Describing Distributions with Numbers

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Chapter 12

• Describing Distributions with Numbers

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Thought Question 1
If you were to read the results of a study
showing that daily use of a certain exercise
machine resulted in an average 10-pound weight
loss, what more would you want to know about the
numbers in addition to the average? (Hint Do you
think everyone who used the machine lost 10
pounds?)
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Thought Question 2
A recent newspaper article in California said
that the median price of single-family homes sold
in the past year in the local area was 136,000
and the average price was 149,160. How do you
think these values are computed? Which do you
think is more useful to someone considering the
purchase of a home, the median or the average?
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Thought Question 3
The Stanford-Binet IQ test is designed to have a
mean, or average, for the entire population, of
100. It is also said to have a standard
deviation of 16. What do you think is meant by
the term standard deviation?
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Turning Data Into Information

• Center of the data
• mean
• median
• mode
• Variation
• variance
• standard deviation
• range
• interquartile range

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Average or Mean

• Traditional measure of center
• Sum the values and divide by the number of values

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Median (M)

• A resistant measure of the datas center
• At least half of the ordered values are less than
  or equal to the median value
• At least half of the ordered values are greater
  than or equal to the median value
• If n is odd, the median is the middle ordered
  value
• If n is even, the median is the average of the
  two middle ordered values

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Median

• Example 1 data 2 4 6
• Median (M) 4
• Example 2 data 2 4 6 8
• Median 5 (ave. of 4
  and 6)
• Example 3 data 6 2 4
• Median ? 2
• (order the values 2 4 6 , so Median 4)

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Case Study
Weight Data
STAT 208 Class SurveySpring, 1997 Virginia
Commonwealth University
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Weight DataMeasures of Center
10 0166 11 009 12 0034578 13 00359 14 08 15
00257 16 555 17 000255 18 000055567 19 245 20
3 21 025 22 0 23 24 25 26 0
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Case Study
Airline fares
appeared in the New York Times on November 5, 1995

• ...about 60 of airline passengers pay less
  than the average fare for their specific
  flight.
• How can this be?

13 of passengers pay more than 1.5 times the
average fare for their flight
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Variance and Standard Deviation

• If all values are the same, then they are all
  equal to the mean. There is no variation.
• Variation exists when some values are above or
  below the mean.
• Each data value has an associated deviation from
  the mean

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Deviations

• A deviation

• what is a typical deviation from the mean?
• small values of this typical deviation indicate
  small variation in the data
• large values of this typical deviation indicate
  large variation in the data

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Variance

• Find the mean
• Find the deviation of each value from the mean
• Square the deviations
• Sum the squared deviations
• Divide the sum by n-1

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Variance Formula
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Standard Deviation Formulatypical deviation from
the mean
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Traditional Summary StatisticsWeight Data

• Mean 158.75
• Standard deviation 35.65

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Quartiles

• Three numbers which divide the ordered data into
  four equal sized groups.
• Q1 has 25 of the data below it.
• Q2 has 50 of the data below it. (Median)
• Q3 has 75 of the data below it.

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QuartilesUniform Histogram
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Obtaining the Quartiles

• Order the data.
• For Q2, just find the median.
• For Q1, look at the lower half of the data
  values, those to the left of the median find
  the median of this lower half.
• For Q3, look at the upper half of the data
  values, those to the right of the median find
  the median of this upper half.

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Weight Data Sorted
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Weight Data Quartiles

• Q1 127.5
• Q2 165 (Median)
• Q3 185

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Weight DataQuartiles
10 0166 11 009 12 0034578 13 00359 14 08 15
00257 16 555 17 000255 18 000055567 19 245 20
3 21 025 22 0 23 24 25 26 0
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Five-Number Summary

• minimum 100
• first quartile 127.5
• second quartile 165
• third quartile 185
• maximum 260

range max ? min 160
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Five-Number Summary Boxplot
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Outliers

• Affect value of the mean and standard deviation
• Median and five-number summary should be used to
  describe center and spread when outliers are
  present

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Number of Books Read for Pleasure Sorted
Med
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Five-Number Summary Boxplot

• Median 3
• interquartile range (iqr) 5.5-1.0 4.5
• range 99-0 99

Mean 7.06 s.d. 14.43
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Number of Books Read for Pleasure
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Key Concepts

• Numerical Summaries
• Center (mean, median)
• Variation (variance, std. dev., range, iqr)
• Five-number summary Boxplots
• Choosing mean versus median
• Choosing standard deviation versus five-number
  summary