14.4 Measures of Variation

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14.4 Measures of Variation

• CORD Math
• Mrs. Spitz
• Spring 2007

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Objectives

• Calculate and interpret the range, quartiles, and
  interquartile range of a set of data

Assignment

• pp. 577-578 1-24 all

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Assignment

• pp. 577-578 1-24 all

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Application

• Pacquita Colon and Larry Neilson are two
  candidates for promotion to manager of sales at
  Fitright Shoes. In order to determine who should
  be promoted, the owner, Mr. Tarsel, looked at
  each persons quarterly sales record for the past
  year. Its on the next slide.

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Here is the info he had access to
Quarterly Sales (thousands of dollars) Quarterly Sales (thousands of dollars) Quarterly Sales (thousands of dollars) Quarterly Sales (thousands of dollars) Quarterly Sales (thousands of dollars) Quarterly Sales (thousands of dollars) Quarterly Sales (thousands of dollars) Quarterly Sales (thousands of dollars) Quarterly Sales (thousands of dollars)
Ms. Colon 30.8 29.9 30.0 31.0 30.1 30.5 30.7 31.0
Mr. Nielson 31.0 28.1 30.2 33.2 31.8 29.8 28.9 31.0
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Application

• After studying the data, Mr. Tarsel found that
  the mean of the quarterly sales was 30,500, the
  median was 30,600 and the mode was 31,000 for
  both Ms. Colon and Mr. Nielson. If he was to
  decide between the two, Mr. Tarsel needed to find
  more numbers to describe this data.

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Application

• The example shows that measures of central
  tendency may not give an accurate enough
  description of a set of data. Often, measures of
  variation, are also used to help describe the
  spread of the data. One of the most commonly
  used measures of variation is the range.

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Definition of Range

• The range of a set of data is the difference
  between the greatest and the least values of the
  set.

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Ex. 1 Use the info in the table to determine
the range in the quarterly sales for Ms. Colon
and Mr. Nielson during the last two years.

• Ms. Colons greatest quarterly sales were 31,000
  and her least were 29,900. Therefore, the range
  is 31,000 - 29,900 or 1,100
• Mr. Nielsons greatest quarterly sales were
  33,200 and his least were 28,100. Therefore,
  the range is 33,200 - 28,100 or 5,100
• Based on this analysis, Ms. Colons sales are
  much more consistent, a quality Mr. Tarsel
  values. Therefore, Ms. Colon is promoted.

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NOTE

• Another commonly used measure of variation is
  called the interquartile range. In a set of
  data, the quartiles are values that divide the
  data into four equal parts. The median of a set
  of data divides the data in half. The upper
  quartile (UQ) divides the upper half into two
  equal parts. The lower quartile (LQ) divides the
  lower half into two equal parts. The difference
  between the upper and lower quartile is the
  interquartile range.

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Definition of the Interquartile Range

• The difference between the upper quartile and the
  lower quartile of a set of data is called the
  interquartile range. It represents the middle
  half, or 50, of the data in the set.

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Ex. 2

• The Birch Corporation held its annual golf
  tournament for its employees. The scores for 18
  holes were 88, 91, 102, 80, 115, 99, 101, 103,
  105, 99, 95, 76, 105, and 112. Find the median,
  upper and lower quartiles, and the interquartile
  range for these scores.

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First Step

• First, order the 15 scores. Then find the
  median.
• 70 80 88 91 95 99 99 101 102 103 105 105
  112 115 139

median
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Second Step

• Find the median of the lower and upper quartiles.
• 70 80 88 91 95 99 99 101 102 103 105 105
  112 115 139

median
lower quartile
upper quartile
The lower quartile is the median of the lower
half of the data and the upper quartile is the
median of the upper half.
The median divides the data in the data in half.
The upper and lower quartiles divide each half
into two parts.
The interquartile range is 105 91 or 14.
Therefore, the middle half, or 50, or the golf
scores vary by 14.
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NOTE

• In Example 2, one score, 139, is much greater
  than the others. In a set of data, a value that
  is much greater or much lower than the rest of
  the data is called an OUTLIER. An outlier is
  defined as any element of the set of data that is
  at least 1.5 interquartile ranges above the upper
  quartile or below the lower quartile.

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NOTE

• To determine if 139 or any of the other numbers
  from Example 2 is an outlier, first multiply by
  1.5 times the interquartile range, 14.

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Ex. 3 The stem-and-leaf plot represents the
number of shares of the 20 most active stocks
that were bought and sold on the New York Stock
Exchange.
Stem Leaf
1 2 2 7
2 3 3 3 4 4 5 6 6 8 8 9
3 0 1 4 6
4 0 6
40 represents 400,000,000 shares.
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Ex. 3 The stem-and-leaf plot represents the
number of shares of the 20 most active stocks
that were bought and sold on the New York Stock
Exchange.
Stem Leaf
1 2 2 7
2 3 3 3 4 4 5 6 6 8 8 9
3 0 1 4 6
4 0 6
The brackets group the values in the lower half
and the values in the upper half. What do the
boxes contain?
values used to find the lower and upper quartiles
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Ex. 3 The stem-and-leaf plot represents the
number of shares of the 20 most active stocks
that were bought and sold on the New York Stock
Exchange.
Stem Leaf
1 2 2 7
2 3 3 3 4 4 5 6 6 8 8 9
3 0 1 4 6
4 0 6
The median is 26.
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c. Find any outliers
Since 46 gt 41.75, 46 is an outlier.
Since 11 lt 41.75, 11 IS NOT an outlier.