Statistical Reasoning for everyday life

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Statistical Reasoningfor everyday life

• Intro to Probability and Statistics
• Mr. Spering Room 113

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4.2 Shapes of Distribution

• CLASS WORK
• Worksheet REVIEW
• ACTIONS are REMEMBERED, WORDS can be FORGOTTEN!
• MAKE an EFFORT, NOT an EXCUSE

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4.2 Shapes of Distribution

• Variation
• Describes how widely data are spread out about
  the center of a distribution.
• ????How would you expect the variation to differ
  between the heights of NCAA Division 1A Mens
  College Basketball Centers and the heights of all
  High School Boy Basketball Players????
• NCAA Division 1A Centers less variation
• High School Boy Basketball Players more variation

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4.3 Measures of Variation

• How do we investigate variation?
• Study all of the raw data
• Range
• Quartiles
• Five-number summary (BOXPLOT or BOX-and-WHISKER)
• Interquartile range
• Semi-quartile range
• Percentiles
• MAD
• Variance Standard Deviation

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4.3 Measures of Variation

• RANGE
• The range of a distribution is the difference
  between the highest and lowest data values.

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4.3 Measures of Variation

• Find the range of the data.
• 4.1, 5.2, 5.6, 6.2, 7.2, 7.7, 7.7, 8.5, 9.3, 11.0
• Range 11.0 4.1
• 6.9

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4.3 Measures of Variation

• Misleading range
• Which Quiz Set has greater variation?
• Quiz Set 1
• 1, 10, 10, 10, 10, 10, 10, 10, 10, 10
• Quiz Set 2
• 2, 3, 4, 5, 6, 7, 8, 9, 10, 8, 9, 10, 6, 5
• Even though Set 1 has a greater range than Set
  2 (9 gt 8). Set 2 has a greater variation
  because Set 1 contains an outlier. Therefore, we
  use quartiles.

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4.3 Measures of Variation

• Quartiles
• Quartiles divide the data into four quarters.
• Lower Quartile (1st Quartile) is the median of
  the data values in the lower half of a data set.
  Exclude the middle value in the data set if the
  number of data points is odd.
• Middle Quartile (2nd Quartile) is the overall
  median
• Upper Quartile (3rd Quartile) is the median of
  the data values in the upper half of a data set.
  Exclude the middle value in the data set if the
  number of data points is odd.

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4.3 Measures of Variation

• Find quartiles

Example 1 Upper and lower quartiles
Data 6, 47, 49, 15, 43, 41, 7, 39, 43, 41, 36
Ordered data 6, 7, 15, 36, 39, 41, 41, 43, 43, 47, 49
Median (2nd Quartile) 41
Upper quartile (3rd Quartile) 43
Lower quartile (1st Quartile) 15

Lower Quartile Q1
Median Q2
Upper Quartile Q3
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4.3 Measures of Variation

• Find quartiles

• Example 2 Range and quartiles
• A year ago, Angela began working at a computer
  store. Her supervisor asked her to keep a record
  of the number of sales she made each month.
• The following data set is a list of her sales for
  the last 12 months
• 34, 47, 1, 15, 57, 24, 20, 11, 19, 50, 28, 37
• Use Angela's sales records to find
• the median
• b) the range
• c) the upper and lower quartiles

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4.3 Measures of Variation

• Answers
• The values in ascending order are
• 1, 11, 15, 19, 20, 24, 28, 34, 37, 47, 50, 57.
• a) Median   (6th 7th observations)
  2             (24 28) 2             26
• b) Range difference between the highest and
  lowest values               57 -
  1               56

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4.3 Measures of Variation

• c) Lower quartile value of middle of first half
  of data Q1                            the
  median of 1, 11, 15, 19, 20, 24                  
            (3rd 4th observations)
  2                            (15 19)
  2                            17
• d) Upper quartile value of middle of second
  half of data Q3                        the
  median of 28, 34, 37, 47, 50, 57                 
         (3rd 4th observations)
  2                        (37 47)
  2                        42
• These results can be summarized as follows

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4.3 Measures of Variation

• Five-number summary
• Consists of the following
• Low Value
• Q1 (lower quartile)
• Q2 (median)
• Q3 (upper quartile)
• High Value

Summary
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4.3 Measures of Variation
Vertical box plot showing normal
distribution FORESHADOWING

• BOXPLOT or BOX-and-WHISKER
• Box plots show variation along the number line.
• Steps for creating a box plot
• Draw a number line that spans the entire data
  set.
• Above the number line, enclose the values from
  the lower to the upper quartile in a box.
• Draw a line through the box at the value
  corresponding to the median.
• Add whiskers extending to the low and high
  values.

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4.3 Measures of Variation

• Example of 5 number summary and box plot.

Lowest Value 1
First Quartile (Q1) 6.5
Median (Q2) 12
Third Quartile (Q3) 19.5
Highest Value 24
So for the data set 1, 4, 9, 12, 12, 16, 23,
24 here is our box plot                      
                                                  
     
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4.3 Measures of Variation
Digest of BOXPLOTS and SKEWNESS
Right-Skewed
Symmetric
Left-Skewed
Q1
Q2
Q3
Q1
Q2
Q3
Q1
Q2
Q3
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4.3 Measures of Variation

• Below is a Box-and-Whisker plot for the following
  data
• 0 2 2 2 3 3 4 5
  5 10 27
• The data are right skewed, as the plot depicts

Min Q1 Q2
Q3 Max
0 2 3 5
27
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4.3 Measures of Variation

• Interquartile range
• i.e. If the five number summary is low 3, high
  23, Q1 4, Q2 12, Q3 19.
• Then the interquartile range is IQR (Q3-Q1)
  (19 4) 15.

Interquartile range The interquartile range is
another range used as a measure of the variation.
The difference between upper and lower quartiles
(Q3Q1), which is called the interquartile range,
also indicates the dispersion of a data set. The
inter-quartile range spans 50 of a data set, and
eliminates the influence of outliers because, in
effect, the highest and lowest quarters are
removed.
Interquartile range upper quartile (Q3) minus lower quartile (Q1)
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4.3 Measures of Variation

• Next Time
• Semi-quartile range
• Percentiles
• MAD
• Variance Standard Deviation
• According to the box-n-whisker above what are the
  values for the 5 number summary
• Low 12
• Q1 22
• Q2 31
• Q3 45
• High 50

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4.3 Measures of Variation

• Classwork
• PRACTICE MAKES PERMANENT
• Pg 174 2-6 even and 25-27 (Letters a, b only)