Chapter 3 Numerically Summarizing Data

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Chapter 3Numerically Summarizing Data

• 3.3
• Measures of Central Tendency and Dispersion from
  Grouped Data

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EXAMPLE Approximating the Mean from a
Frequency Distribution
The following frequency distribution represents
the time between eruptions (in seconds) for a
random sample of 45 eruptions at the Old Faithful
Geyser in California. Approximate the mean time
between eruptions.
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EXAMPLE Computed a Weighted Mean
Bob goes the Buy the Weigh Nut store and
creates his own bridge mix. He combines 1 pound
of raisins, 2 pounds of chocolate covered
peanuts, and 1.5 pounds of cashews. The raisins
cost 1.25 per pound, the chocolate covered
peanuts cost 3.25 per pound, and the cashews
cost 5.40 per pound. What is the cost per pound
of this mix.
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EXAMPLE Approximating the Mean from a
Frequency Distribution
The following frequency distribution represents
the time between eruptions (in seconds) for a
random sample of 45 eruptions at the Old Faithful
Geyser in California. Approximate the standard
deviation time between eruptions.
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Chapter 3Numerically Summarizing Data

• 3.4
• Measures of Location

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The z-score represents the number of standard
deviations that a data value is from the mean.
It is obtained by subtracting the mean from the
data value and dividing this result by the
standard deviation. The z-score is unitless with
a mean of 0 and a standard deviation of 1.
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Population Z - score
Sample Z - score
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EXAMPLE Using Z-Scores
The mean height of males 20 years or older is
69.1 inches with a standard deviation of 2.8
inches. The mean height of females 20 years or
older is 63.7 inches with a standard deviation of
2.7 inches. Data based on information obtained
from National Health and Examination Survey. Who
is relatively taller Shaquille ONeal whose
height is 85 inches or Lisa Leslie whose height
is 77 inches.
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Answer

• Shaquille ONeal Z-Score
• (85-69.1)/2.8 5.67857143
• Lisa Leslie
• (77-63.7)/2.7 4.92592593
• Because ONeal Z-Score gt Lisa s Z-Score,
• We say ONeal is in a higher position than Lisa
  in their Goups.

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The median divides the lower 50 of a set of data
from the upper 50 of a set of data. In general,
the kth percentile, denoted Pk , of a set of data
divides the lower k of a data set from the upper
(100 k) of a data set.
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Computing the kth Percentile, Pk
Step 1 Arrange the data in ascending order.
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Computing the kth Percentile, Pk
Step 1 Arrange the data in ascending order.
Step 2 Compute an index i using the following
formula
where k is the percentile of the data value and n
is the number of individuals in the data set.
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Computing the kth Percentile, Pk
Step 1 Arrange the data in ascending order.
Step 2 Compute an index i using the following
formula
where k is the percentile of the data value and n
is the number of individuals in the data set.
Step 3 (a) If i is not an integer, round up to
the next highest integer. Locate the ith value
of the data set written in ascending order. This
number represents the kth percentile. (b) If i
is an integer, the kth percentile is the
arithmetic mean of the ith and (i 1)st data
value.
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EXAMPLE Finding a Percentile
For the employment ratio data on the next slide,
find the (a) 60th percentile (b) 33rd percentile
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Answer

• A) 60th Percentile
• i) the index I (60/100)51 30.6
• 30.6 in not an integer, we round it up to 31.
  so the data value is 66.1
• 33rd
• i) the index I (33/100)5116.83
• Round it up to 17. So the data value at 17th is
  63.6.

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Finding the Percentile that Corresponds to a Data
Value
Step 1 Arrange the data in ascending order.
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Finding the Percentile that Corresponds to a Data
Value
Step 1 Arrange the data in ascending order.
Step 2 Use the following formula to determine
the percentile of the score, x
Percentile of x
Round this number to the nearest integer.
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EXAMPLE Finding the Percentile Rank of a Data
Value
Find the percentile rank of the employment ratio
of Michigan.
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• The most common percentiles are quartiles.
  Quartiles divide data sets into fourths or four
  equal parts.
• The 1st quartile, denoted Q1, divides the bottom
  25 the data from the top 75. Therefore, the
  1st quartile is equivalent to the 25th
  percentile.

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• The most common percentiles are quartiles.
  Quartiles divide data sets into fourths or four
  equal parts.
• The 1st quartile, denoted Q1, divides the bottom
  25 the data from the top 75. Therefore, the
  1st quartile is equivalent to the 25th
  percentile.
• The 2nd quartile divides the bottom 50 of the
  data from the top 50 of the data, so that the
  2nd quartile is equivalent to the 50th
  percentile, which is equivalent to the median.

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• The most common percentiles are quartiles.
  Quartiles divide data sets into fourths or four
  equal parts.
• The 1st quartile, denoted Q1, divides the bottom
  25 the data from the top 75. Therefore, the
  1st quartile is equivalent to the 25th
  percentile.
• The 2nd quartile divides the bottom 50 of the
  data from the top 50 of the data, so that the
  2nd quartile is equivalent to the 50th
  percentile, which is equivalent to the median.
• The 3rd quartile divides the bottom 75 of the
  data from the top 25 of the data, so that the
  3rd quartile is equivalent to the 75th
  percentile.

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EXAMPLE Finding the Quartiles
Find the quartiles corresponding to the
employment ratio data.
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Checking for Outliers Using Quartiles
Step 1 Determine the first and third quartiles
of the data.
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Checking for Outliers Using Quartiles
Step 1 Determine the first and third quartiles
of the data.
Step 2 Compute the interquartile range. The
interquartile range or IQR is the difference
between the third and first quartile. That is,
IQR Q3 - Q1
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Checking for Outliers Using Quartiles
Step 1 Determine the first and third quartiles
of the data.
Step 2 Compute the interquartile range. The
interquartile range or IQR is the difference
between the third and first quartile. That is,
IQR Q3 - Q1
Step 3 Compute the fences that serve as cut-off
points for outliers.
Lower Fence Q1 - 1.5(IQR) Upper Fence Q3
1.5(IQR)
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Checking for Outliers Using Quartiles
Step 1 Determine the first and third quartiles
of the data.
Step 2 Compute the interquartile range. The
interquartile range or IQR is the difference
between the third and first quartile. That is,
IQR Q3 - Q1
Step 3 Compute the fences that serve as cut-off
points for outliers.
Lower Fence Q1 - 1.5(IQR) Upper Fence Q3
1.5(IQR)
Step 4 If a data value is less than the lower
fence or greater than the upper fence, then it is
considered an outlier.
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EXAMPLE Check the employment ratio data for
outliers.
Q113 th62.9 Q3 38th67.2 Q3-Q14.3 So
(62.9-1.54.3, 67.21.54.3)(56.45,73.65) The
OUTLIER is 52.7
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West Virginia