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1
Message to the user...
Revised 2002 Statistics Show 1 of 3
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2
Statistics...
Measures of Center and Spread

• Median, Mean, Range
• Quartiles
• 5-Number Summary

3
Arithmetic(numeric) Data AnalysisA. Measures of
Center

• FIRST RANK the data list it in order (usually)
  smallest -- largest
• 1. Median (M) of a distribution of n values
• is found in the (n1)/2
• place in the list.
• (It is sometimes referred to as the 50th
  percentile)

• If n (the number of values in the list) is odd,
  then the median is an actual value in the list.
• If n is even, then the median is the average of
  the 2 middle values.
• (It might not be an actual value in the list)

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Arithmetic(numeric) Data AnalysisA. Measures of
Center

• (ex1) Given the following set of data, find the
  median (M)
• 17 19 21 21 21 25 25 25 26 27 28 30
  31 32 32
• There are 15 values in the list (n15)
• The POSITION of the median is
• (151)/2 8th position
• The value of the median, M25

M
5
Arithmetic(numeric) Data AnalysisA. Measures of
Center

• (ex2) Given the following set of data, find the
  median (M)
• 17 19 21 21
• There are 4 values in the list (n4)
• The POSITION of the median is
• (41)/2 2.5th position
• The value of the median, M(1921)/2 40/2 20

M 20
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Arithmetic(numeric) Data AnalysisA. Measures of
Center

• 2. Mean (x) of a distribution of n values
• is the arithmetic average of the n values.
• (the sum of the values)/n

• (ex2) Find the mean (x) of the 4 values
• 17 19 21 21
• so, n4
• and the mean
• x (17192121)/4
• x 78/4 19.5

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Arithmetic(numeric) Data AnalysisB. Measures of
SPREAD

• 1. RANGE the difference described by
  subtracting
• highest - lowest value
• Also sometimes expressed by stating highest
  value to lowest value

• (ex1) The RANGE of the 15 values is
• 32 - 17 15 units
• The values range from 17 to 32

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Arithmetic(numeric) Data AnalysisB. Measures of
SPREAD

• 2. QUARTILES used to measure the spread of the
  data when the MEDIAN is the measure of center.
• a. Rank the data
• b. Examine the data to the left of M
• and find their median
• call this median Q1
• the first quartile
• (also called the 25th percentile)

• (ex1) The 7 values to the left of M are
• 17 19 21 21 21 25 25
• since n 7 for this set of data, its MEDIAN (Q1)
  will be found in the
• (71)/2 4th position
• So Q1 21

Q1
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Arithmetic(numeric) Data AnalysisB. Measures of
SPREAD2. QUARTILES

• (ex1) The 7 values to the right of M are
• 26 27 28 30 31 32 32
• since n 7 for this set of data, its MEDIAN (Q3)
  will be found in the
• (71)/2 4th position
• So Q3 30

• c. Do the same thing for those values to the
  RIGHT of M
• Find their median
• Call it Q3
• the third quartile
• (also called the 75th percentile)

Q3
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Arithmetic(numeric) Data AnalysisB. Measures of
SPREAD2. QUARTILES

• (ex1)
• 17 19 21 21 21 25 25 25 26 27 28 30
  31 32 32
• d. Now the data is split into 4 equal parts
  (quarters)
• The QUARTILES are those MEDIANS found in steps
  (b) and (c)

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Arithmetic(numeric) Data AnalysisB. Measures of
SPREAD

• 3. 5-number summary to describe the spread of
  the data about the MEDIAN is a list of the
  following values (in order from lowest to
  highest)...
• LOWEST Q1 M Q3 HIGHEST

• (ex1) The 5-Number Summary for this set of data
  is
• 17 21 25 30 32
• From this we can tell a lot about the set of
  data, without seeing all 15 values.

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Arithmetic(numeric) Data AnalysisB. Measures of
SPREAD

• (ex1) The 5-Number Summary for this set of data
  is
• 17 21 25 30 32
• Without seeing all of the data, we know
• 50 of the data is 25 or less
• 25 of the data is 21 or less
• 75 of the data is 30 or less
• The data values range from 17 to 32
• etc...

• BOX PLOT a picture of a 5-number summary.
• A box spans the quartiles
• with a line to mark the median
• Whiskers extend to the high and low values.

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Arithmetic(numeric) Data AnalysisB. Measures of
SPREAD

• (ex1)
• 1. BOX PLOT a picture of a 5-number summary.
• A box spans the quartiles
• with a line to mark the median
• Whiskers extend to the high and low values.
• The 5 Summary is
• 17 21 25 30 32
• The box plot can be vertical or horizontal.

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Using the 5 Summary to describe the spread of
the data...

• The 5 Summary is used with the MEDIAN and
  QUARTILES.
• It gives you useful information about the
  original set of data without seeing the entire
  list.
• It is an especially useful description of data
  that is skewed or has outliers.

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End of show 1

• Going on?...
• Statistics
• Show 2 Mean Standard Deviation

REVISED 2002 Prepared by Kimberly Conti, SUNY
College _at_ Fredonia Suggestions and comments to
Kimberly.Conti_at_fredonia.edu