rasual Butler 1.20
• Dorell Wright 1.15
• Qyntel Woods 1.13
• Christian Laettner 1.10
• Steve Smith 1.10
• Shandon Anderson 0.87
• Keyon Dooling 0.75
• Zhizhi Wang 0.75
• Udonis Haslem 0.62
• Alonzo Mourning 0.33

• If I were to change the distribution by
• Adding 100,000 to every player
• Converting to pennies, etc.
• It would not change the shape of the
  distribution!
• Linear Transformations do not change the shape or
  spread of the distribution!

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Effects of a Linear Transformation

• Linear Transformations will not change whether
  your spread whether you
• Multiply every observation by b
• Add a to every observation

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Comparing Distributions

• When comparing distributions, ask your self
• Who are the individuals?
• What are the variables and the units of measure?
• Why was the data gathered?
• When, where, how, and by whom were the data
  produced?
• Also find
• Graphs
• Numerical Summaries
• Interpretation

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Swiss Doctors

• Do male doctors perform more c-sections than
  female? A study in Switzerland found the of
  c-sections performed in a year by male and female
  doctors
• Male doctors Female doctors
• 20 44 5
• 25 50 7
• 25 59 10
• 27 85 14
• 28 86 18
• 31 19
• 33 25
• 34 29
• 36 31
• 37 33
• Who, What, Why, When, where, how and by whom?

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Homework

• Complete assignment and begin to work on the
  Multiple Choice packet.

• Read Section 1.2 Summary.