Describing Data

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Describing Data

• Numerical Summaries

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Numerical Summaries

• Measures of center
• Mean
• Median
• Measures of Spread
• Standard Deviation
• IQR ( Inter-Quartile Range)

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Mean

• A measure of center, or typical value is the
  mean. Also known as the average or x-bar.
• For numeric data values, x_1, x_2, ,x_n the mean
  is defined as,

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Examples

• Babe Ruth had an average of 43.9 home runs per
  year.
• Roger Maris had an average of 26.1 home runs per
  year.
• Barry Bonds avg658/1836.55 HR per year.

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Median

• The median is defined as being the middle value
  in an ordered list, m. To obtain the median
• Order data from smallest to largest values.
• Determine number of data values n.
• If n is odd, the median is located at position
  (n1)/2 in the ordered list.
• If n is even, the median is found by obtaining
  the middle position (n1)/2 and then averaging
  the two middle values found there.

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Example

• Consider a list with an odd number of elements
  like 11, 8, 6, 10, 7 so that n5.
• The ordered list is 6, 7, 8, 10, 11.
• The middle position is (n1)/2 (51)/2 6/23,
  the third position.
• The median is therefore m8.

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Example

• Consider the list 11, 7, 8, 9 with n4 elements,
  an even number.
• The ordered list is 7, 8, 9, 11.
• The middle position is (n1)/2(41)/25/22.5.
• This means the two middle values are 8 and 9.
• The average of 8 and 9 is (89)/2 17/28.5, so
  the median is m8.5.

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Examples

• Median for Babe Ruth is m46, n15 an odd number.
  Middle is at position (151)/28.
• Median for Roger Maris is m24.5, n10 an even
  number. Middle position is (101)/211/25.5.
  Median is average of 23 and 26, m24.5. So here
  median is not a value in the data list.

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Why Need Both Measures?

• Motivating example dataset 12.9, 13.2, 14.4,
  16.5, 17.1.
• Mean is 74.1/514.82.
• Median is m14.4, middle is at (51)/2 3

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Modify Dataset

• Add an additional value to the data, 50.2.
• Mean is now 124.3/6 20.72 (old mean14.82)
• Median is now (14.416.5)/2 15.45 (old median
  14.4)
• Notice that both the mean and median increased by
  the additional large number added to the set.
  This should make sense.

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Modify Again

• Now instead of value 50.2, suppose we take our
  original list 12.9, 13.2, 14.4, 16.5, 17.1 and
  supply an additional value of 374,558,222,999,999,
  999 !!!!
• What happens to the mean?
• What happens to the median?

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Behavior

• The previous example shows that the mean is VERY
  sensitive to weirdo data values.
• The median was not affected very much by the
  weird data or outlier a couch potato.

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Example
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Usage

• When should you use the mean? Or median?
• If your data has substantial skewness or outliers
  you should use the median to describe center.
• If the data is regular, mound shaped with no
  strange features, the mean has a slight
  theoretical advantage.
• Some statisticians suggest to always use the
  median.

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Spread Standard Deviation

• For a data list like
• The standard deviation S is defined as,

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Properties of S

• 0
• Smallest S can be is S0. Means no variation in
  data at all. All values are the same.
• S is very sensitive to weird values and skewness.
• The formula for S shows that it is almost the
  average deviation around x-bar. This is a good
  way to think about S, it is the average amount
  that data values vary around the average, x-bar.

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Interpretation of S

• S is the almost the average deviation of data
  values around x-bar.
• S is the typical amount that data values vary
  around x-bar.
• If S3, we would say that data values typically
  vary by 3 units above or below the average x-bar.

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S Examples

• Babe Ruth home run data, S11.25 home runs. This
  means that Babe Ruths home run values would
  typically vary 11.25 home runs above or below his
  average.
• I expect S to be computed by a computer program
  like StatCrunch. I dont expect this to be
  computed from a calculator.

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Percentiles

• The p-th percentile is defined to be the value
  such that p percent of the data has values less
  than or equal to p.
• For example, the 50 th percentile (the median)
  means half the data has values less than or equal
  to this number.
• I am at about the 90 th percentile for height for
  males.

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IQR

• The IQR ( Inter-Quartile Range ) is defined as
  IQRQ3-Q1.
• Q3 is the value at the 75th percentile.
• Q1 is the value at the 25th percentile.
• FYI, Q2 is the median the 50th percentile.
• These Qs break the dataset into four parts,
  hence name quartiles.

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IQR

• The IQR is the distance or spread in the middle
  half of the data.
• IQR is also a measure of spread like S. If IQR
  gets larger, there is more spread in the data.

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IQR Properties

• IQR is NOT sensitive to outliers or skewness.
  The values are more stable in cases like these
  than the standard deviation S.

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Spread Measures Usage

• Use IQR when data has either skewness or outliers
  or both. It provides a very reasonable measure
  of spread. The standard deviation is potentially
  misleading in this situation.
• The standard deviation, S, is a reasonable
  measure of center for well-behaved datasets.

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Boxplot

• A boxplot is a graphical display of a
  distribution based on only five numbers min,
  Q1, median, Q3, max.
• Values Q1, median, and Q3 form the box. Values
  min and max form the whiskers out from the box.

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Boxplot Example

• Fuel mileage for two-seater and mini-compact cars.

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Example Skewed Right
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Boxplot Example
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Boxplot Example
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Distributions Activity

• Match Histograms to descriptions.
• a) Scores in range 0-100 on an easy statistics
  exam.
• b) Number of cycles required to achieve pregnancy
  for a sample of women attempting to get pregnant.
  Data are self-reported.
• c) Heights of a group of college students.
• d) Numbers of medals won by countries in 1992
  Winter Olympics. Albertville France?
• e) SAT scores for a group of college students.

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Distributions Activity
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Distributions Activity
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Activity Combined Plots
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Bluegill Lengths Lake Mary, MN
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Histogram Example
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Summary

• Numerical and graphical summaries go together.
• A numerical summary without first looking at a
  graph can be very misleading.
• Know the strengths and weaknesses of the
  graphical and numerical summaries you use.